Computational Methods in Transport: Granlibakken 2004 (Lecture Notes in Computational Science and Engineering)

Lecture Notes in Computational Science and Engineering Editors Timothy J. Barth Michael Griebel David E. Keyes Risto M...

Author: Frank Graziani (Editor)

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Lecture Notes in Computational Science and Engineering Editors Timothy J. Barth Michael Griebel David E. Keyes Risto M. Nieminen Dirk Roose Tamar Schlick

48

Frank Graziani Editor

Computational Methods in Transport Granlibakken 2004

With 196 Figures and 23 Tables

ABC

Editor Frank Graziani Lawrence Livermore National Laboratory East Avenue 7000 Livermore, CA 94550, U.S.A. email: [email protected]

Library of Congress Control Number: 2005931994 Mathematics Subject Classiﬁcation: P19005, M1400X ISBN-10 3-540-28122-3 Springer Berlin Heidelberg New York ISBN-13 978-3-540-28122-1 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors and TechBooks using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper

SPIN: 11360605

46/TechBooks

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Contents

Part I Astrophysics Radiation Hydrodynamics in Astrophysics Chris L. Fryer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1 Deﬁning Radiation Hydrodynamics Terms . . . . . . . . . . . . . . . . . . . . . . 3 2 Schemes Used in Astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Astrophysical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 SPH Radiation Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Radiative Transfer in Astrophysical Applications I. Hubeny . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Description of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Absorption, Emission and Scattering Coeﬃcients . . . . . . . . . . . . . . . . 4 Hierarchies of Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 General Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Exact Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neutrino Transport in Core Collapse Supernovae Anthony Mezzacappa, Matthias Liebend¨ orfer, Christian Y. Cardall, O.E. Bronson Messer, Stephen W. Bruenn . . . . . . . . . . . . . . . . . . . . . . . . 1 The Core Collapse Supernova Paradigm . . . . . . . . . . . . . . . . . . . . . . . . 2 The O(v/c) Neutrino Transport Equation in Spherical Symmetry: An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Finite Diﬀerencing of the O(v/c) Neutrino Transport Equation in Spherical Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The General Case: The Multidimensional Neutrino Transport Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Boltzmann Neutrino Transport: The Current State of the Art . . . . . 6 Previews of Coming Distractions: Neutrino Flavor Transformation . 7 Summary and Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 16 17 20 24 26 32 32

35 35 40 42 55 59 63 65 67

VI

Contents

Discrete-Ordinates Methods for Radiative Transfer in the Non-Relativistic Stellar Regime Jim E. Morel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Approximate Radiation-Hydrodynamics Model . . . . . . . . . . . . . . 3 Discretization and Solution Techniques . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 69 73 80

Part II Atmospheric Science, Oceanography, and Plant Canopies Eﬀective Propagation Kernels in Structured Media with Broad Spatial Correlations, Illustration with Large-Scale Transport of Solar Photons Through Cloudy Atmospheres Anthony B. Davis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Extinction and Scattering Revisited, and Some Notations Introduced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Multiple Scattering and Diﬀusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Large-Scale 3D RT Eﬀects in Cloudy Atmospheres . . . . . . . . . . . . . . . 6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88 96 114 122 134 136

Mathematical Simulation of the Radiative Transfer in Statistically Inhomogeneous Clouds Evgueni I. Kassianov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Stochastic RT Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Statistically Inhomogeneous Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Ensemble Averaged Radiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141 141 142 143 144 146 147 148

Transport Theory for Optical Oceanography N.J. McCormick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Aspects Requiring Special Computational Attention . . . . . . . . . . . . . 3 Computational Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Computing Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151 151 156 159 161 161

85 85

Contents

Perturbation Technique in 3D Cloud Optics: Theory and Results Igor N. Polonsky, Anthony B. Davis, Michael A. Box . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Deﬁnition of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Variational Principe to Derive the Radiative Transfer Equation . . . . . . . . . . . . . . . . . . . . . . 4 Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 A Toy Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vegetation Canopy Reﬂectance Modeling with Turbid Medium Radiative Transfer Barry D. Ganapol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Description of the LCM2 Coupled Leaf/Canopy Radiative Transfer (RT) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 LCM2 Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rayspread: A Virtual Laboratory for Rapid BRF Simulations Over 3-D Plant Canopies Jean-Luc Widlowski, Thomas Lavergne, Bernard Pinty, Michel Verstraete, Nadine Gobron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Canopy Radiation Transfer Fundamentals . . . . . . . . . . . . . . . . . . . . . . 2 The Rayspread Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

VII

165 165 165 166 167 168 170

173 173 180 196 210

211 212 219 227 228

Part III High Energy Density Physics Use of the Space Adaptive Algorithm to Solve 2D Problems of Photon Transport and Interaction with Medium A. V. Alekseyev, R. M. Shagaliev, I. M. Belyakov, A. V. Gichuk, V. V. Evdokimov, A. N. Moskvin, A. A. Nuzhdin, N. P. Pleteneva, and T. V. Shemyakina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Statement of a 2D Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . 3 Description of 2D Transport Equation Approximation Methods . . . 4 Description of the Space Adaptive Computational Algorithm for Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Results of Computational Investigations of the Adaptive Method Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

235 235 236 238 238 240 251 254

VIII

Contents

Accurate and Eﬃcient Radiation Transport in Optically Thick Media – by Means of the Symbolic Implicit Monte Carlo Method in the Diﬀerence Formulation Abraham Sz˝ oke, Eugene D. Brooks III, Michael Scott McKinley, and Frank C. Daﬃn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Radiation Transport in LTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Diﬀerence Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Test Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Summary and Directions for Further Work . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

255 255 258 261 268 277 280

An Evaluation of the Diﬀerence Formulation for Photon Transport in a Two Level System Frank Daﬃn, Michael Scott McKinley, Eugene D. Brooks III, and Abraham Sz˝ oke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Equations for Line Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Numerical Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Numerical Results in the Gray Approximation . . . . . . . . . . . . . . . . . . 5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

283 283 285 289 295 304 305

Non-LTE Radiation Transport in High Radiation Plasmas Howard A. Scott . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Non-LTE Energetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Radiation Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Test Case: Radiation-driven Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Linear Response Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

307 307 309 311 316 322 324 325

Finite-Diﬀerence Methods Implemented in SATURN Complex to Solve Multidimensional Time-Dependent Transport Problems R.M. Shagaliev, A.V. Alekseyev, A.V. Gichuk, A.A. Nuzhdin, N.P. Pleteneva, and L.P. Fedotova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 1 Multiple-Group Transport Equation Approximation . . . . . . . . . . . . . 331 Implicit Solution of Non-Equilibrium Radiation Diﬀusion Including Reactive Heating Source in Material Energy Equation Dana E. Shumaker, Carol S. Woodward . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

Contents

3 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IX

355 359 368 369

Part IV Mathematics and Computer Science Transport Approximations in Partially Diﬀusive Media Guillaume Bal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Variational Formulation for Transport . . . . . . . . . . . . . . . . . . . . . . . . . 3 Transport-Diﬀusion Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Generalized Diﬀusion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Local Second-Order Equation and Linear Corrector . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

373 373 375 389 393 398 399

High Order Finite Volume Nonlinear Schemes for the Boltzmann Transport Equation Barna L. Bihari, Peter N. Brown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Discretization of the 3-D Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

401 401 403 405 410 420 421

Obtaining Identical Results on Varying Numbers of Processors in Domain Decomposed Particle Monte Carlo Simulations N.A. Gentile, Malvin Kalos, Thomas A. Brunner . . . . . . . . . . . . . . . . . . . 1 Description of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Ensuring the Invariance of the Pseudo-Random Number Stream Employed by Each Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Ensuring That Addition is Commutative . . . . . . . . . . . . . . . . . . . . . . . 4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . KM-Method of Iteration Convergence Acceleration for Solving a 2D Time-Dependent Multiple-Group Transport Equation and its Modiﬁcations A.V. Gichuk, L.P. Fedotova, R.M. Shagaliev . . . . . . . . . . . . . . . . . . . . . . . 1 Statement of a 2D Transport Problem . . . . . . . . . . . . . . . . . . . . . . . . . 2 KM-method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 MKM-method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

423 423 426 427 430 431 432

435 435 437 438

X

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KM3-method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 Test Computation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

A Regularized Boltzmann Scattering Operator for Highly Forward Peaked Scattering Anil K. Prinja, Brian C. Franke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Generalized Fermi Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Regularized Collision Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

445 445 446 449 452 455

Implicit Riemann Solvers for the Pn Equations Ryan McClarren, James Paul Holloway, Thomas Brunner, Thomas Mehlhorn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Pn Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Solving the Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 High Resolution Flux from Linear Reconstruction . . . . . . . . . . . . . . . 5 Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

457 457 458 459 461 462 463 463 466 467

The Solution of the Time–Dependent SN Equations on Parallel Architectures F. Douglas Swesty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 A Brief Review of The Implicit Discrete Ordinates Discretization Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Iterative Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Speeding Up and Obtaining Convergence . . . . . . . . . . . . . . . . . . . . . . . 5 Parallel Implementation of the Full Linear System Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Parallel Scalability of a 2-D Test Problem . . . . . . . . . . . . . . . . . . . . . . 7 Conclusions and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

469 469 470 472 475 481 483 484 485 485

Diﬀerent Algorithms of 2D Transport Equation Parallelization on Random Non-Orthogonal Grids Shagaliev R.M., Alekseev A.V., Beliakov I.M., Gichuk A.V., Nuzhdin A.A., Rezchikov V.Yu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

Contents

XI

Part V Neutron Transport Parallel Deterministic Neutron Transport with AMR C.J. Clouse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Code Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

499 499 500 508 511 512

An Overview of Neutron Transport Problems and Simulation Techniques Edward W. Larsen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Physical and Mathematical Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Basics of Stochastic and Deterministic Methods . . . . . . . . . . . . . . . . . 4 Stochastic (Monte Carlo) Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Deterministic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Automatic Variance Reduction (Hybrid) Methods . . . . . . . . . . . . . . . 7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

513 513 513 521 522 527 530 531 533

Introduction

There exist a wide range of applications where a signiﬁcant fraction of the momentum and energy present in a physical problem is carried by the transport of particles. Depending on the speciﬁc application, the particles involved may be photons, neutrons, neutrinos, or charged particles. Regardless of which phenomena is being described, at the heart of each application is the fact that a Boltzmann like transport equation has to be solved. The complexity, and hence expense, involved in solving the transport problem can be understood by realizing that the general solution to the 3D Boltzmann transport equation is in fact really seven dimensional: 3 spatial coordinates, 2 angles, 1 time, and 1 for speed or energy. Low-order approximations to the transport equation are frequently used due in part to physical justiﬁcation but many in cases, simply because a solution to the full transport problem is too computationally expensive. An example is the diﬀusion equation, which eﬀectively drops the two angles in phase space by assuming that a linear representation in angle is adequate. Another approximation is the grey approximation, which drops the energy variable by averaging over it. If the grey approximation is applied to the diﬀusion equation, the expense of solving what amounts to the simplest possible description of transport is roughly equal to the cost of implicit computational ﬂuid dynamics. It is clear therefore, that for those application areas needing some form of transport, fast, accurate and robust transport algorithms can lead to an increase in overall code performance and a decrease in time to solution. Besides the multi-dimensional nature of the transport equation, because of the coupling of particle transport to other phenomena the transport equation can in fact be non-linear. Hence, except for a few simple benchmark answers, the transport problem is solvable only via numerical methods. These numerical methods have developed and grown over the years and with the advent of massively parallel architectures, new scalable methods are being sought. Unfortunately, it is still true that in most computer codes, transport is the largest consumer of computational resources. In application areas that use transport, the computational time is usually dominated by the transport calculation. Therefore, there is a potential for great synergy; progress in transport algorithms could help quicken the time to solution for many applications.

XIV

Introduction

Consider for the moment, the details of particular applications where transport plays a role and it is clear the impact of solving the transport equation has on a variety of ﬁelds. In astrophysics, the life cycle of the stars, their formation, evolution, and death all require transport. In star formation and evolution for example, the problem is a multi-physics one involving MHD, self-gravity, chemistry, radiation transport, and a host of other phenomena. Supernova core collapse is an example where 3D, multi-group, multi-angle photon and neutrino transport are important in order to model the explosion mechanism. The spectra and light curves generated from a supernova have generated a wealth of data. In order to make a connection between simulation data and observational data and in order to remove systematic errors in supernova standard candle determinations of cosmological parameters, 3D, multi-group, multi-angle radiation transport is required. The simulation of nuclear reactor science poses a similar set of challenges. In order to move beyond the current state-of-the-art for such calculations, several requirements must be met: (1) a description based on explicit heterogeneous geometry instead of homogenized assemblies; (2) dozens of energy groups instead of two; (3) the use of 3D high-order transport instead of diffusion. These requirements would allow for accurate real-time simulations of new reactor operating characteristics, creating a virtual nuclear reactor test bed. Such a virtual reactor would enable assessments of the impact of new fuel cycles on issues like proliferation and waste repositories. With a 1000times increase in computer power, accurate virtual reactors could reduce the need to build expensive prototype reactors. In the broad area of plasma physics, ICF (Inertial Conﬁnement Fusion) and to a lesser extent MFE (Magnetic Fusion Energy) require the accurate modeling of photon and charged particle transport. For ICF, whether one is dealing with direct drive through photon or ion beams or dealing with indirect drive via thermal photons in a hohlraum, the accurate transport of energy around and into tiny capsules requires high-order transport solutions for photons and electrons. For direct drive experiments, simple radiation treatments suﬃce (i.e. laser ray tracing with multi-group diﬀusion). Although the radiation treatment can be rather crude, direct drive experiments require sophisticated models of electron transport. In indirect drive such as at NIF, laser energy is converted into thermal x-rays via a hohlraum which in turn is used to drive some target. In order to accurately treat the radiation drive in the hohlraum and its attendant asymmetries will require a radiation transport model with NLTE opacities for the hohlraum. The ability to generate NLTE is a tremendous computational challenge. Currently, calculating such opacities in-line comes at a great cost. Typically, the diﬀerence between an LTE transport and NLTE transport calculation is a factor of 5. This fact has sparked research into alternatives such as tabulating steady state NLTE opacities or by simplifying the electron population rate equations so that

Introduction

XV

their calculation is fast. However, all of the alternatives suﬀer from drawbacks which inhibit their widespread use. In the coming years, simulations of NIF (National Ignition Facility) experiments will be crucial in attaining the goal of ignition. The simulations need to be predictive rather than after-experiment ﬁts; therefore, high-order transport coupled self-consistently to other nonlinear physics is a requirement. With a 1000-fold increase in computer power, these types of simulations are feasible. In planetary atmospheres, cloud variability and radiative transfer play a key role in understanding climate. For example coupling clouds to the radiative transfer problem and representing their distribution and size accurately is a diﬃcult problem where such methods as sub-grid scale methods are being applied. Tightly integrated to planetary atmospheres is the problem of radiative transfer and plant canopies and the oceans. For example, the 3D structure of vegetation is a key player in processes eﬀecting carbon sequestration, landscape dynamics and the exchanges of energy, water and trace gases with the atmosphere. These examples are just a small subset of the applications where an accurate and fast determination of particle transport is required. Typically, the numerical methods used to solve the transport equation in a given discipline are communicated to other researchers in that discipline. Rarely are those methods communicated outside of that speciﬁc ﬁeld. For example, nuclear engineers and astrophysicists rarely attend the same meetings. The seven-dimensional nature of transport means that factors of 100 or 1000 improvement in computer speed or memory are quickly absorbed in slightly higher resolution in space, angle, and energy. Therefore, the biggest advances in the last few years and in the next several years will be driven by algorithms. Because transport is an implicit problem requiring iteration, the biggest gains are to be made in ﬁnding faster techniques for acceleration to convergence. Some of these acceleration methods are very application speciﬁc because they are physics based; others are very general because they address the mathematics of the transport equation. In September of 2004, the Computational Methods in Transport Workshop was held at Granlibakken, California with the hope that these issues could be addressed by providing a forum where computational transport researchers in a variety of disciplines could communicate across disciplinary boundaries their methods and their methods successes and failures. The goal of the workshop was to open channels of communication and cooperation between all members of the computational transport community so that (1) existing methods used in one ﬁeld can be applied to other ﬁelds (2) greater scientiﬁc resource can be brought to bear on the unsolved outstanding problems. The idea for the workshop was born at the SCaLeS (Scientiﬁc Case for Large Scale Simulation) meeting held in Washington D.C. in June of 2003

XVI

Introduction

and chaired by David Keyes. In attendance were a group of scientists, engineers, and computer scientists from a wide variety of disciplines whose charter was to demonstrate, from an applications standpoint, the need for new ultrascale computing facilities for Oﬃce of Science missions. A variety of breakout sessions were formed. One of these sessions, chaired by myself and Gordon Olson of Los Alamos National Laboratory, dealt with particle transport. The particle transport group consisted of experts from nuclear engineering, astrophysics, combustion, atmospheres, mathematics, etc. The discussions were lively and informative and it was soon realized what a wonderful opportunity this was to discuss transport issues with persons outside of speciﬁc disciplines. Conversations ﬂowed between experts who would probably never meet in the normal course of doing research. The Computational Methods in Transport Workshop was created out of my hope that a forum could exist where researchers could discuss successes and failures of their methods across discipline boundaries in an invigorating and relaxing atmosphere that would beneﬁt the transport community at large. Both executive and scientiﬁc organizing committees were formed with the goal of putting together a conference designed to stimulate discussion and interaction among attendees. With this goal in mind, talks were selected that gave attendees both an overview of transport issues from a wide variety of ﬁelds along with talks giving specialized detail. In addition, the workshop talks were purposefully chosen to be few in number yet fairly long in length so that speakers could communicate to the audience in a more eﬀective fashion. The papers selected for the present volume have hopefully met both of these goals. Each author has put considerable work into writing a paper meant for an audience who although interested in transport, might not be an expert in plant canopies or supernova. My thanks to each and every author for writing papers of such high quality. The executive committee consisted of David Keyes (Columbia University), James McGraw (Lawrence Livermore National Laboratory), and Stanley Osher (Institute of Pure and Applied Mathematics, UCLA). They provided invaluable advice and guidance of workshop logistics and ﬁnances and I owe them a sincere note of gratitude. The scientiﬁc committee focused on the technical issues of the meeting and it was comprised of individuals drawn from a variety of disciplines who had established themselves in the ﬁeld of computational transport. My warmest thanks to Marv Adams (Texas A&M University), John Castor (Lawrence Livermore National Laboratory), Frank Evans (University of Colorado), Ivan Hubeny (University of Arizona), Tom Manteuﬀel (University of Colorado), and Gordon Olson (Los Alamos National Laboratory) for a job well done. A workshop like the Computational Methods in Transport Workshop would never get oﬀ the ground without ﬁnancial support. In particular, because of the generous support of our sponsors, fourteen students were able to attend the meeting with all expenses paid. The workshop was jointly spon-

Introduction

XVII

sored by Lawrence Livermore National Laboratory and the Institute of Pure and Applied Mathematics at UCLA. Mark Green (IPAM, UCLA), Stanley Osher (IPAM, UCLA), Jim McGraw (LLNL), and Rob Falgout (LLNL) deserve singular thanks for their guidance and ﬁnancial support. A special note of thanks goes to the administrative and support people who have made this workshop a reality. Leigh Faulk (LLNL), Anita Williams (LLNL), Linda Becker (LLNL), Janice Amar (UCLA), Karen Lee (UCLA), Linda Oribello (LLNL), Dave Parker (LLNL), Linda Null (LLNL) and Fred Allen (LLNL) performed nothing short of miracles. Finally, a word of thanks to the editors and staﬀ at Springer-Verlag. In particular, Martin Peters and Thanh-Ha LeThi were of invaluable help and I appreciate their support and patience through this project. Lawrence Livermore National Laboratory July 2005

Frank R. Graziani

Part I

Astrophysics

Radiation Hydrodynamics in Astrophysics Chris L. Fryer1,2 1

2

Theoretical Astrophysics, T-6, Los Alamos National Laboratory, Los Alamos, NM 87545 [email protected] Physics Department, University of Arizona, Tucson, AZ 85721

1 Deﬁning Radiation Hydrodynamics Terms Hydrodynamics codes are used to study nearly every astrophysical phenomena observed. Although coupling radiation and hydrodynamics is much less common, radiation hydrodynamics codes are being used in a growing number of astrophysics problems from energetic out ﬂows of compact remnants such as core-collapse supernovae and active galactic nuclei to the formation of stars and planets to cosmological simulations of the ﬁrst stars. In this paper, we review the current “state-of-the-art” radiation hydrodynamics techniques used in astrophysics. This review will cover both the techniques used in astrophysics (Sect. 2) and the problems that have been studied (Sect. 3). Most of the techniques are well-known in both the transport and hydrodynamics communities. The only surprises may be the rudemenatry level at which most working astrophysics codes do radiation transport. In Sect. 4, we will study in more detail the method used to do radiation hydrodynamics with smooth particle hydrodynamics, as it is less well-known and may prove a powerful tool to solve a number of astrophysics problems. What astrophysicists mean when they say they have a “radiation hydrodynamics” code varies from person to person. So before we begin this review, we must go through a few deﬁnitions. Radiation Hydrodynamics: By “radiation hydrodynamics” we restrict ourselves to those codes that, in some manner or another, solve both the Boltzmann equation and the hydrodynamics equations, coupling the results of both to determine the state of matter in a problem. By solving the Boltzmann equation, we include any technique, no matter how crude. This includes moment closure techniques as well as stochastic and direct discretization techniques. For most applications in astrophysics, this means ﬂux-limited diﬀusion. By solving the hydrodynamics equations, in astrophysics this is almost entirely limited to solving the inviscid (Euler) equations. Similarly, I will loosely deﬁne “coupling these equations” to any scheme that combines the solutions of both these equations. In practice, this is generally done using operator split methods with a single or no iterations to enforce convergence. Working Astrophysics Code: By working astrophysics code, I mean codes that have been developed that have been used to solve an astrophysics

4

C.L. Fryer

problem and the results of these simulations have been published in the astrophysics literature. We do not consider codes that have only appeared in papers describing the technique without actually being run on an astrophysics problem. Parallelized Code: By parallelized code, I mean those codes that actually run on many processors, scaling reasonably well beyond 100 processors. This excludes codes that are parallelized by solving a diﬀerent energy group on each processor (which for most astrophysics problems, limits the code’s use to 16-32 processors). We also exclude codes that can only be used on shared memory machines. Moment Closure vs. Direct Discretization vs. Stochastic Methods: The transport schemes in astrophysics can be loosely classiﬁed into three diﬀerent categories: 1: Moment closure methods where the Boltzmann equation is represented as a series of angular moments and these moments are closed to form a solution. Both Flux-limited diﬀusion and variable Eddington factor techniques ﬁt into this class. 2: Direct discretization of the Boltzmann equation. Probably the most common technique in this category used in astrophysics is the Sn method. This is often called “Boltzmann transport”. 3: Stochastic methods or Monte Carlo Methods. In astrophysics, the use of Monte Carlo techniques has, in the past, been limited to time-independent calculations (e.g. post-process).

2 Schemes Used in Astrophysics Although these deﬁnitions don’t seem restrictive, they exclude a lot of the techniques used in astrophysics: hydrodynamics schemes that put in radiation as a cooling term or use a “leakage” scheme to control the radiation transport through a dense medium (there are a number of problems, for example, in cosmology, where such schemes are suﬃcient to solve the current questions being addressed); and transport schemes that add in material motion, but do not actually solve the hydrodynamics equations (some very sophisticated transport techniques have been used to solve supernova spectra and stellar atmospheres where hydrodynamics plays a secondary role, and hence has been neglected in past studies). Transport Schemes coupled with hydrodynamics schemes include: 1: Pure diﬀusion schemes. These schemes are generally solved as 1-T calculations (one temperature describes both the radiation and matter). Although technically a simpliﬁed solution to the Boltzmann equation, we will not consider codes using these schemes further.

Radiation Hydrodynamics in Astrophysics

5

2: Flux-limited diﬀusion schemes. These schemes are among the simplest of the moment closure techniques. A number of recipes (a.k.a. ﬂux-limiters) are used in astrophysics to connect the diﬀusion and free-streaming limits of the transport equation. These schemes are generally implemented using 2-T (one temperature describing the radiation ﬁeld and one temperature describing the matter) and have been solved both using single and multigroup energy discretization. This technique dominates what is used in astrophysics, especially in multi-dimension. 3: Variable Eddington factor schemes. This technique also uses moments of the Boltzmann equation, but closes at one higher moment using Eddington factors. The technique used to solve for the Eddington factors diﬀers from code to code. Although initially used in pure transport problems, this scheme has been used in both 1-dimensional and 1.5-dimensional (transport along rays) radiation hydrodynamics schemes. 4: Sn transport. In general, most working codes using Sn techniques have been limited to 1-dimensional hydrodynamics codes, but at least 1 paper has been published showing results of a 2-dimensional simulation. For many astrophysics phenomena, pure hydrodynamics codes can provide a ﬁrst order answer to a problem. Codes have been developed ﬁrst modeling hydrodynamics only with transport schemes being added into these codes at a later time. Although this trend is slowly reversing (hydrodynamics schemes are now being coupled to transport codes) because the transport scheme dominates the computing time, at this time, most of the working astrophysics codes arise from hydrodynamics schemes with transport schemes added on top. The current range of hydrodynamics schemes coupled with transport techniques include most of the hydrodynamics techniques used in astrophysics: 1: Lagrangian techniques. In astrophysics, Lagrangian grid codes are used in 1-dimension, but most multi-dimensional Lagrangian codes use smooth particle hydrodynamics. A number of transport schemes have been coupled to 1-dimensional Lagrangian codes. Although coupling transport with smooth particle hydrodynamics takes some thought, it can, and has been done in (at least at the level of ﬂux-limited diﬀusion). 2: Eulerian techniques. These techniques include both ﬁxed grid and adaptive grid reﬁnement methods.A number of transport schemes have been coupled to both 1-dimensional and 2-dimensional Eulerian codes. Fluxlimited diﬀusion has been coupled to 3-dimensional Eulerian codes. Astrophysicists have applied a number of these schemes in 1-dimension. In 2- and 3-dimensions, most of the work is limited to a range of hydrodynamics codes coupled to ﬂux-limited diﬀusion. Below, we discuss speciﬁc codes applied to two of the current astrophysics problems best-studied using radiation-hydrodynamics codes.

6

C.L. Fryer

3 Astrophysical Applications Computational astrophysicists work using small steps, gradually increasing the level of sophistication on a given problem. Early models simulate only the most important physics to compare to broad features in the observations. But as the observations become more detailed, so must the codes. For example, early cosmological structure models modeled only the eﬀects of gravity, focusing on the motion of the dark matter. But as observations of the baryonic matter became more detailed, hydrodynamical eﬀects were added to the computer models. Now, to study the formation of the ﬁrst stars, modelers must include the eﬀects of radiation transport. Although most cosmological studies today can make progress with simplistic radiation routines like leakage schemes, there will come a time when the observational detail requires radiation hydrodynamics techniques. A number of astrophysical problems have already reached this coding requirement. Stellar atmosphere studies, which in the past have done some of the most sophisticated transport schemes in astrophysics using static atmospheres are now studying the eﬀects of convection by coupling a 3dimensional hydrodynamics code with ﬂux-limited diﬀusion [Stein & Nordlund 2003]. Klein and collaborators have studied a number of phenomena using radiation (usually ﬂux-limited diﬀusion) hydrodynamic codes from neutron star accretion [Murray et al. (1995), Klein et al. (1996)] to molecular clouds [Sandford et al. (1982)]. However, we will focus our attention on the two ﬁelds that appear to be most actively pursuing radiation hydrodynamics in astrophysics: accretion disks and core-collapse supernovae. 3.1 Accretion Disks Many astrophysical phenomena are powered by the gravitational energy released when matter accretes onto a compact object: ranging from active galactic nuclei to X-ray bursts to gamma-ray bursts to planet and star formation. Even if the initial velocity asymmetries are small, as this matter falls down onto the compact object, centrifugal forces become increasingly important and the behavior of these phenomena is dominated by the disk of material that forms prior to the accretion phase. The wide variety of applications has led to a number of studies of disk accretion with increasing levels of sophistication. Portions of most of these disks are indeed optically thick (mean free path much less than disk scale height) and modeling the radiation transport correctly is important to many accretion disk problems. Although transport techniques ranging from leakage schemes through Monte-Carlo have been used to post-process disk calculations, the current state-of-the-art for production, radiation-hydrodynamics calculations in disks is limited to ﬂux limited diffusion [Kley 1989, Turner & Stone (2001), Turner et al. (2003)]. These codes

Radiation Hydrodynamics in Astrophysics

7

have mostly been run on shared memory machines with few nodes so it is unclear how well they scale to many processors with limited bandwidth. 3.2 Core-Collapse Supernovae In the past, the supernovae “transport experts” have worked on supernova light-curves. However, the transport schemes used in light-curve calculations are almost entirely limited to solutions of a steady state or uniformly varying medium. Only recently have radiation hydrodynamics codes (generally 1dimensional ﬂux-limited diﬀusion hydrodynamics codes) been applied to this problem [H¨ oﬂich et al. (1993), Blinnikov et al. (2000)]. The results from these calculations are then post-processed to get detailed spectra. The supernova “radiation-hydrodynamics experts” are actually the modellers of stellar collapse (here the neutrino is the transport particle). Corecollapse supernova theorists came to radiation hydrodynamics by ﬁrst focusing on hydrodynamics and including increasingly sophisticated transport techniques. These simulations have been at the cutting edge of computational astrophysics due to a boost from codes built at national laboratories in the sixties [Colgate & Johnson (1960), Colgate & White (1966)]. Core collapse supernovae are powered by the potential energy released during the collapse of the iron core of a massive star down to a ∼50 km proto-neutron star [Fryer (2003)]. This energy leaks out of the proto-neutron star in the form of neutrinos. The neutrino mean free path ranges from a few cm in the proto-neutron star to >100 km in the convective region where neutrino energy is deposited. Figure 1 shows a slice from a 3-dimensional simulation of a core-collapse supernovae. The three circles show the position where the optical depth out of the star is roughly 0.05 for the µ/τ , anti-electron and electron neutrinos (the innermost circle denotes µ/τ ; the outermost circle denotes the electron neutrinos). A number of techniques have been used to study radiation hydrodynamics in core-collapse supernovae. Tony Mezzacappa has developed some very nice diagrams that diﬀerentiate these codes. Using a simpliﬁed version of such a diagram dividing codes by spatial hydrodynamics dimension and tranpsort technique (Fig. 2), we describe some of the key codes developed and their relevant citation (recall that we limit our discussion to codes that have been applied to real problems): 1: In one dimension, a range of transport techniques have been added to both Lagrangian and Eulerian codes. Multi-group, ﬂux-limited diﬀusion has long been used in stellar collapse [Bruenn et al. (1978)]. 2: Very sophisticated have been used in 1-dimensional codes. Discrete techniques (Sn method) have been around for over a decade and are said to be parallel although I have not seen scaling numbers [Mezzacappa & Bruenn (1993)]. The variable Eddington factor technique has also been used in

8

C.L. Fryer

Fig. 1. A slice in the x-z plane of the 3-dimensional collapse of a rotating (rotation axis is the z-axis) star. The shading denotes entropy and the vectors denote velocity (the vector length is the magnitude of the velocity and the direction denotes the direction of ﬂow). The 3 circles denote the positions of roughly the τ = 0.05 surface for µ and τ neutrinos (inner circle), anti-electron neutrinos (intermediate circle) and electron neutrinos (outer circle). The neutrinos are in the diﬀusive limit in the proto-neutron star but pass through the transport regime into the free-streaming limit in the convective region where neutrino heating drives the explosion

1-dimension [Thompson et al. (2003)]. It has only been used in one paper, so it is not clear how well this technique truly works and it is not parallel. 3: Two-dimensional, single-energy ﬂux-limited diﬀusion has been done for over a decade. The ﬁrst simulation using such codes to model collapse through explosion was in 1994 [Herant et al. (1994)]. 4: Two-dimensional, multi-group, ﬂux-limited diﬀusion is less common than in 1-dimension(2:). Indeed, the transport that has been done was not true 2-dimensional transport. The diﬀusion was followed along rays (no lateral transport) [Burrows et al. (1995)]

2

Variable Eddington Factor Discrete Methods

9

5

1.5 Variable Eddington Factor Along Rays Discrete Methods

Multi−Group, Flux−Limited Diffusion

1

4

1.5 Multi−Group Flux−Limited Diffusion Along Rays

3

Single Energy

Multi Energy

Multi Energy Multi Angle

Radiation Hydrodynamics in Astrophysics

Gray Flux− Limited Diffusion

Gray Flux− Limited Diffusion

6

Fully Parallel

I

II

III

Neutrino/Spatial Dimension Fig. 2. Simpliﬁed Mezzacappa diagram of the radiation-hydrodynamics models used in core-collapse studies. The y axis shows the transport technique (singleenergy ﬂux-limited diﬀusion, multi-group ﬂux-limited diﬀusion, and more sophisticated techniques that include angular eﬀects). The x axis denotes level of spatial (and purportedly neutrino dimension). Note that some 2-dimensional techniques only follow transport along rays. The numbers correspond to more detailed descriptions in the text

5: Some of the most intense research has been focused on sophisticated techniques intwo-dimensions. A variable Eddington technique has been developed [Scheck et al. (2004)] that follows transport along rays (no lateral transport). This technique is not extremely parallelizable (diﬀerent nodes are given diﬀerent energy groups to model). A discrete method has also been developed [Livne et al. (2004)]. Similarly, it has only been parallelized by putting diﬀerent energy groups on diﬀerent processors. Also, the one paper using this technique did not appear to run the simulation for many timesteps, and I suspect this technique will not be usable to do the full collapse problem (requiring 100,000 timesteps). 6: Three-dimensional, single-energy ﬂux limited diﬀusion modelling the collapse through explosion was done using SNSPH [Fryer & Warren (2002)]. This code is fully parallel and has scaled nearly linearly up to 512 processors. This list of techniques far outstrips any other astrophysics ﬁeld in radiationhydrodynamics codes. But the limitations are already evident. First note that there are few working codes able to model transport in more than 1 spatial

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C.L. Fryer

dimension. Those that have passed our relaxed criterion for a “working code” are either not parallelizable, not truly multi-dimensional transport schemes or both. Indeed, the only true multi-dimensional radiation hydrodynamics code is limited to single-energy ﬂux-limited transport. Astrophysicists have described much more sophisticated techniques (even at this meeting), but they have not put them into working codes, let alone parallelizable codes. There is a lot astrophysicists can gain from the transport community in making working codes.

4 SPH Radiation Transport One of the few working multi-dimensional radiation hydrodynamics codes in core-collaspe is a technique coupling smooth particle hydrodynamics (SPH) to ﬂux-limited diﬀusion. The coupling, developed in 2-dimensions in 1994 [Herant et al. (1994)] and parallelized in 3-dimensions in 2002 [Fryer & Warren (2002)] has some twists to it and it is worth describing it in some detail. Much of this description is taken from a paper submitted to the Astrophysics Journal by Fryer, Rockefeller, and Warren (2005). 4.1 Transport Scheme In core-collapse supernovae, we model the transport of 3 neutrino species (l = νe , ν¯e , νx where νx corresponds to the τ, µ neutrinos and their antiparticles that are all treated equally). Because neutrino number is the conserved quantity, we transport neutrino number and then determine the energy transport by using the mean neutrino energies. The radiation transport scheme in our SPH code is modeled after the technique to calculate forces in SPH: we calculate symmetric interactions between all neighbor particles. Hence, our ﬂux-limited diﬀusion scheme calculates the radiation diﬀusion in or out of a particle by summing the transport over all neighbors (the equivalent of all bordering cells in a grid calculation). The neutrino transport for particle i is given by: ij j j→i ∇W ij mj /ρj Λνl niνl bi→j − n b (1) dniνl /dt = νl νl νl j

and the corresponding energy transport is: ij j→i j j j→i ∇W ij mj /ρj Λνl iνl niνl bi→j − ξ n b deiνl /dt = νl νl νl νl

(2)

j

where niνl , eiνl are, respectively, the neutrino density and energy density in particle i for species νl, iνl is the mean neutrino energy, and ξ j→i is the redshift correction for jνl as seen by particle i.

Radiation Hydrodynamics in Astrophysics

11

Λij νl is the limiter for the ﬂux-limited transport scheme. The simplest such scheme for 3-dimensions is: ij ij Λij νl = min(c, Dνl /r )

(3)

ij j j i i = 2Dνl Dνl /(Dνl + Dνl ) is the harmonic where c is the speed of light, Dνl mean of the diﬀusion coeﬃcients for the species νl of particles i and j and rij is the distance between particles i and j. This limiter was used by [Herant et al. (1994)] and, for comparison with that work, by [Fryer & Warren (2002)], but we have used a number of other ﬂux-limiters, all of which are valid under this transport scheme [Fryer et al. (1999)]. Beyond some radius in a core-collapse simulation, neutrinos are essentially in the free-streaming regime where transport is not necessary (unless one wants to truly follow the radiation wave as it progresses through the star). We do not model transport beyond this “trapping” radius. Instead we sum up all neutrinos that transport beyond this radius and emit them using a lightbulb approximation. That is, the material beyond this radius sees a constant ﬂux and we can determine the amount of energy a particle gains (dEi /dt) from neutrino interactions simply by using the free-streaming limit:

dEi /dt = Lν (1.0 − e−∆τi )

(4)

where Lν is the neutrino luminosity and ∆τi is the optical depth of a particle i. This assumption is only valid if the total amount of energy imparted ( i dEi /dt) from the neutrinos onto the matter is much less than the tothis, we determine this trapping radius tal neutrino ﬂux (Lν ). To guarantee by evolving it with time such that ( i dEi /dt)/Lν is always less than some value. This value was originally set to 0.1 by Herant et al. (1994), but in recent calculations, we use 0.05. Such a scheme can be easily converted into multi-group, but such modiﬁcations have not yet been done. The scheme scales reasonably well on multiple processors (for a 5 million particle run, the code has scaled nearly linearly up to 256 processors on the Space Simulator Beowulf cluster and up to 512 processors on the ASC Q machine at Los Alamos National Laboratory). In part, this scalability is due to the explicit nature of the transport scheme. In general, explicit transport schemes strongly limits timesteps as the speed of light, not sound, constrains the duration of the timestep. In core-collapse supernovae, this constraint is not too onerous because the sound speed is nearly a third the speed of light anyway, so the explicit transport scheme leads to only a factor of 3 decrease in the timestep. But this explicit ﬂux-limited transport scheme can be used in a much wider variety of problems where the mean free path is very short for the smallest particles. Such scenarios occur in many astrophysics problems.

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4.2 First Test Testing radiation transport schemes in general would, and has, comprised many papers in itself. Here we focus on a simple test comparing the SPH ﬂuxlimited scheme to a 1-dimensional grid-based ﬂux-limited transport scheme. This test does not prove the applicability of ﬂux-limited diﬀusion to the supernova problem, but it does show that our scheme for putting ﬂux-limited diﬀusion into SPH does work. For the initial conditions of this test, we use a spherically-symmetric neutron star atmosphere (material below 1014 g cm−3 ) for a neutron star roughly 130 ms after bounce. We map this structure onto our SPH particles, using the shell setup described in Fryer & Warren (2002). We make a similar setup in a 1-dimensional grid using one zone per shell of SPH particles. With this setup, we minimize the diﬀerences between the density and temperature structure of our 1- and 3-dimensional models. We determine the trapping radius to be 25 km. Below this radius, we set the electron neutrino fraction (Yνe ) to 0.15 with a mean energy of 10 MeV. Above this radius, Yνe is set to zero. Our test focuses on the neutrino transport alone; we evolve only the electron neutrino fraction with time and hold the density, temperature, and electron fraction ﬁxed. We allow no new neutrino emission. The ﬂux arising from

Fig. 3. Neutrino luminosity versus time for a cooling neutron star using a simple 1dimensional ﬂux-limited diﬀusion scheme and the technique discussed here coupling ﬂux-limited diﬀusion to smooth particle hydrodynamics. The initial discrepancy arises from the fact that diﬀusion occurs over all SPH neighbors, allowing transport across the eﬀective (in a 1-dimensional case) of several radial zones (instead of 1 zone in the 1-dimensional transport scheme)

Radiation Hydrodynamics in Astrophysics

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our neutrino trapping radius versus time for both our 1-dimensional grid simulation and our 3-dimensional SPH calculation is plotted in Fig. 3. The initial ﬂux of the SPH calculation is higher, since the neighbors of any particle extend beyond the equivalent of an adjacent cell for the 1-dimensional calculation. In general, the luminosity for both these calculations agree to better than 3%.

References [Blinnikov et al. (2000)] Blinnikov, S., Lundqvist, P., & Bartunov, O., Nomoto, K., Iwamoto, K. Radiation Hydrodynamics of SN 1987A. I. Global Analysis of the Light Curve for the First 4 Months ApJ, 532, 1132 (2000) [Bruenn et al. (1978)] Bruenn, S.W., Buchler, J. R., & Yueh,W. R. Neutrino Transport in Supernova Models – A Multigroup, Flux-Limited diﬀusion Scheme Ap& SS, 59, 261–284 (1978). [Burrows et al. (1995)] Burrows, A., Hayes, J., & Fryxell, B. A. On the Nature of Core-Collapse Supernova Explosions, ApJ, 450, 830–850 (1995). [Colgate & Johnson (1960)] Colgate, S.A., Johnson, H.J., Phys Rev. Letters, 5, 573–576 (1960). [Colgate & White (1966)] Colgate, S.A., White, R.H. ApJ, 143, 626–681 (1966) [Fryer et al. (1999)] Fryer, C. L., Benz, W., Herant, M., Colgate, S.A. What Can the Accretion-induced Collapse of White Dwarfs Really Explain? ApJ, 516, 892 (1999) [Fryer & Warren (2002)] Fryer, C. L., & Warren, M. S. Modeling Core-Collapse Supernovae in Three Dimensions ApJ, 574, L65 (2002) [Fryer (2003)] Fryer, C. L. Stellar Collapse, International Journal of Modern Physics D, 12, 1795–1835 [Fryer & Warren (2004)] Fryer, C. L., & Warren, M. S. The Collapse of Rotating Massive Stars in Three Dimensions, ApJ, 601, 391–404 [Herant et al. (1994)] Herant, M., Benz, W., Hix, W. R., Fryer, C. L., & Colgate, S. A. Inside the Supernova: A Powerful Convective Engine ApJ, 435, 339 (1994) [H¨ oﬂich et al. (1993)] H¨ oﬂich, M¨ uller, E., & Khokhlov, A. Light Curve Models for Type Ia Supernovae - Physical Assumptions, their Inﬂuence and Validity A&A, 268, 570–590 (1993) [Klein et al. (1996)] Klein, R.I., Arons, J., Garrett, J., Hsu, J.J.L.: Photon bubble Oscillations in Accretion-powered Pulsars. ApJ, 457, L85–89 (1996). [Kley 1989] Kley, W. Radiation Hydrodynamics of the Boundary Layer in Accretion Disks. I- Numberical models A&A, 208, 98–110 (1989). [Livne et al. (2004)] Livne, E., Burrows, A., Walder, R., Lichtenstadt, I., & Thompson, T. A. Two-Dimension, Time-Dependent, Multigroup, Multiangle Radiation Hydrodynamics in the Core-Collapse Supernova Context, ApJ, 609, 277– 287 [Mezzacappa & Bruenn (1993)] Mezzacappa, A., & Bruenn, S. W. A Numerical Method for Solving the Neutrino Boltzmann Equation Coupled to Spherically Symmetric Stellar Core Collapse ApJ, 405, 669–684 (1993). [Murray et al. (1995)] Murray, S.D., Woods, D.T., Castor, J.I., Klein, R.I., McKee, C.F.: Radiation Hydrodynamic Models of Eclipsing Low-Mass X-ray Binaries ApJ, 454, L133–136 (1995).

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[Sandford et al. (1982)] Sandford, M.T., Whitaker, R.W., Klein, R.I.: RadiationDriven Implosions in Molecular Clouds ApJ 260, 183–201 (1982). [Scheck et al. (2004)] Scheck, L., Plewa, T., Janka, H.-Th., Kifonidis, K., & M¨ uller, E. Pulsar Recoil by Large-Scale Anisotropies in Supernova Explosions PRL, 92, 0111031–011034 [Stein & Nordlund 2003] Stein, R.F., Nordlund, A.: Stellar Atmosphere Modeling, ASP Conference Proceedings, Vol 288, Aug. 8–12, 2002, Hubeny, Mihalas, Werner (eds.). San Francisco: Astronomical Society of the Paciﬁc (2003) p.519 [Thompson et al. (2003)] Thompson, T. A., Burrows, A., Pinto, P. A. Shock Breakout in Core-Collapse Supernovae and Its Neutrino Signature, ApJ, 592, 434– 456 (2003). [Turner & Stone (2001)] Turner, N. J., & Stone, J. M. 2001, A Module for Radiation Hydrodynamic Calculations with ZEUS-2D Using Flux-Limited Diﬀusion, ApJS, 135, 95–107 [Turner et al. (2003)] Turner, N. J., Stone, J. M., Krolik, J. H. & Sano, T. Local Three-dimensional Simulations of Magnetorotational Instability in RadiationDominated Accretion Disks ApJ, 593, 992 (2003).

Radiative Transfer in Astrophysical Applications I. Hubeny Department of Astronomy, University of Arizona, Tucson, AZ 85721, USA

1 Introduction Radiative transfer is particularly important in astrophysics. One reason is quite understandable: radiation is in most cases the only information we have (and will ever have) about distant objects (exceptions are detected neutrinos from the Sun and supernova SN1987a, and in a near future the gravitational waves). However, there is an even more compelling reason for the a need to deal with detailed radiation transport in astrophysics: In many astronomical objects the radiation is so strong that it signiﬁcantly contributes to the energy and momentum budget of the medium. Therefore, radiation is not only a probe of the physical state, but is in fact an important constituent. In other words, radiation in fact determines the structure of the medium, yet the medium is probed only by this radiation. Unlike laboratory physics, where one can change a setup of the experiment in order to examine various aspects of the studied structures separately, we do not have this luxury in astrophysics: we are stuck with the observed spectrum so we should better make a very good use of it. All that puts heavy demands on proper radiation transfer techniques in astrophysics. Also, there are many diﬀerent situations and physical conditions (e.g., temperature ranges from a few K to more than 109 K; there is a huge range of microscopic processes that give rise to an absorption or emission of a photon), so consequently there is a wide range of numerical techniques used for treating radiation transport in astrophysics. This paper is not aimed to provide a review of all or even the most important numerical techniques; instead it aims at outlining the outstanding problems and ideas behind recent numerical approaches. I also stress that although I talk about transport of photons, many methods can be used for treating transport of neutrinos, which is important for instance in numerical simulations of core-collapse supernovae. Typical objects for which the radiation transport is important are above all stellar atmospheres. In this ﬁeld, most numerical techniques were developed that are now being used in other parts of astrophysics. Other objects are extrasolar planetary atmospheres (the solar system planets as well, but here we have a lot of direct information thanks to various planetary missions). Other important radiation-dominated objects are accretion disks of various kinds—around supermassive black holes (quasars and active galactic nuclei);

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I. Hubeny

neutron stars (X-ray binaries); white dwarfs (cataclysmic variables), and even main-sequence close binaries. Transport of photons (and neutrinos at early stages!) is important in supernovae, and radiative transfer is important in curcumstellar structures like planetary nebulae and H II regions. Finally, radiation transport is also important in global cosmological simulations of large-scale structure of the Universe. The basic textbook of astrophysical radiative transfer is [1] but it does not cover modern and eﬃcient numerical transport techniques. Another textbooks, with emphasis more on hydrodynamical aspect of the problems, are [2], and recent textbook [3]; both contain excellent discussions of radiation transport techniques. Recent proceedings that contains a number of review papers, and to which the reader is referred to for more information about modern methods used in astrophysical radiative transfer, is [4].

2 Description of Radiation In astrophysical formalism, the basic quantity describing the radiation ﬁeld is the speciﬁc intensity of radiation, I(r, t, n, ν), deﬁned such that it is the energy transported by radiation at position r, in a unit frequency range at the frequency ν, across a unit area perpendicular to the direction of propagation, n, into a unit solid angle, and in a unit time interval. The speciﬁc intensity provides a complete description of the unpolarized radiation ﬁeld from the macroscopic point of view. The speciﬁc intensity is related to the photon distribution function, f , as I = hνc f (1) where h and c are the Planck constant and the speed of light, respectively. We note that the same quantity is called angular ﬂux, and is usually denoted as ψ(r, t, n, ν), in the neutron transport theory. The transport (or transfer) equation is essentially the Boltzmann equation for photons, which can be generally written as Df ∂f + (v · ∇)f + (F · ∇P )f = , (2) ∂t Dt coll where v is the velocity, F is the external force, ∇P is the nabla operator in the momentum space, and the term (Df /Dt)coll is the collision term. In case of photons, v = cn, and F = 0 (in the absence of general relativistic eﬀects). The transfer equation can then be written as 1 ∂ + n · ∇ I(ν, r, n, t) = η(ν, r, n, t) − χ(ν, r, n, t) I(ν, r, n, t) . (3) c ∂t where we write the collisional term in terms of the usual absorption coeﬃcient χ and emission coeﬃcient η. The equation looks simple, but this is deceiving.

Radiative Transfer in Astrophysical Applications

17

The point is that χ and η generally depend on radiation intensity as well, moreover in a complex non-linear and non-local fashion. We shall return to this point in the next section. We also note that it is common in astrophysical applications to introduce the optical depth along a given direction as dτ = χds, and the source function as η (4) S≡ . χ

3 Absorption, Emission and Scattering Coeﬃcients The absorption and emission coeﬃcients are introduced phenomenologically, and are deﬁned analogously to the speciﬁc intensity, namely as the energy removed or added to a beam of radiation at unit frequency range, solid angle, area, and time. It is customary to split them into two parts, χ = κth + κsc ,

(5)

η = η th + η sc ,

(6)

and where κth represents the true absorption (a process where a photon is absorbed and subsequently destroyed by a thermal process such as a collisional de-excitation); κsc is the scattering coeﬃcient (a photon is absorbed and subsequently re-emitted). Similarly, η th is the coeﬃcient of thermal emission (a photon is created at the expense of the thermal pool); and η sc is the scattering part of the emission coeﬃcient, corresponding to emission of a photon following previous absorption. Scattering absorption coeﬃcient may be written as nα σα (ν) , (7) κsc (ν) = α

where the summation extends over all species α that do scatter radiation, nα is the number density, and σα the corresponding cross-section. For instance, for hot star atmospheres (spectral types O and B), the dominant, and usually the only scattering process that is considered is the electron (Thomson) scattering, in which case nα σα (ν) = ne σe ; ne being the electron density, and σe the Thomson cross-section. For even hotter objects (e.g., hot accretion disks, T ≈ 105 K and higher), the dominant scattering process is the Compton scattering. For cooler stars (A, F, G), the Rayleigh scattering on atomic hydrogen (and helium) becomes important; for even cooler stars Rayleigh scattering on the hydrogen molecule H2 and other molecules is important, and for coolest stars and substellar mass objects (brown dwarfs and giant planets), scattering on cloud (dust) particles becomes dominant. The scattering part of the emission coeﬃcient may be written as

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I. Hubeny

η sc (ν) =

dν

dΩ I(ν , n )Rα (ν , n , ν, n) ,

(8)

α

where R is the redistribution function, which has the meaning of the probability density that a photon is absorbed at frequency ν and direction n (within appropriate unit intervals around these values), and re-emitted at ν and n. The scattering part of the emission coeﬃcient is the culprit behind most of the numerical complexity of the transfer problem. It makes the problem pathologically implicit because in order to solve the transfer equation we need to know the emission eﬃcient, but it (or at least its scattering part) is a function of the speciﬁc intensity; moreover it couples many frequencies and directions. However, although η sc is the most troubling part of the problem, we shall show that all the other coeﬃcients do also generally depend on radiation ﬁeld, and usually in a complex non-local and non-linear way. From the point of view of microscopic physics, one can write down the absorption and emission coeﬃcients as follows (we only indicate the dependence on frequency, and omit an indication of their dependence on position and time) ni − n∗i e−hν/kT σiκ (ν) [ni − (gi /gj )nj ] σij (ν) + κth (ν) = i

j>i

+

i

ne nκ σκκ (ν, T ) 1 − e−hν/kT , (9)

κ

where the three terms represent, respectively, the contributions of boundbound transitions (i.e. spectral lines), bound-free transitions (continua), and free-free absorption (also called bremsstrahlung). Here, ni is the occupation number (population) of an atom in the energy level labeled i, gi the corresponding statistical weight, and n∗i denotes an equilibrium population of level i corresponding to temperature T , and density ρ. σ(ν) are the corresponding cross-sections; subscript κ denotes the “continuum,” and nκ the ion number density. The relation between the bound-bound cross section σij (ν) and the well-known Einstein coeﬃcients for the for the photo-excitation is σij (ν) = (hν0 /4π)Bij φ(ν); φ(ν) is the so-called absorption proﬁle coeﬃcient, normalized to unity, φ(ν) dν = 1. It represents the conditional probability density that if a photon is absorbed in the transition i → j, it is absorbed in the frequency range (ν, ν + dν). Analogously, the thermal emission coeﬃcient is given by ⎡ 3 2hν nj (gi /gj )σij (ν) + n∗i σiκ (ν)e−hν/kT η th (ν) = 2 ⎣ c i j>i i −hν/kT + ne nκ σκκ (ν, T )e . (10) κ

Radiative Transfer in Astrophysical Applications

19

The three terms again describe the bound-bound, bound-free, and free-free emission processes, respectively. The absorption and emission coeﬃcients are thus described through the corresponding cross sections—given by the atomic physics, and possibly also by the local thermodynamic parameters, temperature and density; and the atomic level populations for all the levels involved in the microscopic processes that give rise to an absorption and emission at frequency ν. Their number may be enormous. Here is the essential complication. One can either assume the Local Thermodynamic Equilibrium (LTE), in which case the level populations are given through the Saha-Boltzmann formula, i.e. ni = n∗i (T, ρ), so that the thermal absorption and emission coeﬃcient are functions of only temperature and density. However, in many cases of interest, the LTE approximation breaks down, and one has to determine the atomic level populations by a set of statistical equilibrium (rate) equations. For a given level i, the rate equation reads (for simplicity, we consider here the time-independent form; the general form would contain additional terms (1/c)(∂ni /∂t) + ∇ · (vni )). ni (Rij + Cij ) = nj (Rji + Cji ) , (11) j=i

j=i

where Rij is the rate of radiative transition from level i to level j, Cij is the analogous collisional rate. Equation (11) thus expresses the principle of statistical equilibrium, namely that the total number of transitions out of level i is equal to the total number of transitions into level i (we note that if this is satisﬁed for all microscopic processes i ↔ j we recover LTE). The essential complication arises due to the fact that while the collisional rates are functions of temperature and perturber density only, the radiative rates are given by 4π σij (ν) I(ν, n) , (12) Rij = dν dΩ hν (and analogously for the downward rate); that is, the level populations that are needed for evaluating even the thermal absorption and emission coeﬃcients are determined through the (yet unknown) radiation ﬁeld, moreover in essentially all frequencies and directions. An approach where at least some important level populations are determined by a self-consistent solution of the statistical equilibrium equation and the radiative transfer equation is called in astrophysics the non-LTE (or NLTE) approach, and its elaboration was in fact the main topic of the ﬁeld of stellar atmospheres theory in the last three decades. Finally, I stress that even in the case of LTE, one still has to deal with with complex non-local and non-linear coupling of radiation ﬁeld and the state parameters. This is because the local temperature is determined through some form of energy balance (radiative equilibrium for hot star atmospheres; radiative + convective equilibrium in cooler stars; plus additional dissipation of mechanical energy in case of stellar chromospheres or accretion disks, etc.).

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In the simplest case of pure radiative equilibrium, which reads ∞ [χ(ν, T )I(ν, n) − η(ν, n, T )] dν dΩ = 0 ,

(13)

0

we see that the temperature structure depends on the radiation ﬁeld in a complex way.

4 Hierarchies of Approximations There is a wide range of diﬀerent radiative transfer problems considered in astrophysics, depending on a degree of sophistication in treating the interaction of radiation and matter. I will brieﬂy summarize the basic types of problems below, although the list is by no means exhaustive. 4.1 Treatment of Absorption and Emission • Formal solution This is the simplest case, in which the emission and absorption coeﬃcients (or the optical depth and the source function) are fully known. In this case the transfer equation is a linear diﬀerential equation, and its solution is easy. The only concern is to ﬁnd a suﬃciently fast and eﬃcient numerical procedure. The topic was recently reviewed for instance in [5] and [6]. The basic use of the formal solution is as an intermediate step of a global iterative method, as we shall discuss later on. • Monochromatic(coherent scattering) problems In this case, χ and S may depend on frequency ν, but there is no coupling to other frequencies ν . In other words, all the frequencies are separate, one one can solve for one frequency at a time. There is a subdivision according to various possibilities of treating the angular coupling: – anisotropic scattering Here, the source function contains a term proportional to sc S (n) ∝ σ(n) p(n , n) I(n ) dΩ , where p is the phase function. – isotropic scattering, where p = 1, and thus the scattering part of the source function is independent of direction and is given by S sc (n) = S sc ∝ I(n ) dΩ ≡ J , where J is the zero-order moment of the speciﬁc intensity, called in astrophysics the mean intensity of radiation (it is called the scalar ﬂux, φ, in the neutron transport theory).

Radiative Transfer in Astrophysical Applications

• Non-coherent scattering problems In this case, the scattering source function contains the term S sc (ν, n) ∝ R(ν , n , ν, n)I(ν , n ) dν dΩ ,

21

(14)

where R is the redistribution function. Typical examples of the situations where a complex frequency and angular redistribution of radiation is important are for instance Compton scattering in high energy astrophysics, resonance scattering in spectral lines, or Mie scattering on cloud particles for coldest stars and giant planets. Simpliﬁed cases include: – angular and frequency redistribution are separate, ¯ , ν) p(n , n) , R(ν , n , ν, n) = R(ν which is a very good approximation in most cases, and which simpliﬁes a numerical solution considerably. – complete redistribution, in which cases the scattering is viewed as two uncorrelated process of absorption and subsequent re-emission, with the same probability densities, ¯ , ν) = φ(ν ) φ(ν) , R(ν where φ(ν) is the absorption proﬁle coeﬃcient. This is usually a good approximation for scattering in spectral lines except strong resonance lines when one has to deal with more general redistribution function. In astrophysical terminology, the latter situation is called partial redistribution, in contrast to the complete redistribution. – coherent scattering, which we mentioned earlier. corresponds to ¯ , ν) = φ(ν ) δ(ν − ν) , R(ν where δ is the Dirac δ-function. 4.2 Treatment of Spatial Dimensions • 1-D plane-parallel geometry In this case one assumes that the medium is composed of plane-parallel, horizontally-homogeneous layers. The problem is thus 1-dimensional, in Cartesian geometry. This approach is usually used for stellar atmospheres. Although it may seem very crude, it actually provides an excellent approximation for atmospheres whose thickness is much smaller that the stellar radius, which is the case for most stellar types except giants and supergiants with extended atmospheres. Also, such an approach is warranted because some other essential pieces of physics, such as departures from LTE, and/or a radiative/convective balance, are important, which

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I. Hubeny

leads to a necessity of treating a large number of frequencies and possibly directions. This approach can be used even in the case of surface inhomogeneities, as long as the radial properties vary much faster than the horizontal properties (which, again, is usually the case). 1-D plane-parallel approach is also often being used for geometrically thin accretion disks. There are several codes for this type of problems; the most widely used are a universal stellar atmosphere and accretion disk program TLUSTY [7–9], and the package PRO2 [10, 11]. 1-D spherical geometry It is used for extended atmospheres, stellar winds, supernovae, H II regions, and others. Although it is still a one-dimensional problem, it is numerically much more demanding than the previous 1-D plane-parallel problem because one has to take into account many more angles (directions), because the radiation intensity is becoming more and more forward-peaked when going outward. The most sophisticated codes of this sort are CMFGEN [12], and PHOENIX [13]. 2-D spherical geometry It is used for rapidly rotating stars, core-collapse supernovae, etc. 2-D cylindrical geometry It is mostly used for accretion disks and core-collapse supernova simulations. A well-described code of this sort is ALTAIR [14, 15]. 3-D Cartesian geometry So far it has been used for snapshots of 3-D hydrodynamic simulations “in the box” for solar convection, modeling viscous dissipation in accretion disks, mass transfer in close binary systems, simulating large-scale cosmological structures, and others. Obviously, the problem is computationally extremely demanding, so the 3-D transfer simulation use a number of various approximations (e.g., [16]).

4.3 Treatment of frequency Ideally, one has to treat as many frequency points as to faithfully reproduce the frequency dependence of the absorption (and emission) coeﬃcient. While the bound-free and free-free processes have relatively smooth cross-sections (although, strictly speaking, the photoionization cross-section do sometimes contain a number of resonances, so they do not be smooth after all) and thus can be reproduced by a relatively small number of discretized frequency points, the bound-bound processes (spectral lines) have very sharply peaked cross-sections. Moreover, a typical stellar spectrum contains literally milions of spectral lines, which should all be consedred explicitly in order to construct an accurate model atmosphere (essentially because the rate equations and the energy balance equation contain integrals of the radiation intensity over the whole frequency range). Model stellar atmospheres (and other astronomical radiation-dominated objects) either ignore the eﬀects of the majority of spectral lines and treat only the most important ones; or do treat, with a varying

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23

degree of approximation, essentially “all” spectral lines. Model atmospheres of the latter kind are called line-blanketed models. I list below several options for treating the frequency dependence of relavant quantities, ordered by increasing accuracy, and by the number of frequency points required. • Gray problem In this case, there is only one frequency; better speaking, one either assumes that the absorption and emission coeﬃcients are frequencyindependent, or one works in terms of frequency-integrated quantities. In the context of stellar atmospheres, such an approach is being used as an interesting pedagogical tool for understanding the temperature structure, and for providing a starting solution for an iteration scheme to solve a general stellar atmosphere problem. • Multi-group/multi-gray is a generalization of the gray model. One introduces several, O(1) – O(10), frequency bins (groups), and averages all the relevant frequencydependent quantities over the individual bins/groups. Again, this approach is no longer being used in the context of stellar atmosphere models; although [17] presented a generalization this approach for NLTE model atmospheres with a treatment of many spectral lines; however, the number of bins was signiﬁcantly higher – O(103 ) to O(104 ). The multi-group approach is however still being used in radiation hydrodynamic studies (or any problem which is already so computationally heavy that one cannot aﬀord to deal with many frequencies), and, in particular, for treating neutrino transport in the core-collapse supernova simulations. In the latter case, the relevant cross-sections are relatively smooth functions of frequency/energy, so that such an approach is a very good approximation anyway. • Opacity Distribution Functions One introduces a number of frequency intervals [typically O(100) - O(103 )]; instead of simply averaging over the intervals, one resamples the opacity in the individual intervals into monotonic functions of frequency, which can then be treated by means of a small number of frequency points [typically O(10) per interval]. One can thus represent the whole frequency space in roughly O(104 ) frequency points. • Opacity Sampling This is essentially a Monte-Carlo treatment of frequency dependence. One selects a set of random frequency points that statistically reproduce a complex frequency behavior of the absorption and emission coeﬃcient. Obviously, the larger the number of frequency points, the higher the accuracy of the solution. In typical (stellar atmosphere) applications, one thus deal has to deal with O(104 ) to few times O(105 ) frequency points. • Full frequency resolution Such an item would be missing in a similar review only a few years ago because it would have been viewed as too computationally demanding.

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However, such approaches are possible now, at least in 1-D. Recent grids of NLTE line-blanketed model stellar atmospheres, such as the grid of O star models [18] that uses Opacity Sampling with a resolution of 0.75 ﬁducial Doppler widths, can be viewed as more or less “full” frequency resolution. This approach needs 2–3 ×105 frequency points; cooler star models would need up to O(106 ) frequency points, but this is becoming possible with current computers. 4.4 Treatment of angular dimensions Brieﬂy, there are two types of approaches. One can either treat an angular dependence of speciﬁc intensity explicitly, or to use either variable Eddington factors (or Eddington tensor in more than 1 spatial dimension), or some other closure relations, and to solve the transfer equation only for the moments of speciﬁc intensity, such as the mean intensity J. The latter approach is very advantageous in the case of isotropic scattering. For the problems of general anisotropic scattering, one usually adopts the former approach and solves the transfer problem for the speciﬁc intensities. In 1-D geometries one typically considers one angular variable (the polar angle measured with respect to the normal to the surface), and assumes azimuthal symmetry of the radiation ﬁeld. In that case, one works in terms of azimuthally-averaged speciﬁc intensity. In 1-D plane-parallel geometry, it is usually suﬃcient to consider a very small number of polar angles (3 is a typical value in stellar atmosphere models). However, model atmospheres with strong irradiation from outside (for instance for very close-in extrasolar giant planets), and moreover with strongly anisotropic scattering phase function, require at least several hundred angles (e.g., [19]). 1-D spherical models do typically require more angles that 1-D planeparallel models, usually there are roughly as many polar angles as the number of radial zones. In multi-D transfer problems the number of angles depend on actual situation; some workers in radiation hydrodynamic models for solar atmosphere use only O(10) angles (see, e.g. [16]); but most transfer studies in multi-D consider O(100) to O(103 ) angles.

5 General Problem 5.1 State Vector The state of a radiation-dominated medium, as discussed above, is fully described by the following general state vector ψ = {I(t, r, ν, n), T (t, r), ρ(t, r), v(t, r), [ni (t, r), i = 1, N L]} ,

(15)

where the state parameters are, respectively, the speciﬁc intensity of radiation, temperature, density, velocity vector, and the atomic (possibly also

Radiative Transfer in Astrophysical Applications

25

molecular) level populations (occupation numbers); here N L is the total number of energy levels for which the statistical equilibrium equation is solved, and for which one thus allows departures from LTE. In some simpliﬁed situations, certain components may be missing. For instance, for static models, v ≡ 0, so that the velocity is skipped from the state vector. Similarly, in LTE, the level populations are known functions of temperature and density, and thus are skipped from the state vector. In we consider the so-called restricted NLTE problem, that is assuming that the temperature, density (and velocity in the case of dynamic models) are known, one is left only with radiation intensity and level populations to be solved for. Finally, if we assume LTE for level populations, and assuming ﬁxed temperature, density, and velocity, the only unknown quantity is the radiation intensity. This is actually the case in many problems in the neutron transport theory and the Earth atmosphere studies. Let us ﬁrst summarize typical numbers of the components of the state vector, in order to appreciate the numerical complexity of the problem. • t – many time steps, depending on the problem. However, the vast majority of astrophysical radiation transport problem is time-independent; sp.dim. ) grid points where N sp.dim. is the number of • r – typically O(100N spatial dimensions; • n – O(100 ) to O(103 ) grid points, depending on problem (see above); • ν – O(100 ) to O(106 ) points, depending on problem (see above); • N L – O(100 ) to O(104 ) levels, depending on problem. Let us now give some examples of actual numbers used in the current state of the art numerical simulations in astrophysics. We will only count the number of individual components of discretized speciﬁc intensity; the number of other components of the state vector is negligible. The total number is a product of number of spatial positions, number of angles, and number of frequencies. In the subsequent examples, we will follow this order. • 1-D static plane-parallel NLTE line-blanketed model stellar atmospheres 102 × few × (few × 105 ) ≈ 108 elements; • 1-D close-in extrasolar giant planets (with anisotropic scattering) 102 × 103 × 103 ) ≈ 108 elements; • 2-D core-collapse supernovae (neutrino transport) (few × 104 ) × 103 × (few × 101 ) ≈ 108 to 109 elements, but per time step; there are O(104 ) time steps. These are already existing simulations. In the (hopefully near) future, one hopes to achieve for instance the following: • 3-D core-collapse supernovae (neutrino transport) (few × 106 ) × 103 × (few × 101 ) ≈ 1010 to 1011 but per time step; ≈ 1015 altogether;

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• 3-D static plane-parallel NLTE line-blanketed model stellar atmospheres 106 × 103 × 105 ≈ 1014 elements. 5.2 Methods of Solution There are several basic possibilities, which I outline below. Again, the list is far from being complete, and is meant for illustration only. The approaches are categorized by means of a degree of sophistication to which the interaction between radiation and matter is treated numerically. • not solving, really This is obviously the simplest, and therefore most popular, approach to the problem. There are several possibilities, including: – optically thin approximation—χ is essentially zero; consequently no interaction of radiation with matter. – diﬀusion approximation—an almost opposite approximation, valid for large optical depths. In this case, one can express the moments of speciﬁc intensity through the gradient of temperature; radiation intensity is thus eliminated from the state vector, but the results are approximate, the more so in the optically thin regime. – ﬂux-limited diﬀusion—an improvement of the diﬀusion approximation, but still inaccurate at small optical depths; – escape probability methods—essentially assume that a photon after its emission is either re-absorbed on the spot, or escape freely from the medium; the branching ratio between these two is a function of optical distance form the nearest boundary. Again, the radiation intensity is eﬀectively eliminated from the state vector. • Monte-Carlo methods These are powerful methods, in particular in multi-D geometries, but there are problems when treating NLTE problem (but see some contributions in this Volume for eﬃcient generalizations of Monte-Carlo methods to NLTE situations). • “Exact” numerical solutions. This is the most important class of solutions, so the next Section will be fully devoted to it.

6 Exact Numerical Solution In astrophysical applications, there are essentially two types of problems: i) solution of a full set of structural equations, generally with a large number of frequency points and a large number of atomic level populations to be solved for—the problem is thus highly non-linear; and ii) solution of a (generally anisotropic) scattering problem with a known structure (temperature, density, and velocity are speciﬁed)—a linear problem.

Radiative Transfer in Astrophysical Applications

27

The typical example of the ﬁrst class of problems is a construction of NLTE (line-blanketed) model stellar atmospheres, where, because of the overall complexity, one is so far restricted to a 1-D geometry. The second class of problems is usually encountered in radiation hydrodynamics problems (stellar wind, accretion disks, supernovae), where one splits the hydrodynamics and radiation (neutrino) transport. We shall consider these two classes in turn. 6.1 Solution of Large Non-Linear Systems A general problem is formulated through a set of non-linear, integro-diﬀerential equations. As is customary, we discretize in space, angle, and frequency, and end up with a set of non-linear algebraic equations. The most straightforward method of solving such a system is a linearization, using the Newton-Raphson method. We write the set of all structural equations formally as F [ψ] = 0

(16)

We start with an educated guess of solution, ψ (0) . (In the case of stellar atmospheres the starting solution is provided either by the LTE-gray model, or by a previously generated NLTE model with slightly diﬀerent basic stellar parameters). The iteration proceeds as ψ (n+1) = ψ (n) − J−1 · F [ψ (n) ] ,

(17)

where J is the Jacobi matrix of the system, Jij = ∂Fi /∂ψj . The method was ﬁrst used to construct NLTE model stellar atmospheres in [20], who used the term “complete linearization”. The number of frequency points was limited to about 100, and the number of energy levels allowed to depart from LTE to about 10. The number of depth points was about 100, so the total number of unknowns was still pretty large, of the order to 104 . However, since the transfer equation couples only three neighboring depths, the overall system was of the block-triadigonal form, so the solution of (17) was reduced to ≈ 100 inversions of 100 × 100 matrices, which made the problem tractable even in the late 1960’s. However, the Jacobian for the modern problems mentioned above would be a matrix 108 × 108 or larger, whose direct inversion is completely out of question. How we deal with such a situation? There are several possibilities; here I will discuss two which are so far used most. The ﬁrst possibility, which is particularly useful for a large number of frequencies and for highly non-linear systems—that is, in the case of NLTE line-blanketed model stellar atmospheres, is to use the idea of the so-called Accelerated Lambda Iteration (ALI) method (for a recent review, see e.g. [6]). The original scheme was developed for simpler, essentially linear problems. To illustrate the basic scheme, let us take a simple case of coherent (monochromatic) isotropic scattering. The source function is

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I. Hubeny

S = (1 − )J + B ,

(18)

where = κth /χ is a probability that an absorbed photon is destroyed before it can be re-emitted; 1 − is the single-scattering albedo. B is the Planck function. We note that an analogous equation holds for the basic NLTE problem, namely transfer of radiation in a gas of two-level atoms. Here we ¯ ¯ only ∞ replace J by J, frequency-averaged mean intensity, that is given by J = J(ν)φ(ν)dν. We also stress that in typical astronomical line-formation 0 problems, is very small, ≈ 10−4 down to 10−6 , which makes the problem numerically more diﬃcult than typical problems in other ﬁelds where ≈ 10−2 or even larger. The formal solution of the transfer equation can be written as J = Λ[S] ,

(19)

where Λ is a linear operator (or matrix, upon discretization). Equation (18) can thus be written as an equation for the source function S, namely

which has a solution

S = (1 − )Λ[S] + B ,

(20)

S = [I − (1 − )Λ]−1 (B) ,

(21)

where I is a unit matrix. Instead of solving the problem directly, one can use an iteration scheme; essentially using a suitable approximate Λ-operator, Λ∗ . Equation (19) is written as J (n+1) = Λ∗ [S (n+1) ] + (Λ − Λ∗ )[S (n) ,

(22)

which leads to the iterative scheme [I − (1 − )Λ∗ ]S (n+1) = (1 − )(Λ − Λ∗ )S (n) + B .

(23)

The operator (matrix) Λ∗ is nothing else than a suitable preconditioner (see below). It can be shown [21] that the diagonal part of exact Λ-operator is an excellent choice. The resulting method is thus equivalent to the Jacobi method. Other possibilities for Λ∗ are the tridiagonal part of exact Λ (or a higher multiband part), or a lower triangular part of Λ; the latter method is essentially the Gauss-Seidel method. One can also use the Successive Overrelaxation (SOR) method, in which the diagonal part is multiplied by 1/ω, with 1 < ω < 2 being a suitable over-relaxation parameter [22]. This method provides a full solution in the case of simple problems (pure radiative transfer without considering other structural equations). In the case of the full problem involving all structural equations, we use the relation J new = Λ∗ S new + (Λ − Λ∗ )S old ,

(24)

with a diagonal or tridiagonal Λ∗ operator, and linearize it. The “old” source function is known, and thus it does not contribute to the linearized equations.

Radiative Transfer in Astrophysical Applications

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The “new” source function is a function of temperature, density, and level populations. When linearizing, (24) eﬀectively eliminates radiation intensities from the state vector. Actually, it turned out that it is extremely advantageous to keep the mean intensity in a few selected frequency points [centers of the strongest lines; a few points just shortward (in the wavelength space) of the photoionizations edges of the most important continua] to be linearized, while all the rest of frequency points are treated using (24). Such a scheme was developed in [9] by the name “hybrid complete linearization/accelerated Lambda iteration (CL/ALI) method”. The method has proved to be very successful for computing model stellar atmospheres and accretion disks ranging from hot disks around solar-mass black holes (T ≈ 109 K down to giant planet atmospheres with T ≈ 100 K. The second possibility is to solve the system (17) by applying the powerful idea of preconditioning. To solve iteratively a general linear system Ay = b ,

(25)

the simplest possibility is a simple relaxation method, known in astrophysical radiative transfer as “Lambda Iteration”, or as a “Source Iteration” method in the neutron transport theory. In this case one solves the system (25) as y(k+1) = y(k) + b − Ay(k) ≡ y(k) + r(k) ,

(26)

where r(k) = b − Ay (k) is the residuum vector in the k-th iteration. As is well known, the method converges extremely slowly when the largest eigenvalue of the matrix I − A is close to unity, which is usually the case in astrophysical applications. The preconditioning method consists in taking another matrix, P , which is close enough to A, and P −1 A is close to the identity matrix, so that the eigenvalues of I − P −1 A are small. The iteration process (26) is modiﬁed to y(k+1) = y(k) + P −1 r(k) ,

(27)

Obviously, the method is advantageous if the matrix P is easy and cheap to invert. A variant of this approach is the so-called double-splitting (or generally a multiple-step splitting), where one apples consecutively two or more diﬀerent preconditioners, viz., y(k+1/2) = y(k) + P1−1 r(k) ,

(28)

y(k+1) = y(k+1/2) + P2−1 r(k+1/2) .

(29)

A choice of suitable preconditioner(s) depend on the problem.

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6.2 Solution of Large Linear Systems Let us consider here a linear scattering problem, but a time-dependent one, with generally anisotropic scattering, and in several spatial dimensions, 1 ∂I(n) ˆ + D(n)I(n) = −χ I(n) + S(n) , c ∂t

(30)

ˆ where the operator D(n) is a general “spatial transport” operator, S(n) is the source term, in which we split the thermal and scattering part, S = S th +S sc . The implicit solution (with backward time diﬀerencing) is written as

c ˆ I(n) = ζ S(n) + I0 (n) , 1 + ζI D(n) (31) ∆t where ζ is given by 1 + χ. (32) c∆t where I0 (n) is the intensity at the previous time step. Although the transfer (31) is linear in the speciﬁc intensity, the fact that the source term couples all directions would mean that one needs to solve a linear system of N A × N S unknowns; N A being the total number of angles, and N S the total number of spatial positions. Since N S is typically of the order of O(104 ) to O(106 ), and N A is a few times 102 , we have to deal with a system for several millions of unknowns, which necessitates using iterative methods. There are again several possibilities. One possibility of solving the transfer equation is to adopt a two-step iteration procedure based on an application of the Accelerated Lambda Operator (ALI) technique, similarly as was recently used for 2-D radiative transfer with anisotropic scattering in the context of strongly irradiated atmospheres of extrasolar giant planets (e.g., [19]). We write the solution of the transfer equations ζ −1 =

I(n) = Λ(n)S (n) , where the total source term S is given by

c S (n) = ζ S(n) + I0 (n) , ∆t

(33)

(34)

and the “formal solution” operator, denoted in keeping with the usual astrophysical notation as Λ, is given by −1

ˆ . Λ(n) = 1 + ζI D(n) The method consists in assuming that the source term is taken as

(35)

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31

S (n) = α(n) S¯ ,

(36)

where S¯ is an angle-averaged source term, S¯ = S (n) dΩ .

(37)

The iterative method consists of two nested iteration loops. In the inner iteration loop we assume that the “anisotropy factor” α is ﬁxed and we solve for a speciﬁc intensity self-consistently with the averaged source term, while in the outer loop we update the anisotropy factor α. The inner loop is solved using the ALI technique, as outlined above. Another possibility is to use the idea of preconditioning. There are several possibilities to choose a suitable preconditioner for solving the system (31). Let us ﬁrst rewrite (31) in a form analogous to (25), viz.

c (38) I0 , [1 + ζI D − ζI S sc ] I = ζI S th + ∆t where I is the vector composed of all the speciﬁc intensities for all directions and positions; D is the position operator that is composed of D(n) for all directions. The structure of the operators D and S is relatively simple: The operator D couples positions, but does not couple directons. That is, D is a matrix which is block-diagonal in angular space, with each block being a sparse matrix in the physical space (because of the nearest-neighbor coupling due to diﬀerentials with respect position). The operator S sc is in contrast diagonal in the physical space (no depth coupling), but the individual block matrices in the angular space are full (all directions are coupled because of the scattering). The global operator (matrix) of the system is thus A = 1 + ζI D − ζI S sc , and the right-hand side (source term) is

c b = ζI S th + I0 . ∆t

(39)

(40)

An attractive possibility is to use for P the operator which is analogous to the original operator, A, namely P = 1 + ζI D − ζI S˜sc ,

(41)

where S˜sc is the scattering part of the source term that is obtained using a lower-order quadrature over angles than that used in the case of the original S sc . For instance, one may use even an S2 quadrature for P , while using an Sn quadrature with a suﬃciently large n for evaluating A. An analogous method is called “transport synthetic acceleration” in the neutron transport theory.

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7 Conclusions Many numerical techniques that were developed in the ﬁeld of astrophysical radiation transport, were also independently developed in other ﬁelds, above all in the neutron transport theory, atmospheric physics, and even in the ﬁeld of plant canopies, as this meeting amply demonstrated. In astrophysics, a large emphasis is given to the problems that solve radiation transport simultaneously with other structural equations, because in many cases the radiation signiﬁcantly inﬂuences the overall structure of a distant object. Thanks to the progress in the fast and eﬃcient numerical methods, and a large increase in the computer power, one is now able to compute models of astronomical objects, such as stellar atmospheres and accretion disks, in an unprecedented degree of completeness and accuracy. However, despite of the recent progress, there is still much space for improvement, in particular by developing more sophisticated iterative methods for treating radiation transport. In this respect, a closer interaction between workers in various scientiﬁc ﬁelds that use transport methods is extremely useful. The present meeting was a ﬁrst signiﬁcant step toward such a goal. I would like to thank the local organizers, in particular to Frank Graziani, for organizing this truly wonderful and enornmously useful workshop.

References [1] [2] [3] [4] [5]

[6]

[7] [8]

[9]

Mihalas, D.: Stellar Atmospheres. Freeman, San Francisco (1978) Mihalas, D., Weibel–Mihalas, B: Foundations of Radiation Hydrodynamics. Oxford University Press, Oxford (1984) Castor, J.: Radiation Hydrodynamics. Cambridge University Press, Cambridge (2005). Hubeny, I., Mihalas, D., Werner, K. (eds): Stellar Atmosphere Modeling. Astronomical Society of the Paciﬁc, San Francisco (2003) Auer, L.: Formal Solution: EXPLICIT Answers. In: Hubeny, I., Mihalas, D., Werner, K. (eds) Stellar Atmosphere Modeling. Astronomical Society of the Paciﬁc, San Francisco, 3–15 (2003) Hubeny, I.: Accelerated Lambda Iteration: An Overview. In: Hubeny, I., Mihalas, D., Werner, K. (eds) Stellar Atmosphere Modeling. Astronomical Society of the Paciﬁc, San Francisco, 17–30 (2003) Hubeny, I.: TLUSTY: A Computer Program for non–LTE Model Stellar Atmospheres. Computer Physics Comm., 52, 103 Hubeny, I., Lanz, T.: Accelerated complete–linearization method for calculating NLTE model stellar atmospheres. Astron. & Astrophys., 262, 501–514 (1992) Hubeny, I., Lanz, T.: Non-LTE line–blanketed model atmospheres of hot stars. 1: Hybrid complete linearization/accelerated lambda iteration method. Astrophys. J., 439, 875–904 (1995)

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[10] Dreizler, S., Werner, K.: Line blanketing by iron group elements in Non–LTE model atmospheres for hot stars. Astron. & Astrophys., 278, 199–208 (1993) [11] Werner, K., Deetjen, J. L., Dreizler, S., Nagel, T., Rauch, T., Schuh, S. L.: Model Photospheres with Accelerated Lambda Iteration. In: Hubeny, I., Mihalas, D., Werner, K. (eds) Stellar Atmosphere Modeling. Astronomical Society of the Paciﬁc, San Francisco, 31–50 (2003) [12] Hillier, D. J., Miller. D. L.: The Treatment of Non–LTE Line Blanketing in Spherically Expanding Outﬂows. Astrophys. J., 496, 407–427 (1998) [13] Hauschildt, P. H., Lowenthal, D. K., Baron, E.: Parallel Implementation of the PHOENIX Generalized Stellar Atmosphere Program. III. A Parallel Algorithm for Direct Opacity Sampling. Astrophys. J. Suppl., 134, 323–329 (2001) [14] Castor, J. I., Dykema, P. G., Klein, R. I.: A new scheme for multidimensional line transfer. II ETLA method in one dimension with application to iron Kalpha lines. Astrophys. J., 387, 561–571 (1992) [15] Dykema, P. G., Klein, R. I., Castor, J. I.: A New Scheme for Multidimensional Line Transfer. III. A Two-dimensional Lagrangian Variable Tensor Method with Discontinuous Finite-Element SN Transport. Astrophys. J., 457, 892– 921 (1996) [16] Stein, R., Nordlund, ˚ A.: Radiative Transfer in 3D Numerical Simulations. In: Hubeny, I., Mihalas, D., Werner, K. (eds) Stellar Atmosphere Modeling. Astronomical Society of the Paciﬁc, San Francisco, 519–532 (2003) [17] Anderson, L.S.: Line blanketing without local thermodynamic equilibrium. I – A hydrostatic stellar atmosphere with hydrogen, helium, and carbon lines. Astrophys. J., 298, 848–858 (1985) [18] Lanz, T., Hubeny, I.: A Grid of Non-LTE Line–blanketed Model Atmospheres of O–Type Stars. Astrophys. J. Suppl., 146, 417–441 (2003) [19] Sudarsky, D., Burrows, A., Hubeny, I., Li, A.: Phase Functions and Light Curves of Wide Separation Extrasolar Giant Planets. Astrophys. J. (in press), astro–ph/0501109 [20] Auer, L.H., Mihalas, D.: Non–LTE Model Atmospheres. III. A Complete– Linearization Method. Astrophys. J., 158, 641–655 (1969) [21] Olson, G., Auer, L. H., Buchler, R.: Rapidly Convergent Iterative Solution of the non–LTE Line Radiation Transfer Problem. J. Quant. Spectrosc. Radiat. Transfer, 35, 431–442 (1986) [22] Trujillo–Bueno, J., Fabiani–Bendicho, P.: A novel iterative scheme for the very fast and accurate solution of non–LTE radiative transfer problems. Astrophys. J., 455, 646–657, (1995)

Neutrino Transport in Core Collapse Supernovae Anthony Mezzacappa1 , Matthias Liebend¨ orfer2 , Christian Y. Cardall1,3 , O.E. Bronson Messer4 , and Stephen W. Bruenn5 1 2 3

4

5

Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6354 CITA, University of Toronto, Toronto, ON, M5S 3H8, CA Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996 National Center for Computational Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6354 Department of Physics, Florida Atlantic University, Boac Raton, FL 33431

1 The Core Collapse Supernova Paradigm Stars more massive than ∼10 M evolve to an onion-like conﬁguration (Fig. 1), with an iron core surrounded by successive layers of silicon, oxygen, carbon, helium, and ﬁnally hydrogen. In addition to iron group nuclei, the core is composed of electrons, positrons, photons, and a small fraction of protons and neutrons. The pressure in the core, which supports it against the inward pull of gravity, is dominated at this stage by the electrons, and the balance between the electron pressure and gravity is only marginally stable. As a result of electron capture on the free protons and nuclei in the core and as a result of nuclear dissociation under the extreme densities and temperatures, electron and thermal pressure support are reduced, and the core becomes unstable and collapses. The velocity of infalling matter in the core increases linearly with radius, characteristic of the “homologous” collapse expected of a ﬂuid whose pressure is dominated by relativistic, degenerate electron pressure. The sound speed, on the other hand, decreases with density and, therefore, radius. Thus, with increasing radius, the infall velocity eventually exceeds the local sound speed; i.e., the infall becomes supersonic. Consequently, during infall the core splits into an inner homologously and subsonically infalling core and an outer supersonically infalling outer core (stage 1 of Fig. 2). Beginning with central densities ∼109−10 g/cm3 , the collapse proceeds through nuclear matter densities (∼1 − 3 × 1014 g/cm3 ). The inner core undergoes a phase transition from a two-phase system of nucleons and nuclei to a one-phase system of bulk nuclear matter. One may view the inner core at this point as one giant nucleus. The pressure in the inner core increases dramatically as the result of Fermi eﬀects and the repulsive nature of the nucleon–nucleon interaction potential at short distances. As a result of this dramatic increase in pressure, the inner core becomes incompressible and rebounds (stages 3–5 of Fig. 2).

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Orbit of Earth around Sun

H, He

H, He

1012cm

He 3x1013cm

He 1010cm

He 1011cm

C, O

O, Ne, Mg

Earth

O, Ne, Mg

2.2x109cm

“Si” “Fe”

Fig. 1. The structure of a core collapse supernova progenitor at the onset of stellar core collapse. The size of the iron core and star are compared with the size of the Earth and its orbit around the Sun, respectively

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2

1

neutrinos

3

6

shock 5

4

7

Fig. 2. The stages of stellar core collapse, maximum compression, bounce, and shock formation. The core separates into an inner, subsonically collapsing and an outer, supersonically collapsing core. The inner core bounce launches a shock wave into the outer core. The shock wave will propagate through the outer iron core and layers above it, expelling much of the layers to produce the supernova

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Any information about the rebounding inner core would be conveyed to the outer core via pressure waves that propagate radially outward at the speed of sound. When these waves reach the point at which the infall is supersonic—i.e., the “sonic point”—they are swept in as fast as they attempt to propagate outward. The net result: No information about the rebounding inner core reaches the infalling outer core, which in turn sets up a density, pressure, and velocity discontinuity in the ﬂow—i.e., a shock wave. This shock wave will ultimately be responsible for propagating outward through the star, disrupting the star in a core collapse supernova. Schematically, the shock wave is launched and energized by the rebounding inner core piston. (In Fig. 2, the shock wave is represented schematically by the orange circle in stages 5–7.) If the shock were to propagate outward without stalling, we would have what has been called a “prompt” explosion. All of the realistic models completed to date suggest that this does not occur. Because the shock loses energy in dissociating the iron nuclei that pass through it as it propagates outward, the shock is enervated. Nuclei exist in the regions shaded with a light blue in Fig. 2. The yellow regions in stages 5–7 are regions of shock- compressed and heated material in which the nuclei are dissociated into nucleons. The shock loses additional energy in the form of electron neutrinos. The copious production of electron neutrinos occurs when the core electrons capture on the newly dissociation-liberated protons. These neutrinos are initially trapped but escape when the shock moves out beyond the electron neutrinosphere. This gives rise to the “electron neutrino burst” in a core collapse supernova, which is the ﬁrst of three major phases of the three-ﬂavor neutrino emission during these events. As a result of these two enervating mechanisms, the shock stalls in the iron core. How the shock is reenergized in a “delayed shock mechanism” is currently the central question in core collapse supernova theory. At the time the shock stalls, the core conﬁguration is composed of a central radiating object: the proto-neutron star (Fig. 3), which will go on to form a neutron star or black hole. The proto-neutron star has a relatively cold inner core, composed of unshocked bulk nuclear matter, together with a hot “mantle” of nuclear matter that has been shocked but not expelled. The ultimate source of energy in a core collapse supernova is the ∼1053 erg of gravitational binding energy associated with the formation of the neutron star. This gravitational binding energy is released after core bounce over ∼10 seconds in the form of a three-ﬂavor neutrino “pulse.” This marks the second phase of the neutrino emission from a core collapse supernova. Electron neutrinos are produced during stellar core collapse by electron capture on protons and nuclei, but after bounce, in the hot proto-neutron star mantle, all three ﬂavors of neutrinos and their antineutrinos are produced and are emitted as the mantle cools and contracts during its “Kelvin-Helmholtz” cooling phase. The neutrinos are emitted from their respective neutrinospheres. The neutrinospheres are deﬁned in a way similar to the way the photosphere

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39

ν-Luminosity Matter Flow

Shock Heating

_ν e + n → p + e-+ νe + p → n + e _ν e + n → p + e-+ νe + p → n + e

Gain Radius

Cooling

Proto-Neutron Star ν-Spheres

Fig. 3. During the shock reheating phase, the stellar core is composed of a central radiating proto-neutron star whose “surface” is deﬁned by the neutrinospheres (represented here by a single sphere) and a region above the neutrinosphere consisting of a net cooling region and a net heating region below the stalled shock, separated by the “gain” radius at which heating and cooling balance. Heating and cooling are mediated by electron neutrino and antineutrino absorption and emission

of the Sun is deﬁned. They are the surfaces of last scattering for each energy and ﬂavor. Equivalently, they are at radii at which the respective neutrino depths are 2/3. In Fig. 3, the neutrinospheres are represented by one surface. The neutrino luminosities during this phase are maintained at their average values ∼1052 erg/s by mass accretion onto the proto-neutron star (the kinetic energy of infall is converted into thermal energy when the material hits the proto-neutron star surface). After explosion is initiated, the accretion luminosity decreases dramatically, and the neutrino pulse enters its third and ﬁnal stage: the exponential decay of the neutrino luminosities characteristic of neutron star formation and cooling.

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The stalled supernova shock is thought to be revived, at least in part, by the charged-current absorption of electron neutrinos and antineutrinos that emerge from the proto-neutron star, a fraction of which are absorbed by protons and neutrons behind the shock. This is known as the “delayed shock” or “neutrino-heating” mechanism, originally proposed by Wilson and Bethe [1, 2]. While the total energy emitted in neutrinos is two orders of magnitude greater than what is required for the generation of an ∼1051 erg explosion, deciphering the precise role of this neutrino heating in the supernova mechanism is, as we will discuss, diﬃcult. Between the neutrinosphere and the shock, the material both heats and cools by electron neutrino and antineutrino emission and absorption. The neutrino heating and cooling have diﬀerent radial proﬁles; consequently, this region splits into a net cooling region and a net heating region, separated by a “gain” radius at which heating and cooling balance. (See Fig. 3.) We refer to the region between the gain radius and the shock as the “gain region.” The neutrino heating in the gain region can be written as 1 Xp Lν¯e 2 1 Xn Lνe 2 Eν¯e . (1) E + ¯a ˙ = a λ0 4πr2 νe F F¯ λ0 4πr2 The ﬁrst (second) term corresponds to the absorption of electron neutrinos (antineutrinos). It depends linearly on the neutrino luminosity and “inverse ﬂux factor,” which is a measure of the isotropy of the neutrino distribution, and quadratically on the neutrino spectrum. In addition to the dependence on the three key neutrino quantities in the heating rate above, the revival of the stalled supernova shock depends on a complex interplay of neutrino heating, mass accretion through the shock, and mass accretion through the gain radius [3]. It is mass accretion through the shock and gain radii that determines the amount of mass in the gain region, the former being a source of mass in the gain region and the latter being a sink. Moreover, the mass accretion through the gain radius serves to both sustain the neutrino luminosities and to undermine the pressure in the gain region, simultaneously—i.e., it serves a supporting and a detrimental role. All three quantities in the neutrino heating rate must be computed accurately [3–8], which requires that we solve the neutrino Boltzmann neutrino transport equations.

2 The O(v/c) Neutrino Transport Equation in Spherical Symmetry: An Illustrative Example To understand the problem at hand and the issues we face in developing a numerical method for neutrino transport in our application, it is for the moment suﬃcient to consider the O(v/c) Boltzmann equation in spherical

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symmetry [9–12]. We will consider the general case later. If we include emission, absorption, isoenergetic scattering of neutrinos by nucleons and nuclei, neutrino–electron scattering, and pair emission and absorption, the Boltzmann equation is ∂(r2 ρF ) 1 ∂F + 4πµ c ∂t ∂m 1 ∂[(1 − µ2 )F ] + r ∂µ 1 ∂lnρ 3v ∂[µ(1 − µ2 )F ] + + c ∂t r ∂µ 1 2 ∂lnρ 3v v 1 ∂(E 3 F ) + + µ − c ∂t r r E 2 ∂E j ˜ = − χF ρ 1 1 2 + E dµ RIS F c h3 c3 1 1 2 − E F dµ RIS c h3 c3 1 1 in ˜ NES −F dE E 2 dµ R + 3 4 F h c ρ 1 1 2 ˜ out −F − 3 4 F dE E dµ RNES h c ρ 1 1 1 2 ˜ em ¯ + 3 4 − −F dE E dµ RPAIR h c ρ ρ 1 abs ¯ ˜ PAIR − 3 4 F dE E 2 dµ R F . h c

(2)

The Boltzmann equation is solved for each neutrino ﬂavor independently. The electron neutrinos and antineutrinos are solved for simultaneously, as are the µ/τ neutrinos and antineutrinos, because each neutrino ﬂavor is coupled to its corresponding antineutrino through the pair emission and absorption channel. The mass derivative term on the left-hand side of the Boltzmann equation describes the propagation of neutrinos with respect to the Lagrangian mass coordinate, m. Outwardly propagating neutrinos have µ > 0, whereas inwardly propagating neutrinos have µ < 0. The ﬁrst µ-derivative term describes the rate of change of the neutrino propagation direction with respect to the outward radial direction as the neutrino propagates inward or outward in mass. The second µ-derivative term describes the aberration in the neutrino propagation direction measured by an observer who is instantaneously comoving with the ﬂuid. Because the ﬂuid is accelerating, two neighboring comoving observers will measure diﬀerent direction cosines. The energy-derivative term

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describes the shift in the neutrino energy measured by comoving observers. This is a Doppler shift resulting from the change in the velocity, with time and/or radius, of an accelerated ﬂuid; two neighboring comoving observers will measure diﬀerent frequencies. On the right-hand side of (2), the ﬁrst two terms describe the change in the neutrino distribution function resulting from the absorption and emission of neutrinos by nucleons and nuclei. The next two terms describe the isoenergetic inscattering and outscattering, respectively, of neutrinos by nucleons and nuclei. The fourth and ﬁfth terms describe non-isoenergetic neutrino– electron scattering, and the last two terms describe pair emission and absorption. F¯ is the corresponding antineutrino distribution function for each neutrino species evolved with (2). The neutrino interactions included in (2) are not complete, but as far as the functional form of the emissivities and opacities is concerned, they are. For example, scattering of neutrinos on nucleons is not isoenergetic when nucleon recoil and correlations are taken into account, but the functional form of the scattering kernel in this case would then simply take on the form of the non-isoenergetic neutrino–electron scattering kernel. For a complete discussion of the state of the art in neutrino emissivities and opacities, the reader is referred to [13].

3 Finite Diﬀerencing of the O(v/c) Neutrino Transport Equation in Spherical Symmetry 3.1 Time Derivative of the Neutrino Distribution Function The ﬁnite diﬀerencing of the time derivative of the neutrino distribution function is straightforward. It is implemented in the following way: ∂F Fi ,j ,k − F i ,j ,k = . ∂t dt

(3)

The quantities with an overbar refer to the previous value of the variables. All other quantities refer to the updated values after the time step dt = t − t. In everything that follows, we adopt the convention that i ≡ i + 1/2, j ≡ j + 1/2, and k ≡ k + 1/2. 3.2 Collision Term The ﬁnite diﬀerence representation of the collision term in equation (2) is

Neutrino Transport in Core Collapse Supernovae

43

ji ,k −χ ˜i ,k Fi ,j ,k ρi jmax 1 2 + 3 3 Ek wl (RIS )i ,j ,l ,k Fi ,l ,k ch c l=1

−

jmax 1 2 E wl (RIS )i ,j ,l ,k F i ,j ,k k 3 3 ch c l=1

+

jmax kmax 1 2 − F ) E (1/ρ ∆E wl i i ,j ,k m m 3 3 ch c m=1 l=1

˜ in ) F × (R NES i ,j ,l ,k ,m i ,l ,m −

jmax kmax 1 2 E F ∆E wl i ,j ,k m m 3 3 ch c m=1 l=1

out ˜ NES × (R )i ,j ,l ,k ,m (1/ρi − Fi ,l ,m ) jmax kmax 1 2 + 3 3 (1/ρi − Fi ,j ,k ) ∆Em Em wl ch c m=1 l=1

˜ em ) (1/ρ − F¯ ) × (R PAIR i ,j ,l ,k ,m i i ,l ,m −

jmax kmax 1 2 E F ∆E wl m m ch3 c3 i ,j ,k m=1 l=1

abs ˜ PAIR × (R )i ,j ,l ,k ,m F¯i ,l ,m

(4)

It is important to note that (a) the collision term in our approach is diﬀerenced implicitly with respect to time and (b) the scattering kernels are not approximated by a truncated Legendre expansion in cos θ. Thus, modulo discretization errors, our approach yields a solution of the original Boltzmann equation. For certain neutrino interactions, the Legendre expansion of the scattering kernels does not converge; i.e., the kernels can be inﬁnite, although integrably ﬁnite (e.g., see [12]). Thus, expanding the kernels in this way is not generally valid. Note that the overbar in the pair creation and annihilation terms denote the antineutrino distribution function, not the value of the distribution function at the previous time step, as in (3). We continue the convention adopted in equation (3), both here and throughout the remaining sections, that quantities without an overbar are deﬁned at the new time step. Thus, in (4) ρ, j, χ, ˜ R... , F , and F¯ are all deﬁned at the new time step. 3.3 Neutrino Propagation in Mass The term µ∂(r2 ρF )/∂m in (2) accounts for the neutrinos that are propagating into and out of a spherical mass shell. In the free streaming limit,

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the advected neutrino number in a time step (as measured by a comoving observer) is large with respect to the neutrino number in the zone. Upwind diﬀerencing of the advection term is appropriate to limit destabilizing errors in the advection ﬂuxes. For discrete direction cosines µj , the direction of the neutrino “wind” is simply given by the sign of µj . (A neutrino with µ = 0 propagates tangential to the mass shell). In diﬀusive conditions, however, the neutrino ﬂux may be orders of magnitude smaller than the prevailing nearly isotropic neutrino density in the mass shell. In this situation, an asymmetric diﬀerencing can easily lead to an overestimate of the ﬁrst angular moment because of improper cancellations among the contributions of the isotropic component of the neutrino ﬁeld. Based on transport coeﬃcients βi,k , we therefore interpolate between upwind diﬀerencing in free streaming regimes (βi,k = 1) and centered diﬀerencing (βi,k = 1/2) in diﬀusive regimes. Moreover, the equilibrium constraint (73) in [14] only holds for βi,k = 1/2. We therefore use transport coeﬃcients deﬁned according to 1/2 if 2dri > λi,k , βi,k = (5) −1 (2dri /λi,k + 1) otherwise . That is, the transport coeﬃcients at zone edges depend on the ratio of the distance dri between zone centers and the neutrino angle and energy dependent mean free path λi,k . We then discretize the propagation term as: µ

µj ∂r2 ρF 2 4πri+1 = ρi+1 Fi+1,j ,k − 4πri2 ρi Fi,j ,k ∂m dmi

(6)

with ρi Fi,j ,k = βi,k ρi −1 Fi −1,j ,k + (1 − βi,k ) ρi Fi ,j ,k

(7)

for outward propagating neutrinos (µj > 0), and ρi Fi,j ,k = (1 − βi,k ) ρi −1 Fi −1,j ,k + βi,k ρi Fi ,j ,k

(8)

for inward propagating neutrinos (µj < 0). 3.4 Change in the Neutrino Direction Cosine from Spatial Propagation For any neutrino, the neutrino direction cosine increases toward unity as it propagates through space. Hence, neutrino propagation causes angular advection relative to a ﬁxed grid of neutrino direction cosines µj . This contribution to the evolution of the neutrino distribution function is described by the term ∂[(1 − µ2 )F ]/r∂µ in (2). The ﬁnite diﬀerence representation is chosen to maintain equilibrium between the radiation ﬁeld and stationary matter under conditions in which such an equilibrium would exist [14, 15]. We set

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45

2 3 ri+1 − ri2 1 ∂[(1 − µ2 )F ] = 3 (ζj+1 Fi ,j+1,k − ζj Fi ,j,k ) . r∂µ 2 ri+1 − ri3 wj

(9)

The diﬀerencing of the coeﬃcients, ζ = 1 − µ2 , is deﬁned by ζj+1 − ζj = −2µj wj .

(10)

The angular integration of the term ∂[(1 − µ2 )F ]/r∂µ produces the zeroth and second angular moments of the neutrino distribution function. Its ﬁnite diﬀerence representation is therefore not as sensitive to cancellations in the diﬀusive limit as the diﬀerencing of the spatial advection term. Upwind diﬀerencing can be justiﬁed. The angular “wind” always points towards µ = 1. However, for reasons of completeness and consistency, we also use centered diﬀerencing in the diﬀusive regime. With angular transport coeﬃcients γi ,k ≡ βi ,k , we interpolate the values of the neutrino distribution function to angular zone edges by Fi ,j,k = γi ,k Fi ,j −1,k + (1 − γi ,k ) Fi ,j ,k .

(11)

3.5 Conservation of Lepton Number and Energy In our attempt to simulate core collapse supernova explosions we are presented with a number of underlying technical challenges that will ultimately dictate the degree to which we are conﬁdent that our simulation outcomes represent reality at all. Among these is the challenge of maintaining conservation of lepton number (for massless neutrinos) and energy in any given supernova model. Conservation of lepton number and energy are no guarantee that a model is correct. Models can be constructed, for example, to conserve total energy but may not accurately model the partition of energy among kinetic energy, internal energy, gravitational energy, etc. But any model that does not conserve lepton number and energy does not satisfy arguably the most important quality control we can use to gauge a simulation’s realism. How should we interpret the prediction of a ∼1051 erg explosion in a model where the total energy varies during the course of the simulation by ∼1051 erg or more? To put matters in perspective as to why ensuring conservation presents such a great technical challenge, we simply have to recall that the ultimate source of energy in a core collapse supernova is the gravitational binding energy of the remnant neutron star. This ∼1053 erg of energy is two orders of magnitude larger than the ∼1051 erg associated with the explosion. Consequently, total energy must be conserved over the course of a simulation to better than one part in 103 . A typical simulation will be carried out over ∼105−6 time steps, which requires that energy be conserved systematically to better than one part in 108−9 per time step. This is a severe requirement, one that is very diﬃcult to satisfy in a realistic supernova model.

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First, we must deﬁne an energy that is conserved. For the radiation ﬁeld— modulo energy losses and gains owing to the neutrino interactions with the matter—the lab frame speciﬁc radiation energy is globally conserved. In one approach, we may begin with the comoving frame speciﬁc radiation energy and ﬂux. In the continuum limit, the lab frame speciﬁc radiation energy conservation is guaranteed by a cancellation of terms in the equations for the comoving frame speciﬁc radiation energy and ﬂux when these equations are added together to give the evolution (conservation) equation for the lab frame speciﬁc radiation energy (the lab frame speciﬁc radiation energy, by the Lorentz transformation, is a sum of the comoving frame speciﬁc radiation energy and ﬂux). In the O(v/c) limit, for example, the cancelling terms arise from both the O(1) and O(v/c) (“observer corrections”: angular aberration and frequency shift) terms in the original Boltzmann equation from which the comoving frame speciﬁc energy and ﬂux equations arise (these are the ﬁrst two “moments” of the Boltzmann equation, deﬁned in Sect. 2). We must in turn ensure that such cancellations occur in the discrete limit to ensure that energy is conserved in our simulations. This requires that we construct the discrete representation of the terms in the Boltzmann equation from which the cancelling terms arise with great care—i.e., the discrete representation of the O(1) and O(v/c) terms in the Boltzmann equation are not independent. This numerical feat has been achieved in the spherically symmetric case [14, 16–19]]. The proliferation of O[(v/c2 )] terms in the Boltzmann equation as we move from one- to three-dimensional models makes this approach increasingly diﬃcult [19, 20]. As a result, other approaches are now being considered to achieve the same end [21]. The integration of the Boltzmann equation over momentum space, spanned by the neutrino direction cosine and energy, and mass gives the conservation laws for neutrino number and energy. We deﬁne J N and H N to represent the zeroth and ﬁrst µ number moments of the distribution function: 1 ∞ F E 2 dEdµ , JN = −1 1

0

HN = −1

∞

F E 2 dEµdµ .

0

Integration of (2) over µ and E with E 2 as the measure of integration gives the following evolution equation for J N : ∂ j 2 ∂J N 2 N + 4πr ρH − E dEdµ + χF E 2 dEdµ = 0 . (12) ∂t ∂m ρ The aberration and frequency shift terms in equation (2) do not contribute because 1 − µ2 vanishes at µ = ±1 and E 3 F is zero for E = 0 and E = ∞. One more integration over rest mass m from the center of the star to its surface gives the evolution equation for the total neutrino number.

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47

Slightly less straightforward is the derivation of total energy conservation. Deﬁne the energy moments E J = F E 3 dEdµ H E = F E 3 dEµdµ E K = F E 3 dEµ2 dµ QE = F E 3 dEµ3 dµ; , By taking the zeroth and ﬁrst angular moments of the energy moment ( E 3 dE{∂F/∂t = O[F ]}) of the Boltzmann equation, the latter weighted by the ﬂuid velocity, v,—i.e., E 3 dEdµ{∂F/∂t = O[F ]} and (v E 3 dEdµ {∂F/∂t = O[F ]}— one ﬁnds ∂lnρ 2v ∂J E ∂ v E 2 E + + 4πr ρH − J − KE KE + ∂t ∂m ∂t r r j 3 E dEdµ + χF E 3 dEdµ = 0 , (13) − ρ

v

∂ ∂H E dv E v E + K − 4πr2 vρK E − 4πr2 ρ J − KE ∂t ∂m r dm ∂lnρ 2v E 3 + −v H + v χF E dEµdµ = 0 . ∂t r

(14)

Combining, we obtain the lab frame energy conservation equation: ∂ E ∂ J + vH E + 4πr2 ρ vK E + H E ∂m ∂t j 3 E dEdµ + χF E 3 dEdµ + v χF E 3 dEµdµ − ρ 1 ∂ 4πr2 ρv H E . − 2 4πr ρ ∂t

0=

(15)

Note that J E + vH E is the lab frame neutrino energy density as expressed in terms of the comoving frame moments J E and H E . Integration of (15) over enclosed mass leads to an equation for total neutrino energy evolution. The ﬁrst term on the RHS is the time evolution of the conserved energy. The second term corrects for surface eﬀects, namely the work at the surface against radiation pressure, ρK E , and the surface luminosity, 4πr2 ρH E . The second line represents the energy exchange between radiation and matter by emission, absorption, and radiation stress. If one aims to check energy conservation to machine precision, it is important to be aware that taking

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the O(v/c) limit in the comoving frame does not lead to the identical physical description as taking the O(v/c) limit in the lab frame. For that reason, we keep higher order terms in the third line of (15) for compatibility with the physical description of the O(v/c) Boltzmann equation (2) in the comoving frame. These higher-order terms can become nonnegligible. We observe that the expressions (∂lnρ/∂t+2v/r)KE and K E 4πr2 ρ∂v/∂m in equations (13) and (14) cancel based on the hydrodynamic continuity equation ∂v ∂lnρ 2v + = −4πr2 ρ . (16) ∂t r ∂m Similarly, the terms involving (J E −K E )v/r cancel. As these cancelling terms are of O(v/c) and proportional to the neutrino density, they grow during a core collapse simulation and become quite large at and after bounce. Hence, it is important for a satisfactory global energy conservation to guarantee their cancellation in the ﬁnite diﬀerence representation as well. Identifying the origin of the terms (∂lnρ/∂t + 2v/r)KE and K E 4πr2 ρ∂v/∂m, we ﬁnd that (∂lnρ/∂t + 2v/r)KE originates from the zeroth moment of the observer correction v ∂lnρ 2v 1 ∂ 3 + E F , (17) − 1 − µ2 µ2 ∂t r r E 2 ∂E in the Boltzmann equation (2), and K E 4πr2ρ∂v/∂m originates from the ﬁrst moment of the space term, µ∂ 4πr2 ρF /∂m, in the same equa E propagation tion. The terms J − K E v/r stem from the zeroth moment of the same observer correction (17) and the ﬁrst moment of the angular propagation term ∂ 1 − µ2 F / (r∂µ) in the Boltzmann equation (2). The requirement of global energy conservation in the lab frame therefore imposes interdependencies on the ﬁnite diﬀerencing of the space propagation term, the angular propagation term, and the terms describing the observer corrections [17]. In particular, to determine the “matched” ﬁnite diﬀerencing for ∂lnρ 2v + (18) A ≡ µ2 ∂t r v (19) B ≡ (1 − µ2 ) r we begin by multiplying the discrete representation of the left-hand side of the Boltzmann equation (2) (sans the terms describing the observer corrections) by 1 + µ¯ vi+1 (in what follows, unless otherwise speciﬁed the indices are i , j , and k ):

Neutrino Transport in Core Collapse Supernovae

F − F¯ cdt 4πµ 2 [¯ r ρ¯i+1 Fi+1 − r¯i2 ρ¯i Fi ] + (1 + µ¯ vi+1 )E dm i+1 2 − r¯i2 ) 1 3(¯ ri+1 + (1 + µ¯ vi+1 )E [ζj+1 Fj+1 − ζj Fj ] 3 2(¯ ri+1 − r¯32 ) w vi+1 )E F¯ (1 + µvi+1 )EF − (1 + µ¯ = cdt vi+1 EF µvi+1 EF − µ¯ − cdt 4πµ 2 + [(1 + µ¯ vi+1 )E r¯i+1 ρ¯i+1 Fi+1 − (1 + µ¯ vi )E r¯i2 ρ¯i Fi ] dm 4πµ2 [¯ vi+1 r¯i2 ρ¯i EFi − v¯i r¯i2 ρ¯i EFi ] − dm 2 − r¯i2 ) 1 3(¯ ri+1 + [ζj+1 EFj+1 − ζj EFj ] 3 2(¯ ri+1 − r¯32 ) w (1 + µ¯ vi+1 )E

+ =

49

(20)

2 − r¯i2 ) 3(¯ ri+1 1 v¯i+1 [µζj+1 EFj+1 − µζj EFj ] 3 2 2(¯ ri+1 − r¯3 ) w vi+1 )E F¯ (1 + µvi+1 )EF − (1 + µ¯

cdt vi+1 µvi+1 − µ¯ − EF cdt 2 − r¯i2 ) 1 3(¯ ri+1 [ζj+1 EFj+1 − ζj EFj ] + 3 2(¯ ri+1 − r¯32 ) w 4πµ 2 + [(1 + µ¯ vi+1 )E r¯i+1 ρ¯i+1 Fi+1 − (1 + µ¯ vi )E r¯i2 ρ¯i Fi ] dm 4πµ2 2 − r¯ ρ¯i EFi [¯ vi+1 − v¯i ] dm i 2 − r¯i2 ) 3(¯ ri+1 1 v¯i+1 [µζj+1 EFj+1 − µζj EFj ] . + 3 2(¯ ri+1 − r¯32 ) w As noted, the total energy equation is obtained when summing equations (13) and (14) and then integrating over m (the integration in µ and E has already taken place). In this sequence of integrations, the term involving A in (13) cancels with the term −4πr2 ρK E dv/dm in equation (14). We must ensure this happens in the discrete limit. Identifying the appropriate velocity gradient term in (21) and focusing on the appropriate integration (in this case, over m), we require that (below, the term involving A comes from the zeroth moment of ﬁrst term in the observer correction (17) after an integration by parts; the term involving the velocity gradient is the next to last term in (21), corresponding to the ﬁrst moment of the spatial propagation term in the Boltzmann equation (2)):

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Ai Fi dmi

i=1,imax−1

−

4πµ2 r¯i2 ρ¯i Fi (¯ vi+1 − v¯i )

i=1,imax−1

=

Ai Fi dmi

i=1,imax−1

−

4πµ2 r¯i2 (βi ρ¯i Fi + (1 − βi )¯ ρi −1 Fi −1 )(¯ vi+1 − v¯i )

i=1,imax−1,j≤jmax/2

−

4πµ2 r¯i2 (βi ρ¯i −1 Fi −1 + (1 − βi )¯ ρi Fi )(¯ vi+1 − v¯i )

i=1,imax−1,j≥jmax/2+1

=

Ai Fi dmi

i=1,imax−1

−

4πµ2 r¯i2 (¯ vi+1 − v¯i )βi ρ¯i Fi

i=1,imax−1,j≤jmax/2

−

2 4πµ2 r¯i+1 (¯ vi+2 − v¯i+1 )(1 − βi+1 )¯ ρi Fi

i=1,imax−2,j≤jmax/2

−

4πµ2 r¯i2 (¯ vi+1 − v¯i )(1 − βi )¯ ρi Fi

i=1,imax−1,j≥jmax/2+1

−

2 4πµ2 r¯i+1 (¯ vi+2 − v¯i+1 )βi+1 )¯ ρi Fi

i=1,imax−2,j≥jmax/2+1

=0,

(21)

which gives [17] Ai = 4πµ2

ρ¯i 2 (¯ r2 (¯ vi+1 − v¯i )βi + r¯i+1 (¯ vi+2 − v¯i+1 )(1 − βi+1 )) dmi i

(22)

for j ≤ jmax/2 and Ai = 4πµ2

ρ¯i 2 (¯ r2 (¯ vi+1 − v¯i )(1 − βi ) + r¯i+1 (¯ vi+2 − v¯i+1 )βi+1 ) dmi i

(23)

for j ≥ jmax/2 + 1. Similarly, deﬁning B according to B≡

2 − r¯i2 3 r¯i+1 v¯i+1 B , 3 3 2 r¯i+1 − r¯i

(24)

and again focusing on the appropriate integration (in this case, over µ), we require that (below, the term involving B comes from the zeroth moment of the second term in the observer correction (17), after an integration by parts; the second term is the last term in (21), corresponding to the ﬁrst moment of the angular propagation term):

Neutrino Transport in Core Collapse Supernovae

0=

51

B Fj wj

j=1,jmax

2 [µj αj+1 Fj+1 − µj αj Fj ]wj w j=1,jmax j = B Fj wj +

j=1,jmax

+

2[µj αj+1 (γFj + (1 − γ)Fj +1 )

j=1,jmax

− µj αj (γFj −1 + (1 − γ)Fj )] B Fj wj = j=1,jmax

+

2[µj αj+1 γ − µj αj (1 − γ)]Fj

j=1,jmax

+

2µj −1 αj (1 − γ)Fj

j=2,jmax

+

(−2)µj +1 αj+1 γFj

j=1,jmax−1

=

B Fj wj

j=1,jmax

+

2γαj+1 (µj − µj +1 )Fj

j=1,jmax

+

2(1 − γ)αj (µj −1 − µj )Fj ,

(25)

j=1,jmax

which gives [17] Bj

2 − r¯i2 µj +1 − µj 3 r¯i+1 v¯i+1 2γαj+1 3 3 2 r¯i+1 − r¯i wj µj − µj −1 . + 2(1 − γ)αj wj =

(26)

3.6 Frequency Shift from Observer Motion For comoving observers, we must account for the observed shift in the frequency of a neutrino measured by two neighboring observers moving with diﬀerent velocities. Motivated by the deﬁnitions (18) and (19), the observer correction, (17), can be written as ∂ 3 ∂F 3 E F . + µ2 A − B E (27) 0=E ∂t E ∂E

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It has been found in [10] and [14] that (27) has an analytical solution. One may write [14] the prefactor in (27) as the time derivative of the quantity 2 2 Rf = r(3µ −1) ρ(µ ) : ∂lnRf = µ2 A − B . ∂t If we transform from the “Eulerian” variable, x = E, to the “Lagrangian” variable, y = E/Rf , (27) becomes ∂ 3 ∂ 3 ∂Rf E F E F 0= + 2 E × Rf ∂t R ∂t ∂E f E ∂ E3F ∂ 3 ∂ [E/Rf ] ∂ 3 E F = E F − . = ∂t ∂t ∂t E E ∂ [E/Rf ] E/Rf For a small section of energy phase space E 2 ∆E = E23 − E13 /3, this relationship leads to ∂ 2 E F ∆E =0. (28) ∂t E/Rf The validity of (28) for arbitrary neutrino distributions F leads to the following interpretation: The neutrinos initially residing in the energy interval E 2 ∆E move, according to the observer correction, along constant E/Rf in the phase space of a comoving observer. This allows us to determine the evolution of any neutrino property. For example, the neutrino speciﬁc energy in this phase space interval, d = E 3 F ∆E, evolves according to ∂E ∂ 3 ∂lnRf 2 E F ∆E d . (29) = E F ∆E = ∂t ∂t E/Rf ∂t E/Rf A ﬁnite diﬀerence representation of (28) and (29) has been given in [14]. Consider a neutrino energy group k , with neighboring groups k + dk, dk = ±1. Equation (28) tells us that the number of neutrinos before the comoving observer correction, Fi ,j ,k Ek2 dEk , is equal to the number of neutrinos after the correction. The distribution after the correction has a diminished number of neutrinos Fi ,j ,k Ek2 dEk − n− i ,j ,k in group k and an additional number + of neutrinos ni ,j ,k +dk in the neighboring group k + dk, Fi ,j ,k Ek2 dEk −

+ Fi ,j ,k Ek2 dEk − n− i ,j ,k + ni ,j ,k +dk = 0 .

(30)

Equation (29) deﬁnes a similar correction for the speciﬁc neutrino energy in group k

+ +dk n + E Fi ,j ,k Ek3 dEk − Fi ,j ,k Ek3 dEk − Ek n− k i ,j ,k i ,j ,k +dk 2 3 (31) = − µj Ai ,k − Bi ,j ,k Fi ,j ,k Ek dEk dt .

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53

where Ai ,k and Bi ,j ,k are given by (22), (23), and (26). Equations (30) and (31) uniquely deﬁne the solution 2 n− i ,j ,k = µj Ai ,k − Bi ,j ,k

dEk E 3 Fi ,j ,k dt Ek +dk − Ek k

− n+ i ,j ,k = ni ,j ,k −dk ,

(32)

− 2 which leads, by the update Fi ,j ,k = F i ,j ,k + n+ − n i ,j ,k i ,j ,k / Ek dEk , to the following ﬁnite diﬀerence representation of the frequency shift term in the Boltzmann equation (2): 1 Ek2 dEk × µ2j Ai ,k −dk − Bi ,j ,k − µ2j Ai ,k

dEk −dk E 3 Fi ,j ,k −dk Ek − Ek −dk k −dk dEk − Bi ,j ,k Ek3 Fi ,j ,k . Ek +dk − Ek

(33)

3.7 Angular Aberration from Observer Motion We are left with the task of ﬁnding a ﬁnite diﬀerence representation for the angular aberration in (2). Because the global energy depends on (1 + ui µj ) Fi ,j ,k , any correction to the neutrino propagation direction also aﬀects energy conservation. Therefore, it is desirable to construct a numerical implementation of angular aberration that conserves the speciﬁc neutrino luminosity µj Fi ,j ,k wj [17], much like our numerical implementation of the frequency shift preserved the speciﬁc neutrino energy. With ζ = 1 − µ2 , the angular aberration term can be rewritten as ∂F ∂ [ζµF ] . (34) = (A + B/ζ) ∂t µ ∂µ As before, we search for an analytic solution. For the quantity Ra = r3 ρ, we ﬁnd dlnRa = A + B/ζ . dt It is now possible to rewrite (34) in terms of the “Lagrangian” variable y = ζ −1/2 µ/Ra instead of the “Eulerian” variable x = µ. After a multiplication by ζµ, (34) becomes:

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∂ [ζµF ] 0 = ζµ + α (A + B/ζ) ∂µ µ ∂ ∂ ∂Ra [ζµF ] + ζ −1/2 µ 2 × ζ 3/2 Ra [ζµF ] = ∂t Ra ∂t ∂µ µ ∂ [ζµF ] = ∂t µ −1/2 ∂ ζ µ/Ra ∂ ∂ [ζµF ] [ζµF ] − . = ∂t ∂t ∂ ζ −1/2 µ/Ra ζ −1/2 µ/Ra

∂F ∂t

µ

And once again, is clear. The neutrinos initially residing the interpretation = ζ µ − ζ1 µ1 are shifted by angular aberration in the interval 1 − 3µ2 ∆µ 2 2 √ along constant µ/ ζRa in the phase space of a comoving observer: ∂ 2 1 − 3µ F ∆µ =0. (35) ∂t ζ −1/2 µ/Ra We may now evaluate the change in other neutrino properties; for example, the speciﬁc luminosity, d = 1 − 3µ2 µF ∆µ, ∂ ∂µ 1 − 3µ2 µF ∆µ = 1 − 3µ2 F ∆µ ∂t ∂t ζ −1/2 /Ra −1/2 ζ µ/Ra =ζ

∂lnRa d . ∂t

(36)

We identify the bin size 1 − 3µ2j ∆µj = wj with our Gaussian quadrature weights. Equation (35) leads to the condition for number conservation:

+ Fi ,j ,k wj − Fi ,j ,k wj − n− i ,j ,k + ni ,j +dj,k = 0 and (36) leads to a prescription for the numerical evolution of the speciﬁc luminosity:

+ Fi ,j ,k µj wj − Fi ,j ,k µj wj − µj n− i ,j ,k + µj +dj ni ,j +dj,k = − (ζj Ai ,k + Bi ,j ,k ) Fi ,j ,k µj wj dt . where dj = ±1. The change in the neutrino distribution from angular aberration is then: − − n (37) /wj Fi ,j ,k = F i ,j ,k + n+ i ,j ,k i ,j ,k with n− i ,j ,k = (Ai ,k + Bi ,j ,k /ζj )

− n+ i ,j ,k = ni ,j −dj,k .

wj ζj µj Fi ,j ,k dt µj +dj − µj

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55

This leads to the following ﬁnite diﬀerence representation of the angular aberration term in the Boltzmann equation (2): 1 wj −dj ζj −dj µj −dj Fi ,j −dj,k (Ai ,k + Bi ,j −dj,k /ζj −dj ) wj µj − µj −dj wj − (Ai ,k + Bi ,j ,k /ζj ) ζj µj Fi ,j ,k . (38) µj +dj − µj We apply the aberration corrections with dj = +1 for µ ≤ 0 and dj = −1 for µ > 0. This is not upwind diﬀerencing and, therefore, runs the risk of producing negative neutrino distribution functions. However, there are three reasons to accept this shortcoming: (i) The angular aberration correction is generally small, with the exception of aberration in the vicinity of strong shocks with large velocity gradients. (ii) As long as the angle-integrated neutrino density is positive, transiently negative contributions from backward directions on the Gaussian angular grid (µ < 0) allow the computer representation of strongly forward peaked neutrino ﬂuxes that would not be representable with positive neutrino distribution functions on a grid with limited angles µmax < 1. (iii) The chosen direction of dj guarantees that no neutrinos are shifted oﬀ the grid. This is a prerequisite for number and energy conservation.

4 The General Case: The Multidimensional Neutrino Transport Equations As the previous sections illustrate, one of the greatest challenges encountered when developing the numerical methods to obtain a physically reliable solution of the neutrino transport equations in core collapse supernovae is to achieve simultaneous lepton number and energy conservation or balance, the former for massless neutrinos and Newtonian gravity, where lepton number is conserved and where the concept of a conserved energy can be deﬁned in multiple spatial dimensions, the latter for the general relativistic case in two and three spatial dimensions. The ﬁrst step in meeting this challenge is to derive the conservative lepton number and energy conservation/balance equations, which at a more fundamental level means ﬁnding the corresponding evolved variables that naturally lend themselevs to a conservative decription of the neutrino radiation ﬁeld. As we will see, both the conservation/balance equations and the fundamental variables can be extracted from the underlying kinetic theory at a point in the theory where the fundamental physical laws are manifestly conservative [20]. The distribution function f represents the density of neutrinos in phase space. The phase space for neutrinos of deﬁnite mass m, Mm , is ﬁlled with ˆ trajectories xµ (λ), pi (λ) , or “states”. As a collection of neutrinos evolves, the number of neutrinos in each state changes due to collisions. If one considers a 6-dimensional hypersurface Σ in Mm , the ensemble-averaged number

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N [Σ] of occupied states crossing Σ is N [Σ] =

fω ,

(39)

Σ

where ω is the surface element. An equation governing the evolution of the distribution function is obtained by considering a closed 6-dimensional hypersurface ∂D bounding a region D in Mm . The net number of occupied states emerging from D is, from (39) and the generalized Stokes’ theorem, N [∂D] = fω = d(f ω) . (40) ∂D

D

This equation relates a volume integral to a surface integral and is key to obtaining conservative formulations of kinetic theory. In fact, in rushing headlong in deriving the general relativistic Boltzmann equation ∂f ρˆ ∂f = C[f ] . (41) pµˆ Λµ¯ µˆ eµ µ¯ µ − Γ νˆ ρˆ ˆµ p ∂x ∂pνˆ from (40), (40) is easily overlooked. Here Λµµ¯ is the Lorentz boost between the orthonormal lab and (ﬂuid) comoving frames and eµµ¯ is the tetrad transformation between the coordinate basis and the orthonormal lab frame. The integrand on the right-hand side of (40), d(f ω), is in conservative form. Manipulating it just enough to bring it into the form d(f ω) = F [f ]Ω, where Ω is the volume element in phase space, yields a conservative formulation of neutrino number kinetics, and manipulating the exterior derivative d(vµ pµ f ω), for an arbitrary timelike vector vµ , yields a conservative formulation of neutrino four-momentum kinetics. Deﬁning the speciﬁc neutrino number ﬂux vector (42) N µ ≡ Lµ µˆ pµˆ f and the speciﬁc neutrino stress-energy tensor T µν ≡ Lµ µˆ Lν νˆ pµˆ pνˆ f ,

(43)

where Lµˆ µ = eµµ¯ Λµµ¯ is the transformation to the comoving frame, the conservative kinetic equations for neutrino number and four-momentum are: 1 ∂ √ √ −gN µ µ −g ∂x −1 ˆi ∂p ˆj ∂p ∂ ∂u 1 − E(p) det det Γ µˆνˆ pνˆ ˆj Lµˆ µ N µ ∂u ∂uˆi E(p) ∂u ∂p = C[f ] ,

(44)

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57

∂ √ 1 √ −gT µν + Γ µ νρ T νρ ν −g ∂x −1 ˆi ∂p ˆj ∂p ∂ 1 ρˆ ∂u ν ˆ µν − E(p) det L νT det Γ νˆρˆp ∂u ∂uˆi E(p) ∂u ∂pˆj = F µ ν N ν + Lµ µˆ pµˆ C[f ] .

(45)

√

where −g is the determinant of the spacetime metric. The adjective “speciﬁc” often denotes a quantity measured per unit mass, but in this context we use it to denote the neutrino ﬂux and stress-energy in a given invariant momentum space volume element. While the distribution function f obeys the Boltzmann equation (41), the speciﬁc neutrino number ﬂux and stressenergy satisfy the conservative equations (44) and (45). In stating that (44) and (45) constitute conservative formulations of neutrino kinetics, we mean that the connection to the corresponding balance equations is transparent. We can use elementary calculus to form the familiar invariant momentum space volume element ∂p 3 1 d3 p = (46) det d u , E(p) E(p) ∂u ˆ where a transformation from orthonormal momentum components pi to some other set of coordinates (e.g., momentum space spherical coordinates ˆi u = {|p|, ϑ, ϕ}) has been performed. Multiplying equations (44) and (45) by (46) and integrating, the terms with momentum space derivatives are obviously transformed into vanishing surface terms. The results are the balance equations ∂ √ 1 µ √ −gN = C[f ]πm , (47) −g ∂xµ 1 ∂ √ νµ ν ρµ √ = −Γ −gT T + C[f ]Lν νˆ pνˆ πm . (48) ρµ −g ∂xµ for total neutrino number and 4-momentum, where N µ and T µν are the neutrino number current and stress-energy tensor, respectively. In the above √ equations, πm = −g0ijk dpi ∧ dpj ∧ dpk //6|p0 | is the invariant volume element on a mass shell corresponding to the mass m in the tangent space [20]. In the previous sections, we presented a detailed account of how one obtains a conservative diﬀerencing for the equation governing the evolution of the speciﬁc neutrino distribution function for spherically symmetric ﬂows. Working with comoving frame neutrino four momenta, the conservation of total lab frame energy required a matching of the ﬁnite diﬀerencing of the radial, angular, and energy advection terms in this equation. One can, of course, take a diﬀerent approach. While the use of comoving frame four momenta for the neutrinos is a prerequisite given the neutrino interactions with the stellar core, which are most naturally and most simply expressed in terms of these

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four momenta, the use of comoving frame neutrino densities (number, energy, . . . ) may not be the best choice. Given that the global lab frame energy is conserved, working with densities in this frame that are expressed in terms of comoving frame neutrino four momenta might be the right combination considering both local neutrino interactions and global energy conservation. Nonetheless, however natural this latter approach might seem, we are still not free of the need to match the ﬁnite diﬀerence representations for various terms in the kinetic equations, as we now show. Consider the relationship between (44) for neutrino number kinetics and the t component of (45) for neutrino 4-momentum kinetics. Speciﬁcally, we need to relate the spatial and momentum divergence terms of these equations to each other and identify terms that cancel. First we relate the spatial and momentum divergence terms of (44) and the t component of (45). Their spatial divergence terms ∂ √ 1 −gLµ µˆ pµˆ f , Nx [f ] ≡ √ µ −g ∂x t ∂ √ 1 T x [f ] ≡ √ −gLt νˆ Lµ µˆ pνˆ pµˆ f , −g ∂xµ

(49) (50)

are related by ∂ Lt νˆ pνˆ Nx [f ] = T t x [f ] − f pµˆ pνˆ Lµ µˆ µ Lt νˆ . ∂x

(51)

Their momentum divergence terms Np [f ]

−1 ˆi ∂p ∂p ˆj ∂ 1 µ ˆ ρˆ ∂u f , (52) ≡ −E(p) det det Γ µˆρˆp p ∂u ∂uˆi E(p) ∂u ∂pˆj t T p [f ] −1 ∂p ≡ −E(p) det ∂u ˆi ∂p ˆj ∂ 1 µ ˆ ρˆ ∂u t ν ˆ × L νˆ p f , (53) det Γ µˆρˆp p ∂u ∂uˆi E(p) ∂pˆj are related by ˆ ∂ui ∂pνˆ ˆ Lt νˆ pνˆ Np [f ] = T t p [f ] + f Γ j µˆρˆpµˆ pρˆ ˆ Lt νˆ . ∂pj ∂uˆi

(54)

Comparison of (51) and (54) with (44) and the t component of (45) shows that the following equation must be valid:

Neutrino Transport in Core Collapse Supernovae

ES + EM = −Γ t µρ Lµ µˆ Lρ ρˆpµˆ pρˆ ,

59

(55)

where

∂ t L νˆ (56) ∂xµ is the “extra” term of (51) relating the spatial divergence terms of the number and energy balance equations, and ES ≡ f pµˆ pνˆ Lµ µˆ

ˆ

ˆ

EM ≡ −f Γ j µˆρˆpµˆ pρˆ

∂ui

∂pˆj

Lt νˆ

∂pνˆ ∂uˆi

(57)

is the “extra” term in (54) relating the momentum divergence terms of the number and energy balance equations. In a computational approach to transport in which a neutrino number distribution equation is solved and a neutrino energy equation is used as a consistency check, care must be taken that (55) is satisﬁed numerically. In particular, a given ﬁnite diﬀerence representation of the spacetime divergence Nx [f ] implies a corresponding ﬁnite diﬀerence representation of positiondependent terms in ES , and the position-dependent terms in EM that cancel with their counterparts in ES must have a matching ﬁnite diﬀerence representation. Similarly, a given ﬁnite diﬀerence representation of the momentum divergence Np [f ] implies a corresponding ﬁnite diﬀerence representation of momentum-dependent terms in EM , and the momentum-dependent terms in ES that cancel with their counterparts in EM must have a matching ﬁnite diﬀerence representation. It is important to note that (44) and (45) are general; i.e., they can be expressed in any spatial coordinate system using any neutrino four-momenta. If one specializes these equations to the comoving frame spatial coordinates and four-momenta used in the spherically symmetric case detailed herein and to the O(v/c) limit, one ﬁnds that ES + EM = 0. The ﬁnite diﬀerence matching constructed in Sect. 6.5 guarantees that this sum is satisﬁed numerically in this case. In the general case, the sum is not zero and the number of terms is signiﬁcantly greater, but the basic task is the same. The essential point is that we now have the complete framework within which to carry it out.

5 Boltzmann Neutrino Transport: The Current State of the Art After four decades of core collapse supernova research, detailed spherically symmetric simulations that now include state of the art neutrino interactions, an industry standard equation of state, and multiangle, multifrequency, Boltzmann neutrino transport in full general relativity have ﬁnally been performed [22]. It is important to remember that we are working in phase space.

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Therefore, simulations that assume spherical symmetry and require a solution of the neutrino Boltzmann equation are actually simulations in three dimensions (radius or mass, direction cosine, and energy), not one dimension (radius). When cast in this light, it is not surprising that such threedimensional multiphysics simulations have taken four decades to complete. This is also an omen of what will be required to perform such simulations in three spatial dimensions (six phase space dimensions). Despite this seemingly overwhelming requirement, we must proceed in a systematic and steadfast manner. A typical result for these simulations is shown in (Fig. 4). The shock radius reaches a maximum and then recedes with time, indicating a failure of the stellar core to explode and initiate a core collapse supernova. The results shown here are for a 13 M model, an “optimistic” case owing to the small size of its iron core (1.2 M , but the outcome is the same for progenitors of 15, 20, 25, and 40 M , all of which have more massive cores). The shock trajectories for both the Newtonian and general relativistic cases are shown. In the general relativistic case, owing to the increased gravitational ﬁeld, the shock radius is always deeper. However, the inclusion of general relativity does not necessarily present a more pessimistic case, as might appear by considering the shock radius results out of context [17]. In Fig. 5, a more complete picture is presented for both the Newtonian and general relativistic simulations. Shown are all of the core proﬁles in density, entropy, electron fraction, velocity, and composition, along with the three key neutrino quan-

Newton+O(v/c) Relativistic

3

Radius [km]

10

2

10

1

10

0

0.1

0.2 0.3 Time After Bounce [s]

0.4

0.5

Fig. 4. The shock trajectories are plotted as a function of time for both the Newtonian and fully general relativistic spherically symmetric models in the case of a 13 M progenitor [22]. The shock recedes with time in both cases, indicative of a failed explosion. This result is typical for all spherically symmetric models

Neutrino Transport in Core Collapse Supernovae 4

1 Velocity, u [km/s]

(b) grid points

0 −1 −2 −3 −4 −5 −6 5

10

20

Baryon Density, log10(ρ [g/cm3])

(a)

x 10

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Newton+O(v/c) Relativistic

14 12 10 8 6

50 100 200 500 1000 2000 5000 Radius, r [km]

5

10

20

50 100 200 500 1000 2000 5000 Radius, r [km]

(c)

(d)

18 0.5

16

Electron Fraction, Ye

Entropy per Baryon, s [kB])

20

14 12 10 8 6 4

0.4 0.3 0.2 0.1

2 0 5

10

20

50 100 200 500 1000 2000 5000 Radius, r [km]

5

10

20

50 100 200 500 1000 2000 5000 Radius, r [km]

(e)

(f)

100 Xh X +X n p X

50 Mass Fraction, X

Luminosity, L [1052erg/s]

0.5

20 10 5 4 3 2 1

0.2 0.1 0.05

α

0.01

0.001

0.0001 5

10

20

50 100 200 500 1000 2000 5000 Radius, r [km]

5

10

20

(g)

(h) Neutrino Energy, Erms [MeV]

Mean Flux Factor, F

1 0.8 ν spheres 0.6 0.4 0.2 0

50 100 200 500 1000 2000 5000 Radius, r [km]

200 150

e Neutrino e Antineutrino

100 75 50 30 20 15 10 7.5 5

5

10

20

50 100 200 500 1000 2000 5000 Radius, r [km]

5

10

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50 100 200 500 1000 2000 5000 Radius, r [km]

Fig. 5. The velocity, density, entropy, electron fraction, electron neutrino and antineutrino luminosity, composition, electron neutrino and antineutrino ﬂux factor and rms energy proﬁles at 100 ms after bounce in Newtonian and general relativistic spherically symmetric models initiated from a 13 M progenitor [22]

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tities mentioned above—luminosity, rms energy, and mean ﬂux factor—for both the electron neutrinos and antineutrinos. It is important to point out that, until these simulations were completed, failure to produce explosions in past models that used approximate treatments for the neutrino transport (e.g., multigroup ﬂux-limited diﬀusion; see [6, 23–26]) could have resulted from either the transport approximations or from the neglect of essential physics. We could not have known a priori which of these possibilities would in fact be realized. We now know that the transport approximations were not the cause of these failures and, as a result, have crossed a threshold in supernova theory: new physics is needed. Nonetheless, the move from multigroup ﬂux-limited diﬀusion to Boltzmann neutrino transport led to important quantitative changes in the spherically symmetric models, as might have been expected given the sensitivity of the neutrino heating to the neutrino luminosities, spectra, and angular distributions in the postshock region. Moreover, in spherical symmetry, the Boltzmann results have been used to tune the ﬂux-limiters to better reproduce the Boltzmann results [18]. Thus, Boltzmann solvers for the two- and three-dimensional cases must be developed in conjunction with solvers using approximate techniques, such as two- and three-dimensional multigroup ﬂux limited diﬀusion [26]. The latter, of course, must be utilized until PetaScale resources (for three-dimensional models) become available. Eﬀorts by several groups are now underway to develop Boltzmann neutrino transport for these cases [20,21,27,28]. The simulations performed thus far have been conﬁned to the Newtonian gravity, O(1) limit, restricted not only by Newtonian gravity but by the exclusion of the O(v/c) terms on the left-hand side of the Boltzmann equation mentioned above [28]. The observer corrections are critical in the evolution of the comoving frame neutrino distributions and in the dynamics of stellar collapse [10–12, 29]. They cannot be excluded from any realistic model, as Figs. 6–7 show. In these spherically symmetric collapses, by the time the central density reaches a value of 1×1014 g/cm3 , the maximum infall velocity, inner homologous core mass, and entropy throughout the inner core are signiﬁcantly diﬀerent for the collapse simulation in which the O(v/c) terms have been neglected [30]. In addition, without the O(v/c) terms, the neutrino distributions at a given energy can signiﬁcantly exceed unity, which is unphysical. The O(v/c) terms are responsible for properly redistributing the neutrinos in energy (to higher energy) as the collapse proceeds. If they are neglected, some neutrinos in energy groups for which f > 1 must be discarded so that f = 1 (or handled in some other ad hoc fashion), and energy and lepton number will not be conserved. Figure 7 illustrates the extent to which the conservation breaks down. At a central density of 1 × 1014 g/cm3 , we have lost nearly 3 × 1051 erg. While discarding the neutrinos is only one ad hoc way to manage the error incurred when the O(v/c) terms are neglected, these results clearly demonstrate the impact these terms have not only on the neutrino distributions, but on the dynamics

Neutrino Transport in Core Collapse Supernovae

63

Fig. 6. The entropy and velocity proﬁles as a function of enclosed mass at a central density of 1014 g/cm3 for spherically symmetric models with and without the O(v/c) “observer corrections” [30]. The diﬀerences between the two cases are striking

of stellar collapse and the conservation of total energy and lepton number. Not only can they not be neglected, they must be included with great care.

6 Previews of Coming Distractions: Neutrino Flavor Transformation It is now an experimental fact that neutrinos have mass and, therefore, mix in ﬂavor. Observations of Solar and atmospheric neutrinos, and experiments at LSND, indicate there may be as many as three independent values of the diﬀerence in the square of the neutrino masses (δm2 ), and four mixing angles, which would require three active and at least one sterile neutrino. Although we await conﬁrmation (or not) of the LSND ﬁndings, the data already strongly suggest that neutrino mixing should be included in core collapse supernova models. Neutrino mixing may signiﬁcantly aﬀect one or more of the following: the supernova mechanism, supernova nucleosynthesis, and terrestrial supernova neutrino detection. As we discussed earlier, all three active neutrino ﬂavors are involved in core collapse supernova dynamics. Electron, muon, and tau neutrinos and their antineutrinos are produced primarily through thermal emission,

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Fig. 7. The total energy versus central density for the same models shown in Fig. 6. The model without the observer corrections has clearly lost an energy ∼3 × 1051 ergs prior to stellar core bounce

nucleon–nucleon bremsstrahlung, and neutral-current neutrino–neutrino annihilation in the hot mantle of the proto-neutron star after core bounce. Owing to the lack of charged-current interactions among the muon and tau neutrinos and antineutrinos, electron neutrino and antineutrinos decouple at lower densities given their larger total interaction cross sections. Decoupling at lower densities and, consequently, lower temperatures results in softer relative spectra for the electron ﬂavor neutrinos [31]. Herein lies the essential relevance of neutrino mixing to the supernova mechanism: If ﬂavor conversion between electron ﬂavor and muon/tau ﬂavor neutrinos were to occur below the supernova shock wave in the neutrino heating epoch after stellar core bounce, the neutrino heating behind the shock, which is mediated predominantly by the charged-current absorption of electron neutrinos and antineutrinos, could be signiﬁcantly increased [32, 33]. The softer electron neutrino ﬂavor spectra would be replaced by the harder muon/tau neutrino ﬂavor spectra in this region. Even if neutrino ﬂavor conversion did not occur deep in the stellar core— in or above the proto-neutron star after core bounce—and, therefore, was not a major factor in initiating the explosion, our ability to use the next Galactic supernova neutrino detection to improve supernova models, better understand the complex dynamics of core collapse supernovae, and cull fundamental nuclear and particle physics from it, would be severly compromised

Neutrino Transport in Core Collapse Supernovae

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if we did not have a way to predict the ﬂavor conversion that could occur between the source supernova and terrestrial detectors. The argument to consider neutrino mixing in the core collapse supernova context, particularly with an eye toward the explosion mechanism, has been made even more compelling recently with the discovery that neutrino mixing may occur deep in the stellar core after bounce at small values of δm2 [34, 35]. This mixing may arise as a result of the neutrino background in and above the proto-neutron star, which would be fundamentally diﬀerent than MSW mixing induced by the matter background, or vacuum mixing. Neutral-current neutrino–neutrino forward scattering increases the neutrino eﬀective mass, much as charged-current electron-neutrino scattering increases the electron-neutrino eﬀective mass in the MSW case. The net result may be near maximal mixing of neutrino ﬂavors in the environment of the protoneutron star after bounce [34, 35], with obvious ramiﬁcations for the supernova mechanism, nucleosynthesis, and neutrino signatures. Pantaleone [36,37] was the ﬁrst to recognize that a background neutrino, in a superposition of ﬂavor states, would lead to an oﬀ-diagonal refractive index in ﬂavor space if neutrino–neutrino interactions are taken into account. This idea was then further developed by Sigl and Raﬀelt [38]. Matter-enhanced neutrino mixing has been included in an approximate way in past (spherically symmetric) models of core collapse supernovae using a simple Landau-Zener prescription [32,33,39]. In such a prescription, the probability of mixing in the resonance region as a function of neutrino energy is computed, and the commensurate fraction of neutrinos is simply converted from one ﬂavor to the other. This prescription is easily implemented in spherically symmetric models, where a resonance region is a spherical shell. While the experimental evidence for neutrino mixing is now clear, and while a number of past exploratory studies have elucidated some of the possible ramiﬁcations neutrino mixing may have for core collapse supernova dynamics, the precise impact of such ﬂavor transformation remains to be determined. Neutrino mixing is a coherent, quantum mechanical phenomenon, unlike the incoherent collisional phenomena included in the Boltzmann kinetic equations discussed earlier. A more complete (quantum kinetic) treatment of neutrino transport in stellar cores beyond (classical) Boltzmann transport will be needed if we are to accurately and fully explore the impact of neutrino mixing on core collapse supernova dynamics.

7 Summary and Prospects Neutrino transport in core collapse supernovae is arguably the quintessential transport application. (1) The energy radiated in neutrinos in the course of a core collapse supernova is 10–100 times the supernova explosion energy. Errors in simulating the neutrino radiation ﬁeld can, therefore, be catastrophic in determining the explosion characteristics etched in the turbulent stellar

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core ﬂuid ﬂow to which the neutrinos are coupled. (2) Core collapse supernovae are still believed to be powered by the intense neutrino ﬂux emitted by the proto-neutron star, through neutrino heating of the postshock material. This neutrino heating is sensitive to the neutrino luminosities, spectra, and angular distributions—that is, to the neutrino distribution functions. (3) The gravitational ﬁelds in and around the proto-neutron star and the explosive stellar core ﬂow deviate signiﬁcantly from nonrelativistic values. This requires an extension of the discrete ordinates method to accommodate angular aberration and frequency shift owing to special and general relativistic eﬀects. (4) Neutrinos are leptons, and for zero neutrino mass at least, lepton number is conserved. Moreover, the deleptonization of the stellar core during collapse determines the initial conditions at shock formation: the initial shock radius and the energy imparted to the shock, which set the stage for the supernova dynamics that follows. Given that both lepton number and energy must be conserved, a ﬁnite diﬀerence representation of the original continuum Boltzmann equations that conserves both must be developed, which presents a signiﬁcant technical challenge, as we have demonstrated here. Despite the challenge, signiﬁcant progress has been made. First, a deep and precise understanding of the above issues has been achieved. Second, a conservative (in lepton number and energy)numerical method has been developed for general relativistic spherically symmetric ﬂows, and a clear path forward, albeit diﬃcult from a practical perspective, has been identiﬁed for the multidimensional case. Stellar core collapse simulations in full general relativity with Boltzmann neutrino transport have been performed, marking a milestone in supernova theory. The hope is we will be able to perform twodimensional special relativistic simulations with Boltzmann neutrino transport within the next two years. Despite the signiﬁcant advances made to date and the clear path forward in the case of zero neutrino masses, the experimental conﬁrmation that neutrinos have mass fundamentally alters the game. The classical Boltzmann kinetic equations are now only half the story. Coherent quantum mechanical mixing of neutrino ﬂavors must be included in the models along with the collisional physics included before. Thus, full quantum kinetic neutrino transport is now being developed in the spherically symmetric case and, based on the outcome of these future simulations, neutrino quantum kinetics in two- and ultimately three-dimensional simulations may need to be developed.

Acknowledgments A.M., C.Y.C., and O.E.B.M. are supported at the Oak Ridge National Laboratory, which is managed by UT-Battelle, LLC for the DOE under contract DE-AC05-00OR22725. A.M., C.Y.C., and S.W.B. are also supported in part by a DOE Scientiﬁc Discovery through Advanced Computing grant through

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the Oﬃce of Science Advanced Scientiﬁc Computing Research, High Energy Physics, and Nuclear Physics Programs.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

Wilson JR. in Numerical Astrophysics, ed. JM Centrella, JM LeBlanc, RL Bowers. Boston: Jones and Bartlett (1985) Bethe HA, Wilson JR. Astrophysical Journal 295:14 (1985) Janka HT. Astronomy and Astrophysics 368:527 (2001) Burrows A, Goshy J. Astrophysical Journal Letters 416:L75 (1993) Janka HT, M¨ uller E. Astronomy and Astrophysics 306:167 (1996) Messer OEB, Mezzacappa A, Bruenn SW, Guidry MW. Astrophysical Journal 507:353 (1998) Mezzacappa A, Calder AC, Bruenn SW, Blondin JM, Guidry MW, Strayer MR, Umar AS. Astrophysical Journal 495:911 (1998) Mezzacappa A, Liebend¨ orfer M, Messer OEB, Hix WR, Thielemann FK, Bruenn SW. Physical Review Letters, 86:1935 (2001) Castor J. Astrophysical Journal 178:779 (1972) Bruenn SW. Astrophysical Journal Supplement 58:771 (1985) Mezzacappa A, Bruenn SW. Astrophysical Journal 405:637 (1993) Mezzacappa A, Bruenn SW. Astrophysical Journal 410:740 (1993) Burrows A, Thompson T. in Stellar Collapse, ed. C Fryer. Dordrecht: Kluwer Academic Publishers (2004) Mezzacappa A, Bruenn SW. Astrophysical Journal 405:669 (1993) Lewis E, Miller W. in Computational Methods of Neutron Transport. New York: Wiley-Interscience (1984) Mezzacappa A, Messer OEB. Journal of Computational and Applied Mathematics109:281 (1998) Liebend¨ orfer M, Ph.D. thesis, University of Basel, Basel, Switzerland (2000) Liebend¨ orfer M, Messer OEB, Mezzacappa A, Bruenn SW, Cardall CY, Thielemann FK. Astrophysical Journal Supplement 150:263 (2004) Mezzacappa A, Liebend¨ orfer M, Cardall C, Messer O, Bruenn S. in Stellar Collapse, ed. C Fryer. Dordrecht: Kluwer Academic Publishers (2004) Cardall C, Mezzacappa A. Physical Review D 68:023006 (2003) Cardall C, Lentz E, Mezzacappa A. Physical Review D submitted (2005) Liebend¨ orfer M, Mezzacappa A, Thielemann F, Messer OE, Hix WR, Bruenn SW. Physical Review D 63:103004 (2001) Wilson JR, Mayle RW. Physics Reports 227:97 (1993) Swesty F, Lattimer J. Astrophysical Journal 425:195 (1994) Bruenn SW, DeNisco KR, Mezzacappa A. Astrophysical Journal 560:326 (2001) Swesty F, Myra E. in Open Issues in Core Collapse Supernova Theory, ed. A Mezzacappa, G Fuller. Singapore: World Scientiﬁc (2005) Cardall C. in Numerical Methods for Multidimensional Radiative Transfer Problems, ed. R Rannacher, R Wehrse. Springer Publishing Company (2004) Livne E, Burrows A, Walder R, Lichtenstadt I, Thompson T. Astrophysical Journal 609:277 (2004)

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[29] Rampp M, Janka HT. Astronomy and Astrophysics 396:361 (2002) [30] Mezzacappa A, Messer O. Astrophysical Journal submitted (2005) [31] Burrows A, Thompson T. in Stellar Collapse, ed. C Fryer. Dordrecht: Kluwer Academic Publishers (2004) [32] Fuller GM, Mayle R, Wilson JR, Schramm D. Astrophysical Journal 322:795 (1987) [33] Fuller GM, Mayle R, Meyer BS, Wilson JR. Astrophysical Journal 389:517 (1992) [34] Qian YZ, Fuller G. Physical Review D 52:656 (1995) [35] Fuller G, Qian YZ. Physical Review D submitted (2005) [36] Pantaleone J. Physics Letters B 287:128 (1992) [37] Pantaleone J. Physics Letters B 342:250 (1995) [38] Sigl G, Raﬀelt G. Nuclear Physics B 406:423 (1993) [39] Mezzacappa A, Bruenn SW. in The Identiﬁcation of Dark Matter, ed. N Spooner, V Kudryavtsev. Singapore: World Scientiﬁc (1999)

Discrete-Ordinates Methods for Radiative Transfer in the Non-Relativistic Stellar Regime Jim E. Morel Mail Stop D413, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 [email protected]

1 Introduction The purpose of this paper is to brieﬂy describe the application of discreteordinates methods to radiative transfer in the non-relativistic stellar regime. We consider the non-relativistic regime to be characterized by material velocities less than or equal to one percent of the speed of light. In most applications, the radiative transfer equations are coupled to the hydrodynamics equations. We ﬁrst describe a radiation-hydrodynamics model that consists of the Euler equations and the radiation transport equation together with approximations for material-motion eﬀects based upon the assumption of non-relativistic material velocities. For simplicity, we next focus on a model of radiative transfer in a static medium that is adequate for the purpose of describing numerical discretization and solution techniques for the transfer equation that can also be applied in the more general context of radiationhydrodynamics.

2 The Approximate Radiation-Hydrodynamics Model The hydrodynamics equations and the radiation moment equations can be expressed to O(v/c) as follows assuming a grey radiation transport approximation with the transfer equation cast in the Eulerian frame: • Conservation of ﬂuid mass: ∂t ρ + ∂i (ρvi ) = 0 ,

(1)

• Conservation of ﬂuid momentum: ∂t (ρvi ) + ∂j (ρvi vj ) + ∂i p =

σt vi F0,i − σa aT 4 − E0 , c c

(2)

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J.E. Morel

• Conservation of total ﬂuid energy: 1 2 1 2 ρv + ρe + ∂i ρv + ρe + p vi = ∂t 2 2 4 σt −cσa aT − E0 + vi F0,i , c

(3)

• Conservation of radiation momentum: 1 σt vi ∂t Fi + ∂j Pij = − F0,i + σa aT 4 − E0 , c2 c c

(4)

• Conservation of radiation energy: σt ∂t E + ∂i Fi = cσa aT 4 − E0 − vi F0,i , c

(5)

where ρ is the ﬂuid density, vi is component i of the ﬂuid velocity, c is the speed of light, p is the ﬂuid pressure, e is the speciﬁc internal energy of the ﬂuid, E is the radiation energy density, Fi is component i of the radiation ﬂux, Pij is element ij of the radiation pressure tensor, E0 is the comoving-frame radiation energy density: E0 = E −

2 vi (Fi − vi E − vj Pij ) , c2

(6)

F0,i is component i of the comoving-frame radiation ﬂux: F0,i = Fi − vi E − vj Pij ,

(7)

σt is the macroscopic total cross section, σa is the macroscopic absorption cross section, and a is the radiation constant. The source terms in (2) through (5) are expressed in terms of comoving-frame radiation variables. These comoving-frame variables are deﬁned in terms of Eulerian-frame radiation variables by (6) and (7). When the comoving-frame variables are eliminated from (2) through (5) using (6) and (7), certain terms of O(v 2 /c2 ) will appear. Even though (2) through (5) are only correct to O(v/c), these terms are not to be neglected because they eliminate spurious equilibrium solutions [1] and maintain consistency between the Eulerian-frame and comoving-frame momentum and energy sources. We consider the nonrelativistic limit to be characterized by v/c ≤ 0.01. We next describe a very simple approximate model for radiation-hydrodynamics with multifrequency radiative transfer in the nonrelativistic limit. The goal of this approximation is to maintain accuracy in an integral sense while avoiding the complexities of the equations that are correct to O(v/c). Since v/c is very small in the nonrelativistic limit, one might be tempted to neglect material motion corrections to the radiative transfer equation entirely. However, over the course of a calculation, this can result in a significant loss of energy conservation because the kinetic energy change in the

Discrete-Ordinates Methods for Radiative Transfer

71

ﬂuid due radiation momentum deposition is not removed from the radiation energy ﬁeld. Thus we use a simple approximation for nonrelativistic radiation-hydrodynamics that has the following properties: • Total energy and momentum are conserved. • Equilibrium solutions are preserved to O( vc ). • The equilibrium diﬀusion limit is preserved to O( vc ). In the nonrelativistic case, there is little diﬀerence between E and E0 , but the relative diﬀerence between F0,i and Fi can be arbitrarily large. This relative diﬀerence is essentially inﬁnite in the equilibrium state because F0,i = 0 while Fi = 43 vi E to O(v/c). Since the system is locally equilibrated in the equilibrium diﬀusion limit, it follows that one can expect a signiﬁcant relative diﬀerence between F0,i and Fi in this limit. Hence it is generally considered important to preserve the equilibrium diﬀusion limit in any approximate treatment of nonrelativistic radiation-hydrodynamics. See [1] for a formal asymptotic derivation of the non-relativistic equilibrium diﬀusion limit. The ﬁrst step in the derivation of our approximate model is modify the hydrodynamics equations and the grey radiation moment equations. Speciﬁcally, we make the following substitutions: σt vi σt F0,i − σa aT 4 − E0 ⇒ F0,i , (8a) c c c σt σt cσa aT 4 − E0 − vi F0,i ⇒ cσa aT 4 − E − vi F0,i , (8b) c c 4 F0,i = Fi − vi E − vj Pij ⇒ Fi − vi E , (8c) 3 The term neglected in (8a) is a purely relativistic term. Although the radiative transfer equation is always relativistic, the hydrodynamics equations are purely classical to O(v/c). Furthermore, this term is zero in the equilibrium diﬀusion limit. Thus it is reasonable to eliminate this term. In (8b) we replace E0 with E in the radiation energy source term. It can be seen from (6) that this should be a good approximation when v/c ≤ 0.01. Furthermore, E0 = E in the equilibrium diﬀusion limit. Finally, we substitute E/3 for Pi,j in (8c). This approximation is made simply to avoid calculating the radiation pressure tensor. This tensor plays a small role in a small term, and the substitution is exact in the equilibrium diﬀusion limit. The next step in the derivation is to replace the modiﬁed grey expressions with frequency-dependent expressions that are equivalent under the grey approximation. In particular, ∞ σt (ν) 4 vi σt F0,i ⇒ I(Ω, ν) Ωi − dΩ dν , (9a) c c 3 c 0 ∞4π

−→ σa (ν) B(ν) − I Ω , ν dΩ dν , cσa aT 4 − E ⇒ (9b) ∞0 4π −→ 4 vi I Ω ,ν F0,i ⇒ Ωi − dΩ dν , (9c) 3 c 0 4π

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−→ −→ where Ω is the photon direction vector, ν is the photon frequency, I Ω , ν hν −1 3 exp kT −1 is the Planck function is the angular intensity, B = 2hν c2 and k is the Boltzmann constant. Expression (9b) represents an exact substitution, and (9c) represents a substitution that is exact to O(v/c). However, (9a) is simply incorrect unless σt is independent of frequency. The transformations between the Eulerian and co-moving frames being used here are only correct for angle-energy-integrated quantities [2]. The transformation relating the co-moving frame ﬂux and the Eulerian frame ﬂux is eﬀectively being used for frequency-dependent ﬂuxes in (9a). Accounting for all of these substitutions, our approximate hydrodynamic equations and multifrequency radiation moment equations can be expressed as follows: ∂t ρ + ∂i (ρvi ) = 0 ,

∞

0

4π

∂t (ρvi ) + ∂j (ρvi vj ) + ∂i p = σt (ν) −→ 4 vi I Ω ,ν Ωi − dΩ dν , c 3 c

1 2 1 2 ρv + ρe + ∂i ρv + ρe + p vi = 2 2 ∞

−→ − σa (ν) B(ν) − I Ω , ν dΩ dν 0 4π ∞ σt (ν) −→ 4 v2 I Ω ,ν vi Ω i − + dΩ dν , c 3 c 4π 0

(10)

(11)

∂t

∞

−

0

4π

1 ∂t Fi + ∂j Pij = c2 σt (ν) −→ 4 vi I Ω ,ν Ωi − dΩ dν , c 3 c

∂t E + ∂ i F i =

−→ σa (ν) B(ν) − I Ω , ν dΩ dν 0 4π ∞ σt (ν) −→ 4 v2 I Ω ,ν vi Ω i − − dΩ dν . c 3 c 0 4π

∞

(12)

(13)

(14)

The radiative transfer equation for static media can be made consistent with these model equations by adding a P1 -type correction term to the right side of the equation:

Discrete-Ordinates Methods for Radiative Transfer

73

1 ∂I −→ −→ 1 1 3 + Ω · ∇ I + σt I = σs φ + σa B(T ) + Ce + Cf,i Ωi , (15) c ∂t 4π 4π 4π where

I

φ=

−→

Ω ,ν

dΩ ,

(16)

4π

Ce = − 4π

σt (ν) I c

−→

Ω ,ν

Cf,i =

σt (ν)I 4π

−→

4 v2 vi Ω i − 3 c

Ω ,ν

dΩ ,

4 vi dΩ . 3 c

(17)

(18)

Note that if we multiply (15) by Ωi /c and integrate over all directions and frequencies, we obtain (13), and if simply we integrate (15) over all directions and frequencies, we obtain (14). We stress that (15) is an approximate Eulerian frame transfer equation that is consistent with the other approximations that we have made in the hydrodynamic and radiation moment equations. The Eulerian frame transfer equation that is correct to O(v/c) is considerably more complicated than (15) [2]. Our approximate radiationhydrodynamics model is completely deﬁned by (10) through (18).

3 Discretization and Solution Techniques In this section, we discuss discretization and solution techniques for the radiative transfer equation. For simplicity, we consider only the radiative transfer equation for a static medium and neglect the hydrodynamics equations. In general, the techniques we discuss here can be applied within a radiationhydrodynamics calculation. A discussion of discretization and solution techniques for the full radiation-hydrodynamics equations is beyond the scope of this paper. The equations we consider consist of a radiative transfer equation: 1 1 ∂I −→ −→ + Ω · ∇ I + σt I = σs φ + σa B(T ) , c ∂t 4π and an equation for the material temperature T (r, t): ∞ ∂T = σa [ φ − 4πB(T ) ] dν , Cv ∂t 0

(19)

(20)

where Cv is the heat capacity and φ is deﬁned by (16). Equations (19) and (20) are discretized in direction using the standard Sn approximation [2,3]. This is basically collocation at a set of discrete directions that correspond to quadrature points on the unit sphere. The quadrature set

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J.E. Morel

associated with those discrete directions is also used to perform directional −→ integrations. For instance, let { Ω n , wn }M n=1 denote an M -point quadrature set, then (19) and (20) are directionally discretized as follows: 1 1 ∂In −→ −→ + Ω · ∇ In + σt In = σs φ + σa B(T ) , c ∂t 4π ∂T = Cv ∂t

∞

σa [ φ − 4πB(T ) ] dν ,

(21)

(22)

0

where In = I

−→ Ωn ,

(23)

and φ=

M

In wn .

(24)

n=1

Equations (19) and (20) are discretized in frequency using the standard multigroup method [2, 3]. The ﬁrst step in this method is partition the frequency domain into G disjoint contiguous intervals or groups:

[ν 12 , ν 32 ), . . . , [νG− 12 , νG+ 12 ) where the minimum frequency is ν 12 and the maximum frequency is νG+ 12 . The g’th group refers to the interval, [νg− 12 , νg+ 12 ). Next (19) and (20) are discretized as follows: 1 ∂Ig −→ −→ 1 + Ω · ∇ Ig + σt,g Ig = σs,g φg + σa,g Bg (T ) , g = 1, G, c ∂t 4π Cv

G ∂T = σa,g [ φg − 4πBg (T ) ] , ∂t

(25)

(26)

g =1

where

νg+ 1

2

Ig =

I(ν ) dν ,

(27)

B(ν ) dν ,

(28)

νg− 1 2

νg+ 1

2

Bg = νg− 1 2

and σg denotes any desired average of σ(ν) over group g: σg = σ(ν) ν∈[νg− 1 ,νg+ 1 ) . 2

(29)

2

The physical characteristics of radiative transfer problems place severe demands upon spatial discretization schemes. In particular, such schemes must:

Discrete-Ordinates Methods for Radiative Transfer

• • • • •

75

be at least second-order accurate, be highly damped in strongly absorbing media, be accurate in optically-thin regions, possess the thick diﬀusion limit, behave well with unresolved spatial boundary layers.

In general, lumped discontinuous spatial discretization schemes are required for radiative transfer. These schemes are usually either ﬁnite-element based [4, 5] or ﬁnite-volume based [6]. While lumping of the removal and source terms is well deﬁned, lumping of the gradient term is also required in multidimensional calculations [7]. There is no general procedure for gradient lumping and it must generally be performed on a case-by-case basis. Gradient lumping remains a research topic. The need to possess the thick diﬀusion limit places additional demands upon spatial discretization schemes. It is important to explain exactly what it means for a spatial discretization scheme to possess the thick diﬀusion-limit. In this limit, the transport solution is diﬀusive and the spatial cells can be arbitrarily thick with respect to a mean-free-path. The diﬀusion length is the spatial scalelength for diﬀusive solutions, and the diﬀusion length can be arbitrarily large with respect to a mean-free-path. Hence, a spatial mesh with cells that are thin with respect to a diﬀusion length but arbitrarily thick with respect to a mean-free-path might be intuitively expected to yield an accurate solution because the spatial variation of the exact solution is well resolved by the mesh. Indeed, a scheme that yields an accurate solution for highly diﬀusive problems whenever the variation of the exact diﬀusive solution is well resolved by the mesh is said to possess the thick diﬀusion limit. However, it is important to recognize that convergent schemes, i.e. schemes with truncation errors that go to zero as the mesh is reﬁned, do not necessarily possess the thick diﬀusion limit. A truncation error analysis yields no information on the behavior of a scheme in this limit. Such analyses indicate that accurate solutions will be obtained if the mesh cells are thin with respect to a mean-free-path. Spatial resolution on the order of a mean-free-path is completely impracical in highly diﬀusive problems. For instance, as a problem is made increasingly diﬀusive, the characteristic width of the system approaches a ﬁxed value when measured in diﬀusion lengths and becomes inﬁnite when measured in mean-free-paths. To determine the behavior of a spatial discretization scheme in the thick diﬀusion limit, one must perform a discrete asymptotic analysis [4, 7–10]. Boundary layers that are a few mean-free-paths thick can exist at the transitions between non-diﬀusive and diﬀusive regions. It can be very diﬃcult to resolve such layers. In general, an adaptive approach is required because the locations of these boundary layers usually vary dynamically. Furthermore, in a multifrequency calculation one may not be able to resolve a boundary layer on a mean-free-path basis for all frequencies. Thus it is desirable to have schemes that behave well if not highly accurately with unresolved spatial

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J.E. Morel

boundary layers. A discrete asymptotic analysis will reveal this behavior in addition to the behavior of the scheme on the interior of a diﬀusive region. The diﬀusion limit for radiative transfer is characterized by: I = B(T ) , ∂T −→ 4acT 3 −→ − ∇· (Cv + 4aT 3 ) ∇T =0 , ∂t 3σr

(30) (31)

where σr is the Rosseland-averaged total cross section:

∞

σr = 0

1 ∂B(ν) dν σt∗ (ν) ∂T

0

∞

∂B(ν) dν ∂T

!−1 .

(32)

Equations (19) and (20) are usually solved via a modiﬁed Newton’s method. To illustrate this technique, we ﬁrst implicitly diﬀerence these equations in time: −→ −→ 1 1 k+ 1 k+ 12 k+ 1 k+ 12 2 − I k− 2 I + + σ I 2 = · I Ω ∇ t c∆tk 1 1 k+ 12 k+ 1 σs φ 2 + σa B k+ 2 , (33) 4π k+ 1 ∞ 1 Cv 2 k+ 1 k+ 12 k+ 12 k+ 12 2 − T k− 2 T = φ dν . σ − 4πB a ∆tk 0

(34)

We let T ∗ denote the latest Newton iterate for the temperature. Then the linearized equations for the next Newton iteration are obtained by evaluating the material properties at T ∗ and linearly expanding the Planck function temperature dependence about T ∗ : B n+1 = B ∗ +

∂B ∗ k+ 1 T 2 − T∗ , ∂T

(35)

where a superscript “*” denotes a quantity evaluated at T ∗ . This is a modiﬁed Newton’s method because the contribution of the material properties to the Jacobian are neglected. With the expansion given in (35), the material temperature can be eliminated from the transport equation. Suppressing the temporal superscript “k + 12 ”, the linearized temporally-diﬀerenced transport equation can be expressed as: ∞ −→ −→ 1 ∗ 1 ∗ σ νχ I = φ + σa∗ (ν )φ(ν ) dν + ξ , (36) · I + σ Ω ∇ τ 4π s 4π 0 where:

Discrete-Ordinates Methods for Radiative Transfer

στ = σt + τ , 1 , τ= c∆tk ∞ ∗ ∗ σa (ν)4π ∂B∂T(ν) dν 0 ν = C∗ , ∞ ∗ v + 0 σa∗ (ν)4π ∂B∂T(ν) dν ∆tk

77

(37a) (37b) (37c)

∗

χ(ν) = ∞ 0

σa∗ (ν) ∂B∂T(ν) ∗

σa∗ (ν ) ∂B∂T(ν

)

dν

,

ξ = σa∗ B ∗ + τ ψ k− 2 − ∞ 1 1 C∗ νχ σa∗ (ν )4πB ∗ (ν ) dν + vk T k− 2 − T ∗ , 4π ∆t 0

(37d)

1

(37e)

and the material temperature is given by ∞ T k+ 2 = T ∗ + 1

0

1 C∗ σa∗ (ν) [φ(ν) − 4πB ∗ (ν)] dν + ∆tvk T k− 2 − T ∗ . (38) ∞ ∗ Cv∗ + 0 σa∗ (ν)4π ∂B∂T(ν) dν ∆tk

Equation (38) is used to calculate the temperatures after the linearized transport equation has been solved. We note that (36) has the form of the steadystate neutron transport equation with σa playing the role of the ﬁssion cross section, ν playing the role of the number of neutrons per ﬁssion, χ playing the role of the ﬁssion spectrum, and ξ playing the role of a inhomogeneous source. The basic technique for solving (36) is the source iteration technique. This method is based upon the fact that all coupling between direction and frequency occurs on the right side of the equation. The operator on the left side of (36) can be discretized in such a way that a block lower-triangular coeﬃcient matrix is obtained for each direction and frequency group where the unknowns in each block are those associated with a single spatial cell. Such matrices are easily inverted in a direct manner. The inversion process for each discrete direction and frequency starts where radiation enters the mesh, proceeds across the mesh, and ends where radiation leaves the mesh. Hence the inversion process is referred to as a sweep. The iterative process can be represented in a simpliﬁed form as follows: ∞ −→ −→ 1 ∗ 1 +1 σs φ + νχ + στ∗ I +1 = σa∗ (ν )φ (ν ) dν + ξ , (39) Ω · ∇I 4π 4π 0 where is the iteration index. The actual iterative process is nested such that the inner iterations converge the scattering sources for each frequency group while the absoprtion rate is held ﬁxed, and the outer iterations converge the absorption rate. The inner source iteration process for group g can be expressed as

78

J.E. Morel −→ −→

+1 ∗ +1 Ω · ∇ Igi + στ,g Igi =

G 1 ∗ i 1 ∗ σs,g φg + νχg σa,g φg + ξg , 4π 4π

(40)

g =1

where i is the inner iteration index. The outer iteration process can be expressed as follows: −→ −→

+1 ∗ +1 Ω · ∇ Igo + στ,g Igo =

G 1 ∗ o +1 1 o ∗ σs,g φg νχg + σa,g φg + ξg , g = 1, G, 4π 4π

(41)

g =1

where o is the outer iteration index. The spectral radius for the inner iteration process (the iteration on a within-group scattering source) is the usual scattering ratio, ρi = σs∗ /σt∗ , and the spectral radius for the outer iteration process (the iteration on the absorption rate) is given by ∞ χσa∗ dν . (42) ρo = ν σa∗ + τ 0 The spectral radii observed in a practical problem are the maximum values of ρi and ρo , respectively, evaluated over the problem domain. Any region in which ρ0 is close to unity will be diﬀusive. If desired, one can dramatically reduce the spectral radius for the inner iteration process by roughly a factor of 4 via an acceleration technique known as diﬀusion-synthetic acceleration [11]. −→ −→ i + 1 Ω · ∇ Ig 2

+ 12

∗ + στ,g Igi

=

G 1 ∗ i 1 ∗ σs,g φg + νχg σa,g φg + ξg , 4π 4π

(43)

g =1

−→

−∇·

1 −→ i + 12 +1 +1 + σa,g δφgi 2 = σs,g φgi 2 − φgi , ∇ δφg 3σt,g + 12

φgi +1 = φgi

+ 12

+ δφgi

.

(44)

(45)

Our experience is that acceleration of the inner iterations is not usually required. On the other hand, acceleration of the outer iterations is often essential. The spectral radius for the outer iterations is often very close to unity, making it essential to accelerate convergence of the outer iterations. The spectral radius approaches unity in the limit of strong ﬂuid-radiation coupling, i.e., small heat capacity and large absorption cross section. The

Discrete-Ordinates Methods for Radiative Transfer

79

technique used to accelerate the outer iterations is a variation on diﬀusionsynthetic acceleration called the linear multifrequency-grey method [12, 13]. An accelerated outer iteration can be represented as follows: −→ −→ o + 1 Ω · ∇ Ig 2

1 ∗ σ φg 4π s,g

o + 12

−→

+

+ 12

∗ + στ,g Igo

=

1 νχg Rao + ξg , g = 1, G, 4π

(46)

−→

− ∇ ·D ∇ δΦ(o +1/2) + [τ + (1 − ν) σa ] δΦ(o +1/2) +1 = ν Rao 2 − Rao ,

(47)

Ra(o +1) = Ra(o +1/2) + σa δΦ(o +1/2) .

(48)

where G

Ra =

∗ σa,g φg ,

(49)

G 1 g , ∗ 3 στ,g

(50)

g =1

D =

g =1

σa =

G

∗ g σa,g ,

(51)

g =1

g =

χg ∗ +τ σa,g G χg g =1 σ ∗ +τ a,g

.

(52)

The linear multifrequency-grey method is generally very eﬀective. It was thought for many years that diﬀusion-synthetic acceleration and its variants were unconditionally eﬀective. However, it was eventually discovered that large spatial discontinuities in the cross sections can signiﬁcantly degrade the eﬀectiveness of diﬀusion-synthetic acceleration [14]. It is reasonable to assume that its variants suﬀer the same deﬁciency. It has recently been shown that solving the neutron transport equation with a preconditioned Krylov technique and recasting diﬀusion-synthetic acceleration as a preconditioner rather than an acceleration scheme results in an apparently unconditionally eﬀective solution technique [15]. Krylov methods are beginning to have an enormous impact upon solution techniques for the transport equation. The traditional accelerated solution

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techniques represent two-level or two-grid methods. As such, they require diﬀusion discretizations that are consistent with the transport discretizations. This generally leads to non-standard diﬀusion discretizations that are diﬃcult to solve. Furthermore, high-frequency error ampliﬁcation for just a single mode can destroy the eﬀectivness of such solution techniques. Preconditioned Krylov methods with traditional acceleration techniques recast as preconditioners are far more forgiving than traditional accelerated solution techniques. We expect to see great progress made in numerical radiative transfer and radiation-hydrodynamics via Krylov methods.

References 1. Lowrie, R.B., Morel, J.E., Hittinger, J.A.: The coupling of radiation and hydrodynamics. Astrophysical J., 521, 432 (1999) 2. Mihalas, D.M., Mihalas, B.W.: Foundations of Radiation Hydrodynamics. Oxford Press (1984) 3. Lewis, E.E., Miller, W.F., Jr.: Computational Methods of Neutron Transport. American Nuclear Society, La Grange Park, Illinois (1993) 4. Morel, J.E., Wareing, T.A., Smith, K.: A linear-discontinuous spatial diﬀerencing scheme for Sn radiative transfer calculations. J. Comp. Phys., 128, 445 (1996) 5. Morel, J.E., Gonzalez-Aller, A., Warsa, J.S.: A lumped discontinuous ﬁniteelement spatial discretization for Sn triangular-mesh calculations in R − Z geometry. To appear in the proceedings of MC2005: an International Meeting on Mathematics and Computation, Supercomputing, Reactor Physics, and Nuclear and Biological Applications, Avignon, France, September 12–15, 2005 6. Adams, M.L.: A subcell balance method for radiative transfer on arbitrary spatial grids. Transport Theory Stat. Phys., 26, Nos. 4 & 5, 385 (1997) 7. M.L. Adams: Discontinuous ﬁnite element transport solutions in thick diﬀusive problems. Nucl. Sci. Eng., 137, 298 (2001) 8. Larsen, E.W., Morel, J.E., Miller, W.F., Jr.: Asymptotic solutions of numerical transport problems in optically thick, diﬀusive regimes. J. Comp. Phys., 69, 283 (1987) 9. Larsen, E.W., Morel, J. E.: Asymptotic solutions of numerical transport problems in optically thick diﬀusive regimes II. J. Comp. Phys., 83, 212 (1989) 10. Adams, M.L., Nowak, P.F.: Asymptotic analysis of a computational method for time- and frequency-dependent radiative transfer. J. Comput. Phys., 146, 366 (1998) 11. Adams, M.L., Larsen, M.L.: Fast iterative methods for discrete-ordinates particle transport calculations. Prog. Nucl. Energy, 40, 3 (2002) (REVIEW) 12. Morel, J.E., Larsen, E.W., Matzen, M.K.: A Synthetic acceleration scheme for radiative diﬀusion calculations. J. Quant. Spectro. Radiat. Transfer, 34, 243 (1985) 13. Larsen, E.W.: A grey transport acceleration method for thermal radiative transfer problems. J. Comp. Phys., 78, 459 (1988) 14. Azmy, Y.Y.: Impossibility of unconditional stability and robustness of diﬀusive acceleration schemes. Proc. ANS Topical Meeting, Radiation Protection and Shielding, Nashville, TN, April 19–23, 1998, Vol. 1, p. 480 (1998)

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15. Warsa, James S., Wareing, Todd A., Morel, Jim E.: Krylov iterative methods and the degraded eﬀectiveness of diﬀusion synthetic acceleration for multidimensional Sn calculations in problems with material discontinuities. Nucl. Sci. Eng., 147, 218 (2004)

Part II

Atmospheric Science, Oceanography, and Plant Canopies

Eﬀective Propagation Kernels in Structured Media with Broad Spatial Correlations, Illustration with Large-Scale Transport of Solar Photons Through Cloudy Atmospheres Anthony B. Davis Los Alamos National Laboratory, Space & Remote Sensing Sciences Group ISR-2, P.O. Box 1663, Los Alamos, New Mexico, USA [email protected] Summary. It is argued that, to directly target the mean ﬂuxes through a structured medium with spatial correlations over a signiﬁcant range of scales that includes the mean-free-path, one can use an eﬀective propagation kernel that will necessarily be sub-exponential. We come to this conclusion using both standard transport theory for variable media and a point-process approach developed recently by A. Kostinski. The ramiﬁcations of this ﬁnding for multiple scattering and eﬀective medium theory are examined. Finally, we describe a novel one-dimensional transport theory with asymptotically power-law propagation kernels and use it to shed new light onto recent observations of solar photon pathlength in the Earth’s cloudy atmosphere.

1 Introduction and Overview We start with a compact (operator-based) formulation of the monokinetic linear transport problem in higher-dimensions of suﬃcient generality for our present needs. Phase-space density (times velocity c), denoted I(x, Ω), is called “speciﬁc intensity” or “radiance” in the parlance of radiative transfer (RT) theory. It is determined by a one-group linear Boltzman equation [1] LI = SI + Q

(1)

L = Ω • ∇ + σ(x)

(2)

where describes the advection and extinction of particle beams while S = σs (x) p(x, Ω → Ω)[·]dΩ

(3)

4π

describes the volume scattering process, and Q(x, Ω) is the volume source term. We will call this the “3D RT equation” (here, in its integro-diﬀerential

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incarnation). We also need boundary conditions (BCs) which can often be taken as vacuum (no incoming radiance) or reﬂective (which is just like scattering but at a surface). Most numerical solutions of this problem use, in one way or another, the equivalent integral equation I = KI + I0

(4)

that naturally incorporates the BCs; here K = L−1 S

(5)

I0 = L−1 Q

(6)

is the transport kernel while

represents the uncollided particles. The integral operator L−1 is the main focus of this paper; we will call it the propagation kernel. Formally, one can write the solution of (4) as I=

∞ I0 = Kn I0 , 1 − K n=0

(7)

the well-known Neumann series. In this paper however, we are not interested in a solutions of fully speciﬁed (deterministic) 3D RT problems. Rather, we are interested in the mean and other statistical properties of solutions averaged over many realizations of the spatial variability: “mean-ﬁeld” RT theory where the parameters of interest are also means, variances, covariances, correlation functions, and so on, evaluated for the optical properties of the media. Typically, assumptions are made in such a way that these ensemble averages at a point will be invariant under a relevant class of translations and rotations: statistical homogeneity and isotropy prevails. These point transformations need not be fully 3D; for instance, they can be only in the horizontal plane. So what we call here a mean-ﬁeld RT solution is often thought of as a model for large-scale averages in the statistically invariant spatial dimensions. The most venerable approach to this challenging problem was pioneered1 by the regretted transport theoretician extraordinaire G. C. Pomraning (1936–1999) along with his students and coworkers. The interested reader is referred to his deﬁnitive text Linear Kinetic Theory and Particle Transport in Stochastic Mixtures [3]. In this framework, tractable problems with scattering are limited to Markovian binary media: two types of material and uniform probabilities of crossing a boundary per unit of length along any 1

This is on the US side of the Cold War. In the former USSR, parallel developments happened starting, as far as I know, with Avaste and Vainikko’s [2] investigation of broken clouds and continue to this day.

Eﬀective Propagation Kernels in Structured Media

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beam. There are only two structural parameters: the volume mixing ratio and the characteristic scale of the clumps of the less abundant material. One ends up here with the likes of two integro-diﬀerential equations for uniform media to solve simultaneously because they are coupled by linear exchange terms. Other statistical RT models, inspired by the phenomenology of turbulence, have been based on scale-wise expansions (as in Fourier space) and closure methods, e.g., Stephens [4]. Yet other methods simply prescribe averaging over solutions of homogeneous problems using 1-point statistics and thus foregoing any impact of spatial correlations, e.g., Barker [5]. For further examples in atmospheric science, primarily motivated by large-scale radiation budget considerations in climate studies, see the recent survey by Barker and Davis [6]. A very desirable outcome of a statistical RT model from any of the above approaches is “homogenization:” compute eﬀective optical medium properties that can be applied to a homogeneous RT problem, but that somehow to capture the eﬀects of the unresolved variability. For instance, Graziani and Slone [7] seek an eﬀective mean free path dependent on structural parameters of Markovian and other media in order to treat them with a single RT equation. An example in cloud radiation is Cahalan’s [32] rescaled optical depth model which is based on straightforward 1-point/single-variable averaging of solutions of the standard 1D RT problem when cloud optical depth is positively skewed (e.g., lognormal-like). Another notable example in atmospheric science is Cairns et al.’s [9] renormalization of all local optical properties which is grounded in a sophisticated Green function analysis.2 of the 3D RT problem and also adapts powerful techniques from contemporary statistical physics. A common trait of the above methods that care at all about covariances [4] or spatial correlations [9–11] is the (often implicit) requirement that the variability scales and the averaging scales be well-separated. In cloudy atmospheres however, and probably also in many other structured media dominated by turbulent reactive ﬂows, spatial correlations are “long-range” (typically power-law wavenumber spectra are observed). It is therefore not obvious that we can observe scales where the statistics can be deemed homogeneous and that one can really talk about variability conﬁned to a certain range of scales. Although Cairns et al. [9] correctly averages the iterated transport kernel Kn in (7) with the spatial (L−1 ) and angular (S) aspects inherently intertwined in (L−1 S)n , the remainder of this paper is essentially an attempt at treating them separately and relating this approximation to the general idea of diﬀusion. There is little theoretical justiﬁcation for this separation beyond connection with L´evy walks, a popular model in both statistical physics [12] 2

Uses the solutions G of (1) when Q is a roaming Dirac δ source; formally, we have G = (L − S)−1 .

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A.B. Davis

and in stochastic processes [13]. Nonetheless, recent observations of multiply scattered sunlight in the Earth’s cloudy atmosphere [14, 15] support this approximation that we revisit further on. In the following Section, we revisit the basic physics of extinction (i.e., σ) and scattering (i.e., σs p(Ω → Ω)), introducing some useful notation in the process. We then examine in Sect. 3 the general properties of averaged freepath distributions in random but spatially-correlated 3D media as determined by L−1 where the angular brackets denote ensemble/spatial averages; we do this from two diﬀerent standpoints, RT theory and point processes. This enables us to state a fundamental limitation of eﬀective medium theory that may or may not impact practical implementations. We also look at the form of L−1 for the speciﬁc kind of variability observed in the Earth’s cloudy atmosphere. In Sect. 4, we return to multiple scattering transport by ﬁrst showing, under quite general conditions, that angular diﬀusion makes iterates of S very close to projections onto the space of isotropic functions U (x) — physically, photon density— in ﬁnite time; it is then relatively easy to obtain the asymptotic behavior of Kn ≈ L−1 n for both inﬁnite and ﬁnite media, assuming complete angular redistribution. In Sect. 5, we display some recently published absorption-based diagnostics of solar photon transport through real 3D clouds and propose a simple 1D RT model based on the relevant family of propagation kernels L−1 that explains these observations. We oﬀer some concluding remarks in Sect. 6. I have already started and will continue to use the language of radiative transfer (photons, extinction, optics, etc.), and furthermore in the frame of atmospheric science. Much of the following is nonetheless applicable to neutron or neutrino transport in engineering or astrophysical systems.

2 Extinction and Scattering Revisited, and Some Notations Introduced 2.1 The Extinction Coeﬃcient The simplest description of matter-radiation interaction is photon depletion when a narrow collimated beam crosses an optical medium, cf. left-hand side of Fig. 1 where it is assumed that δI ≥ 0. We have basically expressed here the ﬂux-divergence theorem for an “elementary” kinetic volume of length δ. Along the horizontal cylinder the net transport is 0; to the left, there is an in-ﬂux I; to the right, an out-ﬂux I − δI. So the divergence integral is simply the diﬀerence from left to right. Operationally, we have δI ∝ I × δ and the proportionality constant, deﬁned as

(8)

Eﬀective Propagation Kernels in Structured Media

89

Fig. 1. Mechanism of optical extinction by a dilute medium of scattering/absorbing particles: (a) elementary cylindrical volume, (b) photon beam’s view down the axis of the cylinder when a variety of cloud droplet sizes are present. Adapted from Fig. 3.7 in [16]

σ = lim

δ→0

δI/I . . . in m−1 , δ

(9)

is the extinction coeﬃcient or simply “extinction.” What is the detailed mechanism of extinction? This is about a population of streaming photons colliding with an essentially static population of massive particles. So all we have to do is estimate the number of particles in the sample volume in Fig. 1: δN = nδAδ. Multiplying this by the (mean) cross-section s and dividing by δA yields the element of probability for an interaction which, by deﬁnition (9), is σδ, and should be 1. Thus, as some position (x), we have σ(x) = n(x) × s . (10) In this sense, extinction is the interaction cross-section per unit of volume, equivalently, the probability of collision per unit of length. Note that the medium as to be suﬃciently dilute (n−1/3 s1/2 ); otherwise (tightly-packed particles), it is wrong to think that a small distance δ in sn × δ makes it a small element of probability. In (10), the cross-section s is for any kind of interaction by any kind of particle. In radiative transfer, it is naturally partitioned between absorption (photon destruction) and scattering (photon re-direction).3 This partition carries over immediately to the transport coeﬃcients: σ(x) = σa (x) + σs (x) .

(11)

In environmental and astrophysical applications, there is often a mixture of optically relevant material particles in the kinetic volume in terms of size (cf. right-hand side of Fig. 1), shape, chemistry, etc. An important quantity in the following is single-scattering albedo: 3

In neutronics, there is a third elementary process: multiplication, which is basically an “anti-absorption;” formally, we model this process with σa < 0 (an isotropic source of particles that is ∝ I). In optics, there is stimulated emission which can dominate in NTLE situations, such as in laser cavities. Because of the tight correlation between the incoming and outgoing photon’s directions, this is best modeled as a negative extinction in (9).

90

A.B. Davis 0 (x)

= σs (x)/σ(x) .

(12)

We can then express the absorption coeﬃcient as σa (x) = [1 −

0 (x)] σ(x)

,

(13)

where 1 − 0 (x) is sometimes called the “co-albedo” for single scattering. As long as the overarching condition of weak dilution is veriﬁed, we can use the linearity of (10)–(11) to simply sum the cross-sections s(s) and s(a) (for scattering and absorption respectively) of diﬀerent types of photonintercepting particles weighted by the associated densities: (a,s) si × ni (x) . (14) σa,s (x) = species i

A variation of this principle is when the “species” are deﬁned by a continuously varying parameter such as size r: ∞ s(a,s) (r) ×

σa,s (x) =

dn (r; x) dr . dr

(15)

0

For spherical particles (such as cloud droplets) of radius r, Mie scattering theory speciﬁes the eﬃciency factor Q(a,s) (2πr/λ) that appears in s(a,s) (r) = πr2 × Q(a,s) (2πr/λ) where λ is the given wavelength, cf. Deirmendjian [17] for standard size spectra dn/dr used in atmospheric optics. Interesting questions arise when a particle type or size can not be considered uniformly distributed at any scale, i.e., that it is impossible to deﬁne a density ni (x) or dn/dr(r; x). I refer to Knyazikhin et al. [18] for an investigation of more general formulations of the extinction problem that capture this case. In the next section, we will proceed to investigate spatial variability eﬀects under the weak assumption that at least the dominant types of particles have well-deﬁned local densities. 2.2 The Scattering Phase Function Figure 2 illustrates the redistribution of radiant energy between diﬀerent beams through scattering. Our goal is to estimate the element of scattered ﬂux δFs . It is surely proportional to the small solid angle into which the scattering occurs δΩ and to the small loss of ﬂux δF0 incurred when the incoming photons cross the sample volume (conditional to scattering rather than absorption); the latter term is equal to the scattering coeﬃcient times the small length δ. In summary, we have δFs ∝ δF0 δΩ = F0 σs δ × δΩ .

(16)

Eﬀective Propagation Kernels in Structured Media

91

Fig. 2. Schematic of scattered radiance. Adapted from Fig. 3.8 in [16]

We deﬁne the scattering phase function as p(Ω0 → Ω) =

lim

δ,δΩ→0

δFs ... in sr−1 δF0 × δΩ

(17)

which will generally depend on position x as well as one or two angular variables: the scattering angle θs = cos−1 (Ω0 •Ω) and, possibly, an azimuthal scattering angle as well. As an example that we will return to latter on, consider everywhere isotropic scattering: p(x, Ω0 → Ω) ≡ 1/4π .

(18)

As for extinction, let us have a closer look at the mechanics of scattering at the individual collision level.4 To isolate the inherent property of the scattering medium, we compute lim

δs,δΩ→0

δFs /F0 ds = σs (x)p(x, Ω0 → Ω) = n(x) × (x, Ω0 → Ω) δs × δΩ dΩ

(19)

where the last expression is obtained by straightforward generalization of (10) to diﬀerential cross-sections. As for extinction in (15), diﬀerential crosssection should be averaged over the population of particles in the sample volume in terms of size and/or type. 4 In the context of RT, it is technically incorrect to think of a photon as being scattered since its identity, as an eigen-mode of the quantized EM equations, is deﬁned in particular by its direction of propagation. However, a quantum hc/λ of radiant energy is transfered between modes at each elastic scattering, and I will continue the tradition of calling this the scattering of “a photon.” In RT per se, “photon” is in fact short for photon beam since we are in the classic limit of geometric optics where all distances of interest are much larger than the wavelength λ. Physical (wave-theoretical) optics are required only to compute transport coeﬃcients at the individual cross-section level, which can indeed be on the order of the wavelength squared. The weak dilution requirement and the usual assumption of uncorrelated random positions then ensures that we can add intensities (energies) rather than amplitudes and phases to model the behavior of photon beams.

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By radiant energy conservation, we have p(x, Ω0 → Ω)dΩ ≡ 1, ∀x, Ω .

(20)

4π

By optical reciprocity, we have p(x, Ω → Ω0 ) = p(x, Ω0 → Ω), hence p(x, Ω0 → Ω)dΩ0 ≡ 1, ∀x, Ω0 .

(21)

4π

In the remainder of this section, I will assume the spatial variability the phase function is implicit, and drop x from its arguments. In most atmospheric applications (high-altitude cirrus clouds made of ice-crystals being the notable exception), it is reasonable to assume that scattering is axi-symmetric around the incoming beam. Hence, p(Ω0 → Ω) ≡ p(Ω0 • Ω) = p(µs ) .

(22)

where the scattering angle θs is given by µs = cos θs = Ω0 • Ω. This enables an expansion of the phase function in spherical harmonics (eigen-functions of S) without the complication of azimuthal terms: p(µs ) =

1 (2n + 1)ηn Pn (µs ) , 4π

(23)

n≥0

where Pn (x) is th nth-order Legendre polynomial. These coeﬃcients are computed from +1 (24) ηn = 2π Pn (µs )p(µs )dµs . −1

Speciﬁc values of the Legendre polynomials can be obtained eﬃciently by recursion but their analytical expressions are best derived from the generating function [19] 1 Φ(x, z) = Pn (x)z n = √ , (25) 1 − 2xz + z2 n≥0 for |x| and |z| < 1. Using 1 Pn (x) = n!

∂ ∂z

n

Φ(x, z)

,

(26)

z=0

we ﬁnd P0 (x) = 1, P1 (x) = x, P2 (x) = (3x2 − 1)/2, and so on. The orthogonality of the Legendre polynomials, +1 Pn (x)Pn (x)dx = −1

δnn , n + 1/2

(27)

Eﬀective Propagation Kernels in Structured Media

93

where δnn is the Kronecker symbol, ensures diagonalization of the scattering operator S in spherical-harmonic space. It is therefore little surprise that virtually all grid-based (non-Monte Carlo) numerical solutions of the 3D RT equation exploit spherical harmonics. We have η0 = 1 by conservation for any phase function, and the only non-vanishing coeﬃcient for isotropic scattering in (18). Also of considerable interest is +1 (28) g = η1 = 2π µs p(µs )dµs , −1

called the “asymmetry factor” or simply the mean cosine of the scattering angle. This last description of g correctly presents the phase function as a probability density function (PDF) in direction space, and this is indeed how p(µs ) is used in Monte Carlo schemes. Any deviation of the phase function from isotropy corresponds to a directional correlation (in principle, of either sign) between incident and scattered photons. I will be introducing several kinds of averages in upcoming section. So those that concern photon scattering and propagation events deserve a special notation, which I borrow from the probability literature: E(·) which stands for (mathematical) expectation of the random variable in the argument. Thus, we can recast the asymmetry factor in (28) as (29) g = E(Ω • Ω0 ) = Ω • Ω0 dP (Ω|Ω0 ) 4π

where dP (Ω|Ω0 ) = p(Ω0 • Ω)dΩ. The “|” delimiter in a PDF separates the random variable from the given (ﬁxed) quantities that need to be highlighted. 2.3 Henyey–Greenstein Models for the Phase Function A very popular 1-parameter model for the single-scattering process in atmospheric optics and elsewhere is the Henyey–Greenstein [20] phase function 1 1 − g2 . (30) p(µs ; g) = 4π (1 + g 2 − 2gµs )3/2 I will use the delimiter “;” to separate, as needed, variables from parameters in argument lists. I will also use subscripts, e.g., pg (µs ) in this case. In spherical harmonics, the Henyey–Greenstein model (30) yields ηn (g) = g n . (31) ∞ n Indeed, 4πp(x; z) is identical to = 2∂Φ(x, z)/∂z + n=0 (2n + 1)Pn (x)z Φ(x, z) from (25); the above coeﬃcients then follow by comparison with (23). This Legendre decomposition is used in virtually all numerical implementations of the scattering operator S, except Monte Carlo (random quadrature)

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schemes where the simulation of the Markov chain requires the generation of random values of µs with the probability measure implicit in (30). In the case of the Henyey–Greenstein phase function, one can use the method of inverting the cumulative probability because the running integral of (30) with respect to µs has a simple expression. Speciﬁcally, one can generate random values of µs with 1 − g2 1 µs = 1 + g2 − , |g| ≤ 1, g = 0 , (32) 2g 1 + g(1 − 2ξ) where ξ is a (pseudo-)random deviate uniformly distributed on the interval (0,1). When g → 0, L’Hˆ opital’s rule in (32) yields the expected result for isotropic scattering: µs = 1 − 2ξ. The azimuthal scattering angle is selected randomly: φs = 2πξ. Sometimes, it is useful to compute RT in reduced dimensionality. For instance, “3D” RT eﬀects have been successfully investigated in 2D: one direction for the mean ﬂux (z) and one other, at right angles, for the spatial variability (x). It is not necessary in this case to propagate photons in the 3rd dimension and the general representation for radiance is I(x, z, θ), where θ is still measured away from the z-axis. The small price to pay for this conceptual simpliﬁcation is that all the geometrical units used so far must be adjusted: • ﬂuxes are now in s−1 m−1 ; • radiances are ﬂuxes/rad since solid angles and direction (θ, φ) are now reduced to regular angles and just θ; • extinction, scattering and absorption coeﬃcients are still in m−1 but, if needed, “cross-sections” are in meters while densities are in m−2 ; and ﬁnally, • phase functions are in rad−1 . To support studies in this kind of RT in “Flatland,” Davis et al. [21] proposed a 2D version of the Henyey–Greenstein phase function: 1 δFs /F0 1 − g2 = p(µs ; g) = lim (33) δτs ,δθ→0 δτs × δθ 2π 1 + g 2 − 2gµs where τs is the element of (scattering) optical distance across the small 2D “volume” where the scattering occurs (think of a 2D version of Fig. 2). Note that the polar diagram of this 2D phase function, as a function of θs = cos−1 µs , is an ellipse of eccentricity 2g/(1 + g 2 ) with the scattering particle at a focus. The asymmetry factor g retains the same meaning in 2D it was given in 3D through (29): mean cosine of the scattering angle. The 2D analog of the spherical harmonic decomposition (23)–(24) on the 3D unit sphere Ξ is a cosine Fourier series analysis on the interval (−π, +π] where θs takes its values. Speciﬁcally,

Eﬀective Propagation Kernels in Structured Media

p(µs ) =

1 ηn cos nθs π 1 + δ0n

95

(34)

n≥0

where

+π p(θs ) cos nθs dθs . ηn =

(35)

−π

For the 2D Henyey–Greenstein phase function in (33), it can be shown that the nth Fourier coeﬃcient in (35) is ηn (g) = g n , just as in (31) for the 3D case. While this is useful in a numerical implementation of S in 2D for a spatiallygridded representation of the 2D radiance, a 2D Monte Carlo scheme would use

π 1−g −1 tan (1 − 2ξ) , |g| ≤ 1 , (36) θs = 2 tan 1+g 2 in lieu of (32). The ultimate dimensionality reduction in RT is when the light particles can travel only on one axis, say, up or down. Here, ﬂuxes (and/or radiances) are simply in s−1 , the extinction coeﬃcient (like density) is in m−1 while cross-sections and the phase-function are dimensionless. Indeed, scattering in 1D amounts to either no change of direction of travel (µs = +1) or reversed direction (µs = −1). In this case, the angular PDFs in (30) or (33) are reduced to a Bernoulli trial: δFs /F0 µs = +1, with probability (1 + g)/2 = . (37) p(µs ; g) = lim µs = −1, with probability (1 − g)/2 δτs →0 δτs In other words, p(±1; g) = (1 ± g)/2 .

(38)

This is in fact the most general phase function in 1D. As in higher dimensions, g is still the mean of µs , the cosine of the scattering angle, even though it can take only 2 values. RT in 1D is of course not a framework for investigating 3D eﬀects. 1D RT is however an analytically tractable benchmark, at least when 0 and g are constant in the 1D medium.5 In this model, the two 1D “radiances” are readily identiﬁed with 3D hemispherical ﬂuxes in the upper- and lower halves (Ξ± ) of Ξ. It dates back at least to Schuster’s seminal 1905 paper [22]. 1D RT has undeniable tutorial value. Amazingly, it is still used in many global climate models in one or another of its evolved but mathematically equivalent forms [23]. In 1D, 2D or 3D, the extremal values of g, +1 and −1 are only of academic interest. They indeed lead to Dirac δ-functions in the respective scattering-angle spaces: (µs , φs ) ∈ Ξ, θs ∈ (−π, +π], µs ∈ {−1, +1}. For g = +1 scattering per se is defeated and the RT problem collapses onto the pure extinction/absorption/emission problem, which is 1D (beam-by-beam). For g = −1, we obtain a somewhat pathological 1D (beam-by-beam) transport model where propagation direction is switched at every scattering event. 5

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3 Propagation 3.1 Direct Transmission in 3D Media as a Problem in Stochastic ODEs There is an apparently elementary calculus problem posed in (9), namely, dI/I = d ln I = −σ(x)ds ,

(39)

which is easily solved by a change of variables dτ (s) = σ(x)ds. We thus deﬁne optical distance as the running integral of σ along the beam Ω0 from some starting point x0 : s τ (x0 → x = x0 + Ω0 s) = τ (s; x0 , Ω0 ) =

σ(x0 + Ω0 s )ds .

(40)

0

To address the problem of cumulative extinction, we will consider (x0 , Ω0 ) to be ﬁxed parameters. The solution of the ordinary diﬀerential equation (ODE) in (39) is therefore I(s; x0 , Ω0 ) = I(0; x0 , Ω0 ) exp[−τ (s; x0 , Ω0 )] .

(41)

This is the well-known exponential law of direct transmission with respect to optical distance. Consider a uniform medium where optical distance is simply τ (x0 → x = x0 + Ω0 s) = σs, ∀x0 , Ω0 ;

(42)

I(s) = I0 exp(−σs) .

(43)

thus This is Beer’s law of exponential transmission with respect to physical distance. It is obviously of more limited applicability than (41). Looking back at (39) and thinking of σ(s; x0 , Ω0 ) = σ(x0 + Ω0 s) as a random variable, we see that this is a problem in stochastic ODEs with multiplicative noise. The relevant questions in stochastic ODE theory are about the statistical properties of the solutions, in this case of exp[−τ (s; x0 , Ω0 )]. Given that we will be interested in situations where the “noise” σ(s; x0 , Ω0 ) has non-trivial correlation properties, we can not draw on the classic treatments [24]. First however, we examine the transport theoretical signiﬁcance of the direct transmission law as a means of predicting the speciﬁcs of particle beam propagation.

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3.2 Photon Free-Path Distributions I have presented scattering as a random choice of new direction of propagation for the photon. There is also an inherent randomness in photon propagation which deserves to be re-examined from a probabilistic perspective. From (41), but dropping the “0” subscripts for simplicity, we can derive direct transmission Tdir (s; x, Ω) = exp[−τ (s; x, Ω)] = Pr{step ≥ s|x, Ω}

(44)

by taking the ratio Iout /Iin = I(s; · · · )/I(0; · · · ). This is the probability of a photon to not suﬀer any kind of collision in an experiment over the ﬁxed distance s, starting at x in direction Ω. Now think of the photon’s free path or “step” to its next collision. As expressed above, Tdir (s; x, Ω) is the probability that this random variable exceeds s. So, thinking now of s as the random step length, its PDF is deﬁned by p(s|x, Ω)ds = dP (s|x, Ω) = Pr{s ≤ step < s + ds|x, Ω} . In terms of the 3D variability of the optical medium, this leads to d P (s|x, Ω) = σ(x + Ωs) exp[−τ (s; x, Ω)] , p(s|x, Ω) = ds

(45)

(46)

using (40) and (44). Consider the case of uniform extinction σ, the only quantity required in the problem at hand. The resulting free-path distribution (FPD) is given by p(s|σ) = σe−σs

(47)

follows directly from above, or using Beer’s exponential law of direct transmission in (43). The cumulative extinction (optical distance) computation in (40) and of direct transmission in (44) is executed repetitively in many numerical solutions of the RT equation, and the Monte Carlo technique is no exception. In uniform media, the method of inverse cumulative probability follows directly from (47): the random length s > 0 of the step between two successive scattering events is given by s = − ln ξ/σ .

(48)

In 3D media, one draws randomly an optical distance to cover τ = − ln ξ > 0 and then one solves iteratively the equation in (40) for s. The power of the diﬀerential formulation in (39) is that the collision accounting is always done in the “safe” regime where interaction probability is small. Then, conditional to survival, the collision probability is again assessed, and so on. The resulting exponential free-path distribution therefore

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follows directly from the inherent “lack-of-memory” in the course of the beam propagation. This is called a Markov property in probability theory. There is in fact another even more tutorial way of deriving the above FPDs (PDFs for s) by returning to the basic deﬁnition of extinction in (9) and exploiting the photon’s lack of memory about its past: whether it collides or not with a particle in the next instant does not depend on how far it has been traveling. So we can mentally divide s into M 1 small segments of equal length and consider the probability of a photon crossing all of them without colliding with a particle. Since the collision probabilities δI/I ≈ σδs where δs = 1/M are independent in each sub-segment, the cumulative survival probability is by deﬁnition Tdir (σs) ≈ (1 − σs/M )M .

(49)

Taking the limit M → ∞ leads back to (41), hence to the FPD in (47). This proof, used in textbooks such as [25], easily generalizes to the case where σ varies along the beam’s path, leading back to (40)–(41) hence to the FPD in (46). 3.3 Mean-Free-Path and Other Moments A fundamental quantity in transport theory, for light quanta or any other type of particle, is the mean free path (MFP) given by ∞ (x, Ω) = E(s|x, Ω) =

sdP (s|x, Ω) .

(50)

0

Other moments of the FPD are also of general interest: ∞ E(s |x, Ω) = q

sq dP (s|x, Ω) .

(51)

0

Reconsider the uniform-σ case in (47). We ﬁnd (σ) = E(s) = 1/σ .

(52)

So the optical distance given in (42), τ = σs, is just physical distance s in units of MFPs. Free-path moments of arbitrary order q > −1 can be computed from the exponential distribution in (47) and we ﬁnd E(sq ) = Γ (q + 1)/σ q = Γ (q + 1)(σ)q

(53)

where Γ (·) is Euler’s Gamma function: ∞ Γ (x) = 0

tx−1 e−t dt .

(54)

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Recall that, for integer-valued arguments, Γ (n+1) = n!, n ∈ N. In particular, the root-mean-square (RMS) free-path is given by " √ √ E(s2 ) = 2/σ = 2 E(s) . (55) It is larger than the MFP in (52), as required by Schwartz’s inequality. Schwartz’s well-known inequality is equivalent in probability theory to the statement that variance, D(s) = E([s − E(s)]2 ) = E(s2 ) − E(s)2 ,

(56)

is non-negative. Jensen’s inequality in probability theory is less known in general. It is usually stated as E[f (X)] ≥ f [E(X)]

(57)

for any random variable X on the support of f and for any convex function f (i.e., f > 0 if f is everywhere twice diﬀerentiable on the support of the PDF of X). The “=” in (57) is obtained only in two situations: 1. f is linear in X; 2. X is sure (its variance is zero). Schwartz’s inequality is a special case of Jensen’s with f (X) = X 2 being a convex function on the real axis R. Jensen’s inequality, or its converse for concave functions, will be repeatedly invoked further on. 3.4 Enhanced, Non-Exponential Steps in Spatially Correlated Media I’ll demonstrate here that, in media with variable extinction, the MFP is always larger than in a uniform medium associated with the mean extinction, equivalently, with the same overall number of particles according to (10). I’ll also show that the eﬀective FPD is always wider-than-exponential, even if the actual MFP is used. Detailed proofs are provided by Kostinski [26] and Davis and Marshak [27], respectively from the standpoints of non-Poissonian point processes and variable extinction ﬁelds. The importance of spatial correlations in the extinction ﬁeld σ(x) is emphasized in both studies, and echoed here. Non-Uniform Extinction Field Approach Let M ⊆ R3 denote the 3D optical medium of interest. Using shorthand from probability theory, we deﬁne Pr(dσ) = Pr{x ∈ M : σ ≤ extinction at x < σ + dσ}

(58)

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for the 1-point variability of the extinction coeﬃcient. Technically, the bracketed entity is a sub-set of M. In the random ﬁeld theory used here, we will not need to distinguish “probability” per se from the normalized measure of the set (in the sense of Lebesgue) where the values of interest occur. In a uniform medium of extinction σ0 , Pr(dσ) = δ(σ − σ0 )dσ .

(59)

As a simple example of a variable medium, take a Bernoulli (binary-value) random extinction Pr(dσ) = [f1 δ(σ − σ1 ) + f2 δ(σ − σ2 )]dσ, f1 + f2 = 1 .

(60)

We will however usually be dealing with continuous distributions of σ, hence the diﬀerential notation. “Ensemble” averages over this “disorder” in the optical medium will be denoted by angular brackets, · ; so, for instance, we have ∞ σ = q

σ q Pr(dσ) .

(61)

0

Statistical quantities of interest in photon transport are the ensemble-averaged moments of the (random) FPD, equivalently, the moments of the ensembleaverage FPD p(s) = dP (s) /ds . (62) Speciﬁcally, we may want ∞ sq dP (s)

E(s ) = q

(63)

0

where

∞ dP (s) /ds =

∞ p(s|σ) Pr(dσ) =

0

σ exp[−σs] Pr(dσ) .

(64)

0

So (63) is actually a double integral on probability measures: ﬁrst on the spatial disorder, then on the propagation (as written above), or vice-versa. Thinking of τ (s) = σs (65) when extinction σ is random and distance s ﬁxed, we are interested in the statistical properties of the mean direct transmission law e

−τ (s)

∞ =

∞ Tdir (s|σ) Pr(dσ) =

0

exp[−σs] Pr(dσ) . 0

(66)

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101

In short, how does it diﬀer from e−τ (s) = e−σ s ? From (66), we recover the mean FPD in (64): d −τ (s) (67) dP (s) = e ds . ds Three general results can be derived about the mean or “eﬀective” FPD in (64) or (66)–(67). 1. It is exponential only if the medium is homogeneous, a consequence of Jensen’s inequality for f (X) = e−X . This is the converse of the result derived in (43), thus making exactly exponential transmission and homogeneity equivalent statements about an optical medium: d (68) σ(x) ≡ constant ⇐⇒ Tdir ∝ Tdir . ds An important corollary here is that eﬀective medium (or homogenization) theory is of limited value as an approach for 3D RT in the presence of unresolved spatial variability. It is probably wise to customize the eﬀective parameters of the uniform medium to deliver on some important aspect of the problem at hand, rather than to think of it as a truly “equivalent” medium. 2. The mean MFP is always larger than in the hypothetical homogeneous medium with an extinction equal to the mean extinction; considering (10), we are thus putting a number/mass-conservation constraint on the variability. This translates to = E(s) = 1/σ ≥ 1/σ .

(69)

The inequality follows from Jensen’s in (57) for the convex function f (X) = 1/X. The local quantity 1/σ(x) is probably not equal to the MFP deﬁned in (50) for any direction Ω. However, it is clearly a better guess than the mean extinction σ . I will call the local quantity 1/σ(x) the pseudo-MFP. 3. The mean FPD is always wider-than-exponential in the sense that its higher-order moments are always under-estimated by assuming an exponential distribution, even if we make a judicious adjustment for item 2 by using the actual MFP: E(sq ) = Γ (q + 1)σ −q ≥ Γ (q + 1)σ −1 q = Γ (q + 1) q , q > 1 . (70) This follows from Jensen’s inequality for f (X) = X q for q > 1, which is also true for6 −1 < q < 0 (as a means of emphasizing small s values rather than large ones). For moments of order q ≤ −1, the PDF (of X = 1/σ in this case) must vanish suﬃciently fast at 0. 6

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General proofs draw on characteristic-function theory in probability [28]; see Davis and Marshak [27] for details. Figure 3 illustrates the three results with the simple Bernoulli variability model in (60), noting that no account is taken (yet) for correlation lengths as in stochastic RT in Markovian Media. The Key Role of Spatial Correlations What is the point of the mathematical exercise summarized in the previous subsection? What makes an average of the FPD over an ensemble of extinction values relevant to 3D RT? Can we dismiss the strong corollary that limits the scope of eﬀective medium theory? These questions can only be addressed by considering the spatial correlations of the random ﬁeld σ(x). Indeed, recalling that Ξ is the unit sphere, we really should have started with s Pr(dτ |s) = Pr{x ∈ M, Ω ∈ Ξ : τ ≤

σ(x + Ωs )ds < τ + dτ }

(71)

0

for the variability of optical distance over a given physical distance s, rather than (58). Implicitly, we have assumed that the development from (58) to Fig. 3 applies to s 1 σ s (x, Ω) = τ (s; x, Ω)/s = σ(x + Ωs )ds , (72) s 0

where we used the deﬁnition in (40) and introduce an over-score to denote line-averaged quantities.7 This is just the line-average of σ(x) along a ﬁnite portion of the beam {x, Ω}. Note that σ 0 (x, Ω) ≡ σ(x), ∀Ω ,

(73)

as long as σ(x) has some degree of continuity, i.e., that the local H¨ older (a.k.a. regularity) exponent h(x) in |σ(x + r) − σ(x)| ∼ rh(x)

(74)

veriﬁes h(x) > 0 almost everywhere. Apart from somewhat heavier notations, the arguments of Sect. 3.4 leading to the inequalities in (69)–(70) carry over to σ s , as a random ﬁeld with statistical properties that will generally depend parametrically on s. However, the equalities expressed in formulas (69) and (70) to the left of the “≥” carry over as good approximations only if we add one extra condition. Speciﬁcally, we require that the 1-point statistics (i.e., PDFs) of σ s depend only weakly Recall that averages over the 3D disorder are denoted · while averages over the photon transport are denoted E(·). 7

Eﬀective Propagation Kernels in Structured Media (a)

103

ensemble-average (or effective) FPD, its two components (equal weights), and two exponential approximations

1 ensemble-average FPD 2 1

= 1.8 (f = 1/2) 2

= 0.2 (f = 1/2) 1

Exponential FPD for < > Exponential FPD for MFP

0.1

MFP

1/ 0.01 0

(b)

3.0

2

f /f 2

MFP

8

10

1

50/50 75/25 88/12 94/6 97/3

2.5

< >

4 6 step size, s

panels (a) and (c)

2.0

1.5

uniform media

1.0 1

3

5

7 2

(c)

9

1

10.0 1

8.0

2

= 0.2 = 1.8

f = f = 1/2

q

1

2

6.0

4.0

q

E(s ) / (q+1)MFP

/

2.0 exponential laws 0.0 -1

0

1

2

3

4

5

q

Fig. 3. Remarkable inequalities for propagation in a binary mixture of extinctions, before correlations are considered. (a) The actual FPD is compared with its two exponential components for σ1 = 0.2 and σ2 = 1.8 with f1 = f2 = 1/2 in (60); the short steps are dominated by the dense half and the long ones be the tenuous half. Two exponential approximations based on σ = f1 σ1 + f2 σ2 = 1 and on the actual MFP 1/σ = f1 /σ1 +f2 /σ2 = 2.77... are also plotted. (b) The actual MFP is compared to the prediction 1/σ based on mean extinction as σ2 /σ1 increases from 1 to 10 and as the mixing ratio f2 /f1 varies; the special case used in panels (a) and (c) is highlighted. (c) Statistical moments E(sq ) of the actual FPD are compared with the exponential prediction Γ (q+1)1/σq based on the actual MFP. The underestimation of moments at both higher-order (q > 1) and negative order (q < 0) is a direct consequence of the wider-than-exponential nature of the actual FPD. Note that, although their plotted ratio is ﬁnite, both moments are in fact divergent for q ≤ −1. Adapted from Fig. 1 in [27]

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on s over a signiﬁcant range of s-values starting of course at 0. As it turns out, this mandates that the extinction ﬁeld has correlations over that same range of scales, at least in its 2-point statistics such as the (2nd-order) structure function, [σ(x + r) − σ(x)]2 where r, is a given displacement vector. See the appendix in Davis and Marshak [27], with A. Benassi, for a detailed proof. This is just a formal way of making a quite natural assumption, one that all instrument designers make to some extent for the purposes of signalto-noise management. We are simply saying that a (less noisy) estimate of a spatial average is almost as good as a quasi-point-wise value, only with some tolerable (and maybe correctable) loss of extreme values. And this in turn requires that the optical medium has correlations over the range of scales for which (74) applies. As shown by Davis and Marshak, the welldocumented turbulent/fractal structure of terrestrial clouds guarantees that such correlations exist. Indeed, clouds are “scaling” in the sense that |σ(x + r) − σ(x)|p ∼ rζ(p)

(75)

over a signiﬁcant range of scales r (2 to 3 orders of magnitude at least). The l.-h. side of (75) is called the “pth-order structure function” and ζ(p) is known to be a continuous concave function as long as the dependance on p of the prefactors on the r.-h. side can be neglected (yet another consequence of Jensen’s inequality). The exponents ζ(p) can in fact be obtained from the spatial/ensemble statistics of h(x) in (74), and vice-versa, using Frisch and Parisi’s [29] multifractal formalism. This kind of scaling is associated with long-range correlations and leads to a correspondingly weak dependence on s of the PDF of σ s in (72); see Fig. 4 for computations based on synthetic turbulence data. That is not the end of the story. It is important to assess the resilience or fragility of Beer’s exponential transmission law to perturbation by 3D variability. Under what conditions is the MFP signiﬁcantly larger than 1/σ ? And when are the qth-order moments (q > 1) of s signiﬁcantly larger than the prediction of the exponential distribution? Intuitively, this calls for two conditions: 1. that the amplitude of the 1-point variability is suﬃcient (cf. Fig. 3b); 2. that at least some of the scales of correlated 2-point variability are commensurate with the MFP. In item 2, we are thinking about the actual MFP and not the biased estimate 1/σ . Recall that the actual MFP can become much larger than 1/σ if there are signiﬁcant regions of low extinction (consider Fig. 3b when σ1 becomes very small). Davis and Marshak call condition #2 “resonant” variability noting that it is a rather broad resonance, easily achieved in terrestrial clouds and cloud systems. So non-exponential transmission laws are expected to be the rule

Eﬀective Propagation Kernels in Structured Media

(b)

2.5

1

2.0

(x)

1.5

1.0

0.5

0 1 2 3 4 5 6 7 8

0.1

log s 2

0.01 median

PDF estimates (normalized histgram for 10 realizations)

Log-normal variability model

mode

(a)

105

0.001 0.0 0

1000

2000

3000

4000

x

5000

6000

7000

8000

0

0.5 mean±sigma

1

1.5

2 s

2.5

3

(x)

Fig. 4. Example of a transect through a random multifractal extinction ﬁeld with lognormal statistics. This data is synthetic but the 1- and 2-point statistics are typical of in-cloud extinction variability [30] and such stochastic models are used routinely in 3D RT cloud studies. (a) Sample realization σ(x) with a log-normal PDF generated by exponentiating fractional Brownian motion [31] with a wavenumber spectrum in k−5/3 ; the mean (solid line) is set to unity and the std. dev. is ±1/3 (dashed lines). (b) Illustration of the “1-point scale-independence” property [27]: PDFs of segment averages σ s (x) in (72) do not depend on segment length for all but the most extreme values. Adapted from Fig. 6 in [27]

rather than the exception. Other situations can however arise, at least in theory. At a given amplitude, even quite large amplitude variability can be “too fast” or “too slow” to generate the interesting non-exponential transmission laws. To wit, • if the extinction ﬁeld is varying so fast that every photon can sample essentially all the variability between almost every scattering, emission or absorption event, then surely only the ensemble-mean extinction really matters because that is (to high accuracy) the outcome of (72); • in contrast, if the extinction ﬁeld varies so slowly that from its creation to its absorption, escape or detection each photon samples basically just one value of σ, then the ensemble-average transmission law is irrelevant to the transport. The former is the too-fast scenario, a.k.a. the atomistic mix (in stochastic RT theory), and the pertinent approach to assess the bulk transport is to use the mean properties in a homogeneous computation. The latter is the too-slow scenario and an average over homogeneous multiple-scattering computations weighted by the 1-point statistics of the extinction ﬁeld. This procedure is known in cloud radiation studies as the Independent Pixel/Column Approximation [8, 33].

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Non-Poissonian Point-Process Approach In this subsection, I adopt Kostinski’s [26] perspective on extinction as a point process. I’ll start with uniform optical media. Inﬁnitesimal particles are distributed at random, uniformly in space, according to some density n. The photons in a given light beam interact with these particles simply where they are, so extinction events 1. are a statistically homogeneous process, i.e., which does not depend on position; 2. for small enough volumes, the probability of intercepting more than one photon is vanishingly small; and 3. events in non-overlapping volumes are statistically independent. These properties deﬁne a Poisson point-process [34]. So the discrete probability of obtaining exactly N ≥ 0 photon interactions (extinction events) over a given distance s is pN (s; m) = pN (m) =

mN × e−m N!

(76)

where the sole parameter is the mean of N at given s, m = E(N |s) .

(77)

The variance D(N |s), deﬁned as in (56), of a Poisson deviate is equal to its mean.8 We also re-emphasize that, since we are averaging here over the photon transfer events, we use the notation with E(·)’s. Now, setting N = 0 (no events whatsoever over the segment of given length s), we get p0 (s) = exp(−m). By deﬁnition, this is direct transmission Tdir (s). We can therefore make the identiﬁcation m = τ (s) = σs

(78)

in (43), recalling that we found σ (in this case constant) to be particle density n times the collision cross-section per particle s. This is reminiscent of the interpretation of optical distance τ (s) as distance s in units of MFPs; more precisely here, τ (s) is the average number of collisions m suﬀered over distance s, at the average rate of one per MFP. Of course, at the single photon level, only one such interaction is enough to remove it from the beam, however, a small number of lucky ones can travel across large (optical) distances, that is, several MFPs. We now turn to “lumpiness” in non-uniform optical media, translating to spatial correlations between the particles, hence between the extinction 8

This might sound dissonant to √ some readers on dimensional grounds. (Should that not be standard deviation = D? That is until we recall that this random variable is a pure number: we are just counting events.

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107

events. In point process theory, correlation is deﬁned as the deviation from Poissonian behavior in the joint probability of ﬁnding exactly one particle in each of two small volumes δV1 and δV2 at a distance r: Pr{N1 = N2 = 1; n, r} = n2 δV1 δV2 [1 + η(r)]

(79)

where n is the mean density of the particles. Because of their independence in a uniformly random medium, the Poissonian prediction is η(r) = 0 while η(r) = 0 corresponds to some kind of particle correlation in the medium. In short, we retain above properties #1 and #2 but relax #3. Another, more practical, way of deﬁning η(r) uses the “pair correlation” N (0)N (r) . We have η(r) = N (0)N (r) /N 2 − 1

(80)

where N is the number of particles in the small9 test volume δV . We have N = nδV and we consider two such volumes at a distance r. If the events N (0) and N (r) are independent, then N (0)N (r) = N (0) N (r) = N 2 , hence η(r) = 0. Note that, as in (61), we are averaging here over the disorder of the particle distribution, hence the use of · ’s. Following Landau and Lifschitz’s [35] general analysis of ﬂuctuations and correlations in gases and liquids near a phase transition, it can be shown that, in a small test volume δV , one has δN 2 = N + ηN 2

(81)

where δN = N − N and η is the volume average of η(r) over δV . For example, we have r 3 η(r) = 3 r2 η(r )dr (82) r 0 " 3 with r = 3δV /4π in the case of a spherical volume. Any deviation of η from 0 in (81) takes event variance δN 2 away from the Poissonian value of N . Returning to photon transport through a particulate medium, how do we now estimate pN (s) in the presence of spatial correlations? For simplicity, Kostinski proposes to use Mandel’s formula from statistical optics [36]: ∞ pN (s) = 0

1 pN (s|m) Pr(dm) = N!

∞

mN e−m Pr(dm)

(83)

0

where Pr(dm) is the element of probability that the sole parameter of the (discrete) Poissonian distribution in (76), now a (continuous) random variable, falls between m and m + dm. Accordingly, we have transmuted the “;” separator into a “|” in the argument of pN under the ﬁrst integral. 9

As nδV becomes small, we have N = 0, 1 depending on whether a particle is present or not, the former becoming the dominant event as soon as nδV 1.

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For N = 0, this is equivalent to the computation of e−τ (s) = exp(−τ ) Pr(dτ |s) in (66) since τ is also equal to m = σs, σ being the random quantity at present. So the three general properties of e−τ (s) and its derived quantities stated in Sect. 3.4 will follow from this alternative model for photon transport. The close formal analogy between (66) and (83) is traceable to the close connection between collision statistics, (direct) transmission, and FPDs. The essential diﬀerence between the continuum transport-theoretical and random point-process approaches is that, coming from the latter less-familiar perspective, the question of spatial correlations (i.e., droplet clumping tendencies) arises immediately, at any rate, before the question of how to average over the spatial disorder. In the more familiar continuum approach, we ﬁrst argued for averaging the (partial) kernel and then argued that the outcome, as strikingly diﬀerent as it is from an exponential, is only relevant if the optical medium has spatial correlations that overlap with the MFP. The central role of the MFP is not that obvious in the point-process framework. 3.5 Results for Gamma-Distributed Extinctions, and Point-Process Equivalents We assume here a Gamma-distribution for τ = m in (83) or, equivalently, for τ = σs in (64) at a given distance s. The two parameters of this distribution are the mean, σ s, and10 2 τ (s) 2 σ /(σ − σ )2 in the continuum approach = a= m 2 /(m − m )2 in the point-process approach (τ (s) − τ (s) )2 (84) where we recognize a variance in the denominator. So the degenerate (uniform extinction, Poissonian events) case is retrieved in the limit a → ∞: as in (59), no variance at all. The PDF for τ (s), hence for m, reads as a a 1 ×τ a−1 exp[−aτ /σ s] dτ . (85) Pr(dτ |s) = p(τ ; σ s, a) dτ = Γ (a) σ s The second argument —and ﬁrst parameter— is the mean of τ (s). By writing it as σ s the given step size s appears. The Gamma-distribution has the exponential (or Laplace) law of extinction variability as a special case when a = 1: Pr(dτ |s) = p(τ ; σ s, 1)dτ = exp[−τ /σ s]dτ /σ s , (86) not to be confused with (Beer’s) exponential law of direct transmission. As another example, reconsider Figs. 4 where we see a synthetic in-cloud variability with a = 9 in (84), a reasonable amount of skewness (log-normal 10

a is the squared inverse of the standard non-dimensional variability parameter, i.e., std.-dev./mean.

Eﬀective Propagation Kernels in Structured Media

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PDF), and realistic 2-point correlations (associated with the prescribed k −5/3 wavenumber spectrum). The above choice of variability model for τ (s) is not arbitrary. We have adopted the convenient as well as reasonably accurate parameterization by Barker et al. [37] of the observed variability of optical depth (measured vertically) in high-resolution satellite images of a wide variety of cloud scenes. Figure 5 reproduces their evidence for Gamma-distributions for τ (s), where s is the thickness of the cloud layer using LandSat imagery with ≈30 m pixels and ≈60 km swaths. We note that Barker et al.’s determination of cloud optical depth for each LandSat pixel is based, as usual in cloud remote sensing, on a 1D RT model and this procedure is known to underestimate the variance [38]; so the inferred values of a are likely to be upper bounds. In the continuum approach from Sect. 3.4, this choice of variability model applied to (66) yields Tdir (s; σ , a) = e−τ (s) =

1 (1 + σ s/a)a

(87)

for the mean transmission law which is plotted in Fig. 6b. As expected, the exponential law exp(−σ s) is recovered (using L’Hˆ opital’s rule) in the degenerate-σ limit a → ∞. Otherwise, direct transmission is eﬀectively power-law, with diverging moments E(sq ) for q ≥ a. Assuming a > 1, the MFP exists and is given by a (88) E(s) = σ −1 = σ −1 . a−1 Notice the excess over the standard prediction using σ −1 . It is easy to verify here the general result that E(s) = 1/σ using the PDF in (85). In the discrete point-process approach from Sect. 3.4, the same choice of variability in (85) now reads as a a 1 × ma−1 exp[−am/σ s]dm Pr(dm|s) = p(m; σ s, a)dm = Γ (a) σ s (89) where σ s = m . Applying this to (83) leads to a so-called negative binomial distribution for N . More precisely, we have pN (s) = pN (σ s, a) =

Γ (N + a) (σ s)N 1 × × . (90) N N! Γ (a)(a + σ s) (1 + σ s/a)a

The case a ∈ N − {0} = {1, 2, . . . } is known as the Pascal distribution and it arises naturally from the theory of Bernoulli processes. Speciﬁcally, the question is about the probability of having exactly a ≥ 1 successes with probability p and N ≥ 0 failures with probability q = 1 − p, ending with a

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Fig. 5. Empirical support for Gamma-distributed extinction variability in the Earth’s cloudy atmosphere, reproduced from Figs. 1–2 in [37] with permission. (above) Cloud optical depth ﬁelds retrieved from more or less cloudy LandSat images: upper row, very cloudy (1.6 a 22.5); middle row, partially cloudy (0.4 a 1.3); lower row, sparse clouds (0.2 a 0.8). Note that optical depth is proportional to extinction averaged vertically over the thickness of the cloud layer. (below) The inferred Gamma distributions (in one-to-one correspondence with the above images) are generally in good agreement for the whole optical depth PDF: observed histograms in solid lines; Gamma-PDF predictions (based only the observed mean and variance) in dashed lines. Figure 6a illustrates in more detail the whole family of Gamma distributions

Eﬀective Propagation Kernels in Structured Media Gamma Probability Density Functions s = (s) = 1, a = 1/ [ (s)] with

(a)

(b)

2.0

111

Ensemble-averaged Free-Path Distributions (FPDs) for Gamma-distributed optical distances (fixed ) 10

0

1.0

p(s)/

0.5

0.0 0.0

0.5

1.0

1.5

2.0

2.5

10

-1

10

-2

dir

1/2 1 3/2 2 4 8

1.5

dP/d (s)

= |(d/ds)|/

a

10

-3

10

-4

10

-5

3.0

a 1/2 1 3/2 2 4 8

0.1

(s) = s (step s is constant)

1.0

10.0

s

Fig. 6. Gamma-distributed optical distances for a given physical distance: (a) PDFs of τ from (85) for a given s and selected variability parameters, (b) the resulting mean transmission laws from (87) plotted in log-log axes to highlight the power-law tails

success, when p = 1/(1 + σ s/a) ≤ 1. Equation (90) can thus be expressed formally as N −a σ s 1 × (91) × pN (s) = (−1)N N a + σ s (1 + σ s/a)a even if a > 0 is non-integer.11 For N = 0, we naturally ﬁnd the same non-exponential transmission laws as in (87) and Fig. 6b. However, looking back at (49), where we assumed a = M 1 for a ﬁxed value of σ s, we can interpret “success” probability p = 1/(1 + σ s/a) ≈ 1 − σ s/a as that of a particle being transmitted through the 1/ath part of s for a given σ . Thus, for N = 0 “failures” to be transmitted a successive times through that ath portion of optical distance σ s, we indeed ﬁnd (87) for the direct transmission. So clearly the most interesting situation is when a ≈ σ s, i.e., when any given physical distance s is divided roughly into as many distinguishable parts or “clumps” as its (mean) value in optical units. The mean of N is still E(N ) = σ s in (90). Mimicking the form of (81) to emphasize the relation to pair-correlations, we ﬁnd for the variance of N D(N ) = E(δN 2 ) = E(N ) + ηE(N )2

(92)

where δN = N − E(N ) with η = 1/a ≥ 0 . n

(93)

11 binomial coeﬃcients i are deﬁned by the identity (p + q)n = n usual n The n−i i p q where p + q = 1 in the well-known application to combinai=0 i torics; “negative” binomial coeﬃcients are deﬁned by the identity (1 − q)−r = ∞ −r i (−q) for r > 0. i=0 i

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Accordingly, pN (s) in (90) is sometimes called the “over-dispersed” Poisson distribution. It is nonetheless remarkable that in (93) the point-process approach ties the 2-point statistical quantity η in (81) and the 1-point statistical variability parameter a from (84). In density-based stochastic modeling, one can generally choose the PDF and the correlation structure independently. The above results generalize at once the Poissonian (a = ∞, D(N ) = E(N )) case in (76) with m = σ s and the a = 1 case in (86) that was used by Kostinski [26] as a rather extreme example of sub-exponential transmission: Tdir (s; σ , 1) = p0 (σ s, 1) =

1 . 1 + σ s

(94)

This is indeed the critical value of a at which the MFP becomes (logarithmically) divergent and it hails from the special case of (90) with a = 1 known as the geometric distribution, pN (σ s, 1) =

σ s 1 + σ s

N ×

1 . 1 + σ s

(95)

That is the probability of exactly N ≥ 0 failures before one success when each Bernoulli event is a success with probability p = Tdir = (1 + σ s)−1 in (94). 3.6 From Positive to Negative Correlations Physically, what is it about the spatial correlations that is causing the systematic deviations from exponential transmission? Even if there is a single direction of propagation, we are always computing a projection of the particlelight interaction cross-sections parallel to the beam (cf. Fig. 1). There is naturally random overlap in these projections. What the spatial correlations eﬀectively do is to cause more overlap in the projections, hence more photons are transmitted. The sub-exponential laws we found above are the statistical consequence of this clustering. From there, it is of interest to consider the possibility of negative correlations in the spatial ﬂuctuations of the extinction ﬁeld (in the continuum approach) or in the pair-correlation function (in the point-process approach). In the extinction ﬁeld picture, this means that a ﬂuctuation above the mean is more likely than not to be very quickly followed by a ﬂuctuation below the mean, and vice versa; in a sense, this is “fast” variability with a tendency to overshoot the mean. In the pair-correlation picture, this means that the scattering/absorbing particles essentially repel each other, leading to even more uniformity in space than obtained by random (Poisson) positions. The outcome in (80) is η(r) < 0 at least for small values of point-separation distance r. From there, η(r) in (81)–(82) will also be < 0, at least for small enough δV (r) = (4π/3)r3 .

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What are the consequences of negative correlations for photon transmission? Shaw et al. [39] use the pair-correlation model to show that a super-exponential (faster-than-exponential) transmission law will follow from η(r) < 0. The continuum approach has more trouble here because negative correlations are just diﬀerent incarnations of the uniform-σ hypothesis. The question of how relevant negatively-correlated media are to atmospheric optics is, at present, entirely open [27]. The balance of evidence however favors further consideration of the positive correlations related to droplet or cloud clustering. The known mechanisms that cause cloud particles to repel each other, as listed by Shaw et al. [39], indeed require either very close proximity (threatening the important dilution requirement in any transport theory based on geometric optics) or else rather unusual circumstances (e.g., still air, electric charge separation). Negative correlations can occur also at macroscopic scales: certain types of cloud layers (e.g., marine stratocumulus) are systematically topped by clear layers, trains of orographic clouds downwind from a mountain range will also yield negative correlations at regular distances. However, in both these cases, any chance of occurrence of another cloud further along in the vertical or horizontal direction will restore positive correlations, which already dominate at the micro-scale. 3.7 Summary, Discussion, and Outlook On the one hand, we have established the deep non-Poissonian nature of photon transport in variable optical media and, on the other hand, we have underscored the importance of having spatial correlations over MFP scales to obtain non-exponential FPDs. An inescapable consequence of deviations from spatial uniformity of the cloud droplets is that the mean (or eﬀective) photon transport kernel is non-exponential. More precisely it is sub-exponential in the case of positive correlations (clumping tendencies). In the atmosphere, spatial correlations in cloud structure exist over a vast range of scales horizontally as well as vertically, although clearly in qualitatively diﬀerent ways. This range goes from centimeters at least up to the thickness of the troposphere (≈10 km), and often much more in the horizontal. There are both positive and negative correlations, but at scales that matter for photon transport they are overwhelmingly positive. Two-point spatial correlations, even of the right sign and over the right range of scales, are of little consequence unless extinction (and, hence, the pseudo-MFP) also vary widely enough in the sense of the 1-point statistics (i.e., the PDF). Consider the following two scenarios: • In a single dense un-broken cloud layer, the MFP is small with respect to the physical thickness of the layer. This is equivalent to saying that the layer is optically thick and that (literally) an exponentially small amount of direct sunlight gets through.

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• Now imagine a complex situation with broken clouds and possibly also multiple layers (not necessarily all optically thick), and add to that a partially reﬂective surface. In this case, the optical quasi-vacuum between the clouds dominates the average 1/σ of the local pseudo-MFP. So this estimate of the overall MFP can be huge, possibly larger than the physical thickness of the whole cloudy region. In particular, abundant direct sunlight can reach the ground in spite of the presence of clouds. In the following section, we will see that • in the former case, radiation is transported by standard diﬀusion, each photon executes a convoluted trajectory. This path is a classical (Gaussian) random walk with an inner cut-oﬀ at the small MFP scale and an outer cut-oﬀ determined by the ﬁnite thickness of the cloud layer. • in the later case, radiation is transported by anomalous diﬀusion where most steps in the random walk (inside clouds) are small but some steps (between clouds and/or surface) can be huge. This is as predicted by the basic transport computations presented above in the presence of positive correlations and, in this case, the photons execute L´evy-type random walks [40] where the tail of the step distribution is power-law. Before moving on, we note that Borovoi [41] objected to Kostinski’s [26] linkage of non-Poissonian droplet distributions and sub-exponential transmission laws in the light of his previous [42] investigation that used standard transport theory and favored eﬀective medium theory (hence modiﬁed exponential transmission laws). Kostinski’s reply [43] is worth reading in that it (1) reasserts the relevance of the spatial correlations, and hence of nonexponential transmission laws, and (2) claims that the point-process model is more general than standard transport theory because it allows for negative correlations. In [27], we agree with Kostinski on the former claim, coming from the transport theoretical perspective used by Borovoi (cf. Sects. 3.4–3.4), but we have reservations about the latter claim. First, the atmospheric scenarios for negative correlations are marginal if not implausible (as argued above); secondly, empirical evidence weighs in for sub-exponential transmission laws (cf. Sect. 5); thirdly, I have unpublished results showing that density-based computations can handle negative correlations and lead to super-exponential transmission laws (the variability of course has to violate the continuity conditions required in the above). Most importantly however, the author views this as a healthy debate on the fundamentals of 3D RT.

4 Multiple Scattering and Diﬀusions 4.1 Multiple Forward Scatterings Let Ω0 be the initial direction of propagation in R3 , or position on Ξ, for a particle beam in a medium with conservative ( 0 = 1) axi-symmetric

Eﬀective Propagation Kernels in Structured Media

115

scatters. The medium may be variable but we will assume the scattering phase function is the same everywhere. By symmetry, the average direction E(Ωn ) is Ω0 for any number of scatterings n. By taking Ω0 as the polar axis (µ0 = 1), we can use θn = cos−1 (Ω0 •Ωn ) = cos−1 (µn ) to measure the (greatcircle) distance on Ξ between initial and current directions of propagation. From (29), we know that E(Ω0 • Ω1 ) = · · · = E(Ωn−1 • Ωn ) = g ∈ [−1, +1] .

(96)

I now show by induction that E(µn ) = E(Ω0 • Ωn ) = g n .

(97)

Indeed, the only component of interest is (Ωz )n = µn ; all others vanish upon averaging, by symmetry. From spherical trigonometry, " " (98) µn+1 = µn µs + 1 − µ2n 1 − µ2s cos φs where the azimuthal angle φs in the second term is uniformly distributed on [0,2π) and uncorrelated to θs and θn ; so it averages to zero. Therefore E(µn+1 ) = E(µn µs ). Like in the propagation part of particle transport, there is a (discrete) Markovian property for scattering: transition probability is independent of present state. This leads to12 E(µn+1 ) = E(µn )E(µs ) = E(µn )g, ∀n ,

(99)

and thus completes the proof. Equation (97) enables us to quantify accurately the decay of directionality in a light beam embedded deeply in a uniform turbid medium, sometimes called “blooming:” Ωn can be anywhere on Ξ with almost equal probability as soon as we have, say, E(µn ) ≈ 1/e. This happens at a critical scattering order n∗ ≈ −1/(ln g). For small enough values of (1 − g), this reads as n∗ ≈ 1/(1 − g)

(100)

which roughly deﬁnes the number of forward-peaked scattering events required for the photon to loose almost all memory of its original direction of travel. Another way of showing that directional memory is short (i.e., lost in ﬁnite time) is to view the typical great-circle distance from the origin, θn = cos−1 µn , as a discrete-time diﬀusion on the sphere. By identifying 12

An interesting corollary of this recursion formula is that, if the phase function is Henyey–Greenstein, then the angular distribution of n-times scattered radiance (in the absence of boundary and 3D eﬀects) is Henyey–Greenstein with g(n) = g n . Since (under these same conditions) vectorial addition of random directions is determined by spherical convolutions of the PDFs that are phase functions, this implies that Henyey–Greenstein functions are the spherical equivalent of Gaussian PDFs: invariant, modulo scaling, under convolution.

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E[cos θn ] ≈ 1 − E[θn2 ]/2 on the tangent plane13 of Ξ at µ = 1 and, remarking that g n = [1 − (1 − g)]n ≈ 1 − n × (1 − g) when (1 − g) 1, we obtain E[θn2 ] ≈ 2(1 − g)n .

(101)

So 2(1 − g) plays the role of a diﬀusivity constant. Based on (101), where do √ we expect θn to be for n = n∗ in (100)? Approximately at 2 rad (hence ≈ 90◦ ) on average, meaning almost anywhere on Ξ for a given realization. In summary, photons emanating from a collimated beam in a scattering medium with a forward-peaked phase function have a collective memory of their original direction, but it is a relatively short term memory. The higher the number of scatterings, the more isotropic the corresponding radiance, independently of spatial considerations (boundaries, internal variability). The critical number of scatterings needed to redistribute radiance in direction space is estimated to be ≈ (1−g)−1 . In a general multiple-scattering problem we can deﬁne the equivalent number of isotropic scatterings as niso = n/n∗ ≈ (1 − g)n .

(102)

4.2 Impact of Forward Scattering on Propagation: The Transport MFP (without Fick’s Law) We are now in a position to look at the spatial consequences of the shortterm memory of propagation direction due to forward-peaked but conservative ( 0 = 1) scattering, as described in Sect. 4.1. The light particles are executing directionally-correlated random walks based on a FPD that need not be speciﬁed beyond the fact that the MFP must exist. We again assume the photons all leave the origin in direction Ω0 at time n = 0. In the discrete-time picture, the particle is then displaced from position x0 = 0 to x1 = x0 + s0 Ω0 , where s0 is the initial step size. Holding Ω0 ﬁxed, we have (103) E(x1 ) = E(s0 )Ω0 = Ω0 , using the deﬁnition of MFP in (50) or (52). From x1 , and conditional to not being absorbed, the photon moves on to x2 = x1 + s1 Ω1 . It is clear that E(x2 − x1 ) = E(s1 )E(Ω1 • Ω0 )Ω0

(104)

since the propagation and scattering are independent, another consequence of the Markovian property of transport. Now, after the ﬁrst scattering and second step, it follows from (96) that E(x2 ) = [E(s0 ) + E(s1 )E(µs )]Ω0 = (1 + 13

0 g)Ω0

,

(105)

This 2nd-order (Gaussian) approximation is at the core of the “small-angle” approximation in RT theory used extensively in imaging and lidar studies [44].

Eﬀective Propagation Kernels in Structured Media

117

where it is assumed that the 1st and 2nd FPDs have the same MFP and that an absorption may have occurred at x1 with probability 1 − 0 ≥ 0. By iteration, the independence of step-sizes and step-directions leads under the same assumption to lim E(xn ) • Ω0 =

n→∞

∞

n 0 E(µn )

=

n=0

∞

(

n 0 g)

n=0

=

1−

0g

.

(106)

So, after many scatterings, the cumulative eﬀect of forward scattering is simply to boost the initial ballistic motion by a factor (1 − 0 g)−1 . The distance in (106) is the well-known transport MFP t =

1−

0g

.

(107)

Note that we have made no assumption about the FPD beyond the existence of a MFP; in particular, no spatial homogeneity assumption per se was made, only constancy of the MFP (in some spatial/ensemble average sense if required). The emergence of the transport MFP in (107) as the residual impact of short-term directional memory on propagation is in sharp contrast with the usual derivations which come along with the diﬀusion/P1 approximation to the full 3D RT problem in (1)–(3). This classic macroscopic approach to the transport problem is encapsulated in Fick’s law [45], relating particle density and current: ct ∇U (108) F=− 3 where ! ! U (t, x) 1/c = I(t, x, Ω)dΩ (109) F(t, x) Ω 4π

deﬁnes particle (in our case photon) density and current (or net vector ﬂux), anticipating the generalization in Sect. 5.2 for time dependence. Fick’s law (108) is the constitutive relation that closes the macroscopic transport problem started with continuity equation obtained by expressing particle conservation: ∂U + ∇ • F + cσa U = 0 , ∂t

(110)

where the two last terms come from the angular integral of the steady-state 3D RT equation in (1)–(3) in the absence of sources. At least to a ﬁrst approximation, the total path L of a photon is n × . Notice that (for 0 = 1) the same answer is obtained if we reckon on the actual MFP and number of scatterings, or on the eﬀective number of isotropic scatterings niso in (102) and the transport MFP t in (107). This makes physical sense since L/c is simply the time since the photon was emitted and

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it should not depend on whether we count actual scatterings and MFPs or eﬀectively isotropic scatterings and transport MFPs. In summary, photons in an isotropically scattering medium (g = 0) lose track of their direction of propagation at every step (average length ), but it takes their positively Ω-correlated counterparts about (1 − g)−1 forwardpeaked scatterings to “forget” their original direction.14 In the process, this causes them to travel (on average) that much further by drifting in the original direction. So t can be interpreted as the eﬀective MFP for obtaining one isotropic scattering. Figure 7 illustrates these results in two spatial dimensions. 4.3 Asymptotics of Standard and Anomalous Diﬀusion in Finite Media The overarching goal of the research program surveyed in this paper so far is to decompose ensemble-average (or large-scale) photon transport in 3D media into processes that are simpler to model, namely, diﬀusions. Photons diﬀuse in direction space and loose track of their original direction in ﬁnite time but cumulate over this time interval an extended displacement in the original direction, hence the origin of the transport MFP. To a ﬁrst approximation, one can view photons executing random walks (isotropic reorientation at each step) with a mean step given by t and other aspects of the step distribution controlled by the FPD. Figure 8 illustrates schematically random walks of solar photons from the top of the atmosphere to the ground or back to space; notice how horizontal and vertical gaps in the cloudiness promote very large steps. Adopting a Lagrangian perspective on photon transport, let x0 = 0 be the point of departure of the random walk: xn+1 = xn + s ,

(111)

where s is a random step drawn from the given FPD, hence xn =

n

si .

(112)

i=0

We are now in one of two situations. If D(s) = E(s2 ) < ∞, then E(x2n ) ∼ 2t n , 14

(113)

Although it is a regime mostly of academic interest, our computations work also for −1 ≤ g < 0. In particular, at g = −1 we ﬁnd t = /2. It takes only 1/2 of a MFP (and scattering) to loose directional memory when propagation direction is exactly reversed at each event (For a given particle, all happens on a given line if g = −1, irrespective of dimensionality, but there is still an original direction.)

Eﬀective Propagation Kernels in Structured Media (a)

(b)

Free-Path Distribution

Scattering Phase Function

1

2

0.6 15 0.4

10

0.2

5

0

0 0

1

2

3

4

(coarse-grained) pdf

pdf (coarse-grained)

20

0.03

100 0.02

Prob{-20< ³ 0.70

20} —>

… x 112 = 78 events

80 60

0.01

40 20 -120

free path, s

(c)

140 120

0 -180

5

2

Prob(d ) = [(1 g )/(1+g 2gcos )] (d /360) for asymmetry factor g = E(cos ) = 5/6

-60 0 60 scattering angle,

120

0 180

histogram (7x16=112 events)

25

with MFP = E(s) = 1

histogram (7x16=112 events)

Prob(ds) = exp[ s/ ] (ds/ ) 0.8

119

x , n = 0,…,16 n

-12

-6

0

6

12

n 0

0.0

0 1 2

6

…

0.5

16

16

1.0 n

E(z ) =

n

n

g

(1 g)z

z , n = 0,…,16

3

1

2

i

i=0

9

1.5 16

12

–1

n=0 15

2.0

n

cos (g ) 1

2

n

g = E(cos ) n

Mean-free-path = E(s) = 1 2.5

Fig. 7. Rescaling of the mean-free-path due to forward-peaked scattering in a uniform medium. (a) The exponential law of extinction that dictates the distribution of free paths s: Pr{step ≥ s} = exp(−s/ ) with unit mean ( = E(s) = 1). (b) The scattering kernel p(θs ) from (33) is used, the 2D counterpart of the Henyey– Greenstein model in (30); this PDF describes the distribution of scattering angle θs , with an asymmetry factor g = E(cos θs ) = 5/6 = 0.8333 · · · . In panels (a-b), error bars are based on expected (Poissonian) means and variances in number of events per bin for a total of 7 × 16 = 112 samples. (c) Seven 2D particle trajectories starting straight down at the origin, all nmax = 16 scatterings long. The l.-h. scale is in MFPs; the r.-h. z-axis uses “transport” MFPs from (107). In the lower l.-h. corner, an indication of the average direction of travel is plotted for n = 1, . . . , 16 and ∞; in the upper l.-h. corner, the corresponding theoretical average positions are indicated for orders-of-scattering n = 1, . . . , 16, ∞ and again for n = 1, 2, 16 obtained empirically, with st. dev.’s. After a number of anisotropic scatterings, the particle may just as well have been scattered isotropically once, but the MFP for such a scattering is longer by a factor 1/(1 − g) = 6. Adapted from Fig. A1 in [40]

since we can always relate numerically the variance of the FPD to the transport MFP. However, there is another possibility (D(s) = ∞) and another outcome: α (114) E(xα n ) ∼ t n ,

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Fig. 8. Solar photons propagating in the cloudy atmosphere. We consider a wavelength where there is no absorption by cloud droplets nor gases. Each “step” in the photon’s random walk is in fact rescaled, as in (107), so that the scattering can be considered isotropic. Notice how the upper (cirrus) cloud layer can let light through directly as well as produce a diﬀuse illumination of the lower layers; at the same time, it reﬂects up-welling light back towards the ground.

where α = min{0 < q < 2 : E(sq ) = ∞} . q

(115)

This important quantity is known as the L´evy index and it dictates the farﬁeld behavior of the eﬀective FPD we have adopted from (67): dP 1 (116) ds ∼ s1+α , s → ∞ . We will require α > 1 so that t = E(s) < ∞. The process described by (113) is called diﬀusion15 and we will be more speciﬁc in calling it “standard” diﬀusion.16 The process described by (114) is called “anomalous” diﬀusion. 15

This nomenclature is fully consistent with the diﬀusion approximation obtained in a variety of ways from the traditional Eulerian perspective used in transport theory. 16 For the classic Eulerian derivation of (113), we substitute (108) into (110) with σa = 0. This yields the prototypical parabolic PDE ∂t U = D∇2 U , where

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Note that, in spite of their formal similarity, the origins of (113) and (114) are fundamentally diﬀerent. The origin of (113) is the well-known law of probability that variances of sums of independent random deviates add, and we simply replaced the (ﬁnite) variance of the FPD by a dimensionally equivalent parameter (2t ). The origin of (114) is necessarily diﬀerent since E(sα ) = ∞, although it is a slow (logarithmic) divergence. So we are slightly abusing the notation E(·) in (114) and instead interpret it as an “estimator” in sampling theory: we simply sum n random variables s taken to the power α from a PDF with property (116), and then divide by n. For a ﬁxed n, this ratio will increase slowly as log n. The sum itself, will grow as n log n times some scaling factor for the PDF which, apart from a numerical factor, can be taken as α t . In (114), we have summarized this argument and neglected the unimportant logarithmic term as is customary in scaling analysis. The task now is to convert the above information about unbounded random walks in general into more useful information about random walks by photons in ﬁnite optical media. A ﬁrst question we have is about the number of (eﬀectively isotropic) scatterings niso,T suﬀered by photons before crossing the medium (e.g., solar light transmitted to the ground). This is a random number of course but we can estimate approximately its mean by re-assessing (114), extended to include the case (113), as to what is given and what is averaged. If we identify the expectation (or estimation) on the l.-h. side with H α then we can solve for a rough estimation of E(niso,T ) ∼ (H/t )α .

(117)

We have yet to convert the above scale ratio into an optically meaningful quantity. Letting τc denote the ensemble-average cloud optical thickness, we have (118) H/t → (1 − g)σ H = (1 − g)τc at non-absorbing wavelengths, i.e., when 0 in (12) is unity. As it turns out, E(niso,T ) and even E(nT ) are not readily observable quantities while the transit time (a continuous random variable) is.17 A more convenient assessment of transit time is therefore total photon path length LT which is roughly (t ) times nT (niso,T ). For means, we have E(LT ) ≈ E(nT ) ≈ t E(niso,T ) .

(119)

D = c t /3 is the diﬀusivity (assumed constant), to which we apply the initial condition U (r, 0) = δ(r). The well-known solution for t > 0 is U (r, t) = exp(−r2 /4Dt)/(4πDt)3/2 . Recall now that, with this normalization, density U is simply the probability of ﬁnding the diﬀusing particle at position r at time t. Thus ∞ E(r2 |t) = 0 r2 U (r, t)dr = Dt where we can identify the continuous time t with pathlength t n in (113) divided by c. 17 For transit time, we can use either a pulsed laser or the absorption by a uniform gas (cf. Sect. 5) while for estimating order of scattering statistics we would need to use an absorption feature of the scattering particles, here, the cloud water droplets.

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Assembling the information in the last three equations, we ﬁnd E(LT ) ∼ [(1 − g)τc ]α−1 H .

(120)

Another observable and radiometrically important quantity is the probability T of photon transmission through the medium, irrespective of the transit time. To access this deceptively simple property, we need to invoke a lesser-known law for random walks on a half-space. If the random walking particles all leave the plane z = 0, how many steps does it take to return to the plane of departure? It is easy to see the analogy with photon reﬂection from a semi-inﬁnite optical medium, and the answer is [46, 47] √ (121) Pr{(discrete) return time ≥ n} ∼ 1/ n in the regime n 1. This remarkable PDF has inﬁnite variance and inﬁnite mean (only moments of order <1/2 converge). Davis and Marshak [40] brought together the straightforward estimation of E(niso,T ) in (121) and (117) to estimate transmission probability Pr{return time ≥ E(niso,T ) } ∼ (H/t )−α/2 .

(122)

In more optical terms, we have T ∼ [(1 − g)τc ]−α/2 .

(123)

With so many layers of approximation, it is hard to take the generalized (standard and anomalous) photon diﬀusion theory we have just exposed seriously as a model of radiation transport in the Earth’s cloudy atmosphere. However, the theory does make speciﬁc predictions for observable quantities in the time-domain (120) and in steady-state (123). In the next section, we show using both new simulations and new observations that the phenomenology of generalized photon diﬀusion does seem to capture key aspects of real 3D RT in the real atmosphere.

5 Large-Scale 3D RT Eﬀects in Cloudy Atmospheres 5.1 Overcoming the Challenge of Observing 3D Eﬀects in Steady-State RT The theoretical result in (123) is clear, and it has been the same message from any other domain-average model [48]: 3D variability (here, measured by 2 − α > 0) systematically enhances photon transmission through the atmosphere to the ground. It is not however straightforward to demonstrate observationally that 3D RT eﬀects have an impact on very large domains. Indeed, it is easy to obtain large-scale radiances from satellites (they are essentially the raw data), but ﬂuxes are elusive (they call for angular conversion

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models). Furthermore, it is non-trivial to estimate what the domain-average optical depth is and that is necessary to make an independent prediction based on 1D RT. When using satellites with large-enough pixels, we tend to use said 1D RT models in inversion mode, which of course only gives us the “eﬀective” optical depth that is needed to drive the 1D RT model to give us the observed radiance or ﬂux.18 It is then tempting to compare this estimate of τc with independent ground-based and in-situ estimates, and many studies have done that. However, the space-time sampling volumes of surface and airborne sensors is radically diﬀerent from their space-borne counterparts. Then we can try to convince ourselves that the obvious 3D RT eﬀects (shadows, over-illumination, etc.) are small-scale and of varying sign. So, upon largescale —and moreover angular— averaging, they will surely all but vanish. Not so! In recent years, increasing sophistication in multi-spectral, multi-angle and/or multi-resolution methods have shed new light on this diﬃcult problem in cloud remote sensing [38]. Interestingly, the most compelling direct observational evidence for the impact of 3D cloud structure on large-scale atmospheric radiation processes comes from time-domain RT [49]. This is not very surprising for the physics and engineering communities which have been forever using “impulse responses” to probe systems with complex, possibly unknown, internal structure. The α-dependence in the asymptotic ensembleaverage 3D RT result in (120) for pathlengths of transmitted photons gives us a hint. The real surprise is however that this time-dependent 3D RT can be performed observationally using abundant and free solar photons rather than power-hungry pulsed lasers. 5.2 Time-Dependent 3D RT . . . with Solar Photons Let us now open up the deterministic 3D RT problem described in (1)–(3) to time dependence: I(x, Ω) → I(t, x, Ω). The advection operator is changed accordingly: 1 ∂ + Ω • ∇ + σ(x) . (124) L = Ω • ∇ + σ(x) → c ∂t Without loss of generality (because of the superposition principle), we can consider the source term as delta-in-time: Q(x, Ω) → Q(t, x, Ω) = q(x, Ω)δ(t). We note that even in steady-state RT sources and radiances all have raw units of “per second” because, after all, this is transport theory and particles take time to move. Here we have a ﬁnite number of particles q(x, Ω) dx dΩ that are released all at once and we monitor how they spread in time, until they are absorbed or have escaped (“absorbed” by a boundary). 18

Circularity notwithstanding, this eﬀective τc is all that is required anyway in many climate-driven applications.

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A deep beauty of general time-dependent transport theory is that this is not really a new class of problems [50,51]. Indeed, ﬁrst we make the convenient change of variables in (and of units for) I and Q: I(t, · · · ) → I(L, · · · ) where pathlength L = ct, and similarly for Q. Now we take the Laplace transform of I with respect to path L: ∞ ˜ ···) = I(k,

e−kL I(L, · · · )dL ,

(125)

0

where k is at present just the Laplace conjugate of path L (in units of m−1 ). We also Laplace transform the time/path-dependent 3D RT equation: for the advection operator in (124), L˜ = Ω • ∇ + σ(x) + k ,

(126)

˜ x, Ω) = q(x, Ω). So I˜ simply obeys a steady-state 3D RT equation while Q(k, with an enhanced extinction term. Not any kind of enhancement. Speciﬁcally, because the scattering process in (3) in unaﬀected, it is the absorption coeﬃcient σa (x) in (13) that is uniformly boosted by an amount k. Physically, this amounts to adding to the structured 3D scattering/absorbing medium a uniform gas that only absorbs photons at a rate of k per unit of length along any beam. This formal mapping of the temporal Green function problem in 3D RT to a steady-state 3D RT problem with (more) gaseous absorption —and ˜ · · · ) × p(L, · · · ) for sometimes simply (125) with the notation I(L, · · · ) = I(0, the pathlength PDF— is called the “equivalence theorem.” Now suppose we are interested in computing the moments E(Ln ), n = 1, 2, 3, . . . , at some position (x, Ω) in photon phase-space. By deﬁnition, we have ∞ ∞ n n (127) E(L | · · · ) = L I(L, · · · )dL / I(L, · · · )dL . 0

0

From (125), we can obtain these integrals by successive orders of diﬀerentiation in the Laplace-transformed quantities: n d 1 n ˜ I(k, · · · ) − . (128) E(L | · · · ) = ˜ ···) dk I(0, k=0 We assume, just for simplicity at present, that the only absorption is by the well-mixed (uniform) gas characterized by k. Then, in principle, one can derive the statistical moments of pathlength from the behavior of steady˜ · · · ), in weak-absorption limit (k → 0). state radiance, denoted here by I(k, Finally, we note that (128) carries over to any integration over the position x or angular Ω variables.

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5.3 Some Recent Observations We are extremely fortunate in atmospheric science that the above program can be implemented thanks to the sharp spectral features of molecular oxygen that occur near the peak (in photon number) of the solar spectrum. The spectro-radiometric technology required for robust pathlength moment estimation has been maturing over the past 15 or so years and so has the analysis methodology, which is not quite as simple as in (128) because of ﬁnite spectral resolution eﬀects, cf. Min and Harrison [52, 53]. Even though the phenomenology of anomalous photon diﬀusion from Sect. 4.3 is not well known, there is a growing awareness that pathlength statistics convey information about the spatial variability of clouds, cf. recent review article by Stephens et al. [49]. Several groups have investigated mean photon pathlength in transmission (i.e., using ground-based instruments), mostly in conjunction with optical depth, with [14, 54] or without [55, 56] reference to the anomalous diﬀusion/L´evy walk model. We now examine some recently published observations and analyses. Figure 9 is a composite of ﬁgures from the recent paper by Scholl et al. [15]. It shows O2 -based photon pathlength observations and ancillary cloud diagnostics; I refer the reader to the original paper for all instrumental and data analysis details. In the upper panel we see an evolving cloud episode using a sophisticated mm-radar proﬁler [57]. Over the hour-long observation period (extracted from a much longer one), two well-deﬁned cloud layers become gradually thinner, more tenuous, and more disjoint. The upper layer between 9 and 10 km in altitude is a cirrus (ice-crystal) cloud. Cirrus layers are typically highly textured (think “angel-hair” clouds) and generally semiopaque in the sense that sunlight cannot get directly through but, thanks to the strongly forward-scattering phase function, one can still see the location of the bright solar source; at the same time, cirrus layers are powerful diffusers of sunlight. The lower cloud deck, below 2 to 1 km, starts as a dense boundary-layer cloud, probably a strato-cumulus, that are invariably made of liquid water droplets. It is clearly producing in-cloud drizzle and maybe even ground-level rain between 12:14 and 12:22 Z; after that episode, it rapidly breaks up. In the middle panel we see the mean pathlength E(LT ) of transmitted solar photons (in units of cloud-system thickness H) plotted versus rescaled optical depth (1 − g)τc ; the pathlength statistics were derived from a highresolution spectro-radiometer fed by fore-optics with narrow ﬁeld-of-view (0.86◦ ) centered on the zenith. The theoretical curves are inspired by the power-law relation predicted in (120). Speciﬁcally, a more detailed diﬀusiontheoretical formula for E(LT )/H by Davis and Marshak [58], valid only in the standard α = 2 case but including pre-asymptotic corrections, is 1 4 + 3 E(LT )/H = × (1 − g)τc × 1 + (129) 2 2 1+

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Fig. 9. Solar photon pathlengths and coincident cloud structure at Cabauw (the Nederlands) on May 22, 2003, between 12:00 and 13:00 Z. (upper ) Time-series of reﬂectivity proﬁles from the up-looking KNMI 35 GHz radar [57] which is sensitive to cloud droplets; three distinct periods are deﬁned (color-coded) as the two well-deﬁned cloud layers thin and break up. (middle) Mean pathlength cumulated inside the cloudy region (base of lowest cloud to top of highest) versus rescaled cloud optical depth (1 − g)τc : observations with their uncertainties and theoretical predictions parameterized by α, as explained in the text. (lower ) RMS-to-mean ratio for pathlength versus (1 − g)τc : observations with their uncertainties and a theoretical prediction for the α = 2 case, as explained in the text. (Adapted from Figs. 11–13 in [15], with permission)

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where19 = 2χ/(1 − g)τc

(130)

becomes small as τc increases without bound. Scholl et al. simply took the right-hand side of the above expression for E(LT )/H to the power (α−1). This ansatz gives the correct asymptotic behavior in (120) and also a physically reasonable value of 3χ/2, which is numerically equal to or slightly larger than unity, for τc → 0 (even though diﬀusion is not a good model for transport in that limit of thin media) and α → 1. As expected, we see the observed data pointing towards α-values for the most part signiﬁcantly less than 2. Also, the eﬀective α value thus retrieved from the data decreases as expected when the clouds break up. " The lower panel shows the ratio E(L2T )/E(LT ) versus (1 − g)τc . We see that this RMS-to-mean ratio for transmitted photon pathlengths is only weakly dependent on optical thickness. This is as predicted by Davis and Marshak [58] in the α = 2 case, cf. solid curve which is based on (129) and a like expression for the RMS pathlength (only with a diﬀerent prefactor and correction term). According to the observations, this weak dependence seems to generalize to situations where α < 2. The 2nd-order moment of transmitted solar photons was previously investigated observationally by Min et al. [54], also thanks to enhanced instrumental capability. When comparing pathlength mean and variance, these authors found compatibility with the Davis and Marshak predictions for the α = 2 case but rather as an extreme situation. However, their data was normalized and plotted diﬀerently than in the lower panel of Fig. 9; so there does not seem to be any contradiction. In the following, I describe a new 1D transport theory with a power-law propagation kernel that accounts for all the observed properties of transmitted solar photons. 5.4 A New 1D Theory From the outset of this paper, we have used the standard Eulerian framework of RT based on the 3D linear transport equation and describing all the dependent and independent variables, the relevant coeﬃcients, the BCs, and so on. We then drifted towards an ever more Lagrangian outlook, describing in detail —at times graphically— how individual photon beams interact with homogeneous and variable optical media: propagation, scattering, and so on. We now return to an Eulerian perspective on RT in order to formulate a more general class of 1D models inspired by our ﬁndings on the systematic eﬀects of 3D spatial variability on the propagation process. This new model 19

In the diﬀusion approximation, parameter χ in (130) arises in the (mixed) boundary conditions applied to the parabolic PDE that determines photon density U inside a uniform slab. This PDE results from combining (110) with (108) and χ t is the so-called “extrapolation length” [45]. Values of χ = 2/3 or χ = 0.71 have been used, depending on the quantity of interest and the required accuracy [58].

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will of course neatly explain the new observations described in the previous subsection. First, we review standard 1D RT modeling when the sources are azimuthally symmetric (equivalently, azimuthally-averaged). We thus wish to determine the radiance function I(z, µ) of two variables 0 < z < H and −1 ≤ µ ≤ +1 that obeys ⎤ ⎡ +1 dI 0 (131) = σ ⎣−I(z, µ) + ˚ p(µ , µ)I(z, µ )dµ ⎦ , µ dz 2 −1

with the following very speciﬁc BCs I(0, µ) = F0 /π, 0 < µ ≤ +1 (diﬀuse irradiance) I(H, µ) = 0, −1 ≤ µ < 0 (vacuum)

(132)

where F0 is the incoming ﬂux. Immediate inspection of the above problem tells us that the usual change of variables, from physical to optical depth, dz → dτ = σdz, and (133) H → τc = σH in the BCs , is in order.20 This amounts to using the (constant) MFP = 1/σ as the unit of length and it is indeed one of the fundamental length scales of the problem, the other being cloud thickness H. In (131), we use the azimuthally-averaged phase function 1 ˚ p(µ , µ) = 2π

2π

p(µµ +

"

1 − µ2

"

1 − µ2 cos φ)dφ .

(134)

0

If discrete ordinates are used, there is no need to Fourier decompose the problem into as many decoupled 1D RT equations; only the discrete values of ˚ p(µ , µ) are required. If spherical harmonics are used, then one must invoke [59] 1 ˚ p(µ , µ) = (2n + 1)ηn Pn (µ)Pn (µ ) . (135) 4π n≥0

In the case of the Henyey–Greenstein phase function, we recall from (31) that ηn = g n . To formulate the integro-diﬀerential problem in (131)–(132), we can also use the integral form I = K∞ I + I0 from (4) where % H +1 dz −1 ˚ pg (µ , µ)e−σ(z −z)/|µ | [·]dµ /|µ |, µ > 0 0 zz +1 K∞ (σH, 0 , g)[·] = σ . 2 dz −1 ˚ pg (µ , µ)e−σ(z−z )/|µ | [·]dµ /|µ |, µ < 0 0 (136) This is useful even if extinction σ and other parameters, 0 and ˚ p(µ , µ), were to depend on z. 20

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In the arguments of the kernel K∞ we have anticipated the usual change of variables, from physical to optical units of depth, in (133). The BCs in (132) dictate the forcing term, I0 (z, µ) =

F0 exp(−σz/µ) , π

(137)

corresponding to diﬀuse irradiation at the cloud top (z = 0) and an absorbing lower boundary (z = H). As recalled in the Introduction, this problem is formally or computationally solved by iteration. But why the subscript “∞”? The exponential terms in the transport kernel (136) and the uncollided ﬂux (137) have very speciﬁc physical meanings: we are looking respectively at the FPD (grouping the exponential with the σ) and the probability of transmission from the boundary to a distance s = z/µ where the ﬁrst scattering (or absorption) happens. We propose simply to replace these as follows. First, the direct transmission term exp(−σz/µ) in (137) becomes P (s) = Tdir (s; σ , a) =

1 (1 + σ s/a)a

(138)

from (87) with s = z/µ. Second, the FPD term σ exp(−σ|z −z|/|µ |) in (136) becomes d σ (139) p(s) = Tdir (s; σ , a) = ds (1 + σ s/a)a+1 with s = |z − z|/|µ |. This leads to the more general class of 1D integral transport equation to solve based on the kernel Ka (σ H,

0

× ⎧ −(a+1) ⎪ σ (z −z) ⎨ H dz +1 ˚ p (µ , µ) 1 + [·]dµ /|µ |, µ > 0 g a|µ | z −1 , −(a+1) ⎪ σ (z−z ) ⎩ z dz +1 ˚ p (µ , µ) 1 + [·]dµ /|µ |, µ < 0 a|µ | 0 −1 g 0 , g)[·]

= σ

2

(140) where a > 0. The choice of arguments again reﬂects that we can make the natural change of spatial variable in (133): dz → dτ = σ dz and H → τc = σ H. However, τ can no longer be interpreted as the physical depth z measured in units of MFP because the MFP is no longer 1/σ and is indeed systematically longer, cf. (88). The uncollided radiance term is also changed, in this case, to −a F0 σ z , (141) I0 (z, µ) = 1+ π a|µ| with a > 0. As previously shown, all these expressions give the proper limits for a → ∞ as long as we identify the mean extinction σ with the uniform (a.k.a. in probability as the “degenerate”) value σ.

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What results are we particularly interested in? The total transmission is one of them, namely, 2π Ta (τc , 0 , g) = F0

1 µI(H, µ)dµ

(142)

0

as it measures the mean particle current through the medium. In atmospheric RT, this is the amount of spectral radiation that reaches a dark surface, such as the ocean. To compute the radiant energy sent immediately back to space, as well as total ﬂux reaching a partially reﬂective surface, we need the albedo 2π Ra (τc , 0 , g) = F0

0 |µ|I(0, µ)dµ ,

(143)

−1

In remote sensing applications, we are limited to sampling radiance in direction space rather than obtaining the above integrals. So we shall turn our modeling interests towards “zenith” radiance at ground level (here, z = H) and “nadir” radiance at the top-of-the-atmosphere (here, z = 0): respectively, Ia↓ (τc ,

0 , g)

=

πI(H, +1) F0

(144)

Ia↑ (τc ,

0 , g)

=

πI(0, −1) , F0

(145)

and

where we have opted for so-called “Bidirectional Reﬂection Function” units.21 If, beyond the above steady-state quantities (for any 0 ≤ 1), we are interested in statistics of pathlength L (usually for the conservative case 0 = 1) then we can apply the corollary of the equivalence theorem in (128) using the simple change of variables 0 → k = (1 − 0 )σ , hence 0

= 1 − kH/τc .

(146)

This means that the expression in (128) for the nth-order pathlength moments (n = 1, 2, 3, . . . ) now reads as q 1 ∂ I(· · · ) (147) E(Ln | · · · )/H n = τc −n − I(· · · ) ∂ 0 0 =1 21

In these convenient “BRF” units, there is no diﬀerence in numerical values between albedo —or transmittance in the case of (144)— and radiance if it is isotropic (a.k.a. Lambertian). If not, which is of course the generic case, then we can interpret the radiance as an eﬀective albedo or transmittance for the particular direction of interest.

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where we have normalized the pathlengths by H and let (· · · ) represent both photon phase-state variables (z, µ) and parameters of the optical medium (τc , 0 , g). We have also dropped the tildes and the explicit dependence on k( 0 ) in (128) since it is now just a parameter of the 1D RT problem. Recall that the recipe in (147) can be applied to any of the above angularly integrated or sampled quantities, namely, I(· · · ) → Ta (·) or Ia↓ (·) at z = H or else Ra (·) or Ia↑ (·) at z = 0 where (·) now represents only optical parameters. After using (147), the H-normalized pathlength moments for boundary ﬂuxes or z-axis radiances will depend only on τc and g. The Lagrangian asymptotic analysis of the previous section suggests this dependence should collapse onto a power-law in (1 − g)τc for large enough values. But what is the connection between a > 0 and α < 2 from Sect. 4.3? Recalling that α is the critical (logarithmically divergent) moment of the FPD, we see that α = min{a, 2} ,

(148)

irrespective of the phase function’s properties (i.e., g) which can only impact low orders of scattering. The remaining and important question is how do we numerically solve the problem in (4) with (140) and (141)? It will be interesting to see how discrete ordinates and spherical harmonics apply to the new class of 1D transport problems in (140)–(141). What is clear at present is that there is an intimate connection between the exponentials in (136)–(137) and the 1D diﬀerential equation formulation in (131) —and this diﬀerential formalism plays a key role in obtaining some of the classic numerical solutions [60]. There is no integro-diﬀerential equivalent of (140)–(141) where power-law kernels appear. Expressions with pseudo-diﬀerential operators22 may exist, but this remains an open question. See Buldyrev et al. [61] for a positive answer to this question for a related (strictly L´evy) type of propagation kernel. If one is only interested in the boundary ﬂuxes (142)–(143), then the easiest is certainly to use a basic Monte Carlo algorithm. Bearing in mind the general theory of the method, all we need to do as far as the random particle trajectory is concerned is to replace the single line of code that executes (48) by a , (149) s = (ξ −1/a − 1) × σ where, if H and τc are the given quantities, we use σ = τc /H. If one is only interested in the outgoing radiances (144)–(145) then Monte Carlo is still the easiest way to go although we will also need the probability of direct transmission to the upper or lower boundary according to (138), with s = z and H − z respectively, to compute weights in the local estimation technique [62]. Last but not least, the Monte Carlo method enables direct 22

A good example is (−∇2 )γ , for 0 < γ < 1 which can be implemented easily in Fourier space in the form of a high-pass ﬁlter in k2γ .

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estimation of the normalized moments in (147) while tracing the particle trajectories under the assumption that 0 = 1. Figure 10 shows numerical results for lower-boundary (i.e., transmission) quantities for ﬁve incarnations of the 1D transport model: the standard case (a = ∞), and 4 new cases (a = 1.2, 1.4, 1.6, 1.8) where the MFP is still ﬁnite. In all cases, the scattering kernel p(Ω • Ω) was Henyey–Greenstein with g = 0.85. Panel (a) shows Ta (τc , 1, g) versus (1 − g)τc in log-log axes; we see that the anomalous asymptotic scaling in (123) is indeed realized with α from (148) at large enough τc . The same remark applies to Ia↓ (τc , 1, g) in panel (b) that furthermore shows the characteristic linear increase with optical depth at small values. A maximum in Ia↓ (τc , 1, g) is reached between (1 − g)τc ≈ 1/3 (a = ∞) and ≈ 1 (1 < a < 2). Panels (c) and (d) display in lin-lin axes respectively the mean and RMS-to-mean ratio for pathlengths based on Ia↓ in (147). Where the asymptotic regime starts can be determined visually by examining Figs. 10a–b: we see that the scaling predicted for Ta (τc , 1, g) in (123) seems to begin at (1 − g)τc ≈ 10 (i.e., τc ≈ 70 for g = 0.85) for the selected values of a = α between 1.2 and 1.8; this threshold is practically oﬀ the chart in Fig. 10c (which was designed to have the same span as the middle panel of Fig. 9). It is well-known —and we clearly see— that asymptotic behavior starts much sooner for the standard exponential kernel (a = ∞), near (1 − g)τc ≈ 2 (i.e., τc ≈ 13 for g = 0.85). We understand the delayed transition to asymptotic scaling in transmission and in mean pathlength when a decreases from ∞ to unity as a consequence of the growth of the optical (physical) thicknesses of the top and bottom boundary layers for ﬁxed τc (H and σ ). Indeed, according to (88) and (106), the relevant transport MFP is t (a) =

a H E(s) = × , 1 − 0g a − 1 (1 − 0 g)τc

(150)

a natural estimate for the boundary layer thickness, that increases without bound as a → 1. As a gets close to unity, the two boundary layers have invaded the whole domain; physically, the presence of the boundaries can be felt at all levels in the medium because of the long tail of the propagation kernel in (139). If we retain the often quoted criterion [63], from the a = ∞ case, for the onset of asymptotic behavior, 2×

2 t (∞) = 1, H (1 − 0 g)τc

then here we have:

(151)

2a (152) a−1 which ranges from 4 to 12 for 1.2 ≤ a ≤ 1.8, more-or-less as observed in the numerical simulations. (1 −

0 g)τc

Eﬀective Propagation Kernels in Structured Media (a)

1

10

100

0.1 a a a a a

0

=1

g = 0.85 0.01 0.01

0.1

1

mean optical depth, 9

for transmitted (zenith) radiance

0.1

L

Transmittance, T

(c)

mean optical depth, 1

10

0

8

6

/F

zen

0.01 0.01

2 1/2

=1

g = 0.85

L

normalized zenith radiance, I

0.1

0

0.1

1

10

rescaled mean optical depth, (1–g)

100

4 3 2 1

0

=1

g = 0.85 5

(d) 100

a a a a a

80

5

0

/ L for transmitted (zenith) radiance

10

60

10

15

rescaled mean optical depth, (1–g)

0

1

1

40

0

100

optical depth, 0.1

20 a a a a a

7

rescaled mean optical depth, (1–g)

(b)

133

100

mean optical depth, 1.7

0

20

40

60

80

100

1.6 1.5 1.4 1.3 1.2 1.1 1.0

a a a a a

0.9 0.8

0

=1

g = 0.85

0.7 0

5

10

15

rescaled mean optical depth, (1–g)

Fig. 10. Results from Monte Carlo solutions of the new 1D integral transport equation in (4) using (140)–(141) with 0 = 1 and g = 0.85. (a) Log-log plot of transmission in (142) as a function of (1−g)τc for a = 1.2, 1.4, 1.6, 1.8, and ∞. (b) Same as (a) but for zenith radiance in (144). In both panels, we see the predicted scaling in τc −α/2 for large enough values. (c) Lin-lin plot of E(L)/H from (147) with n = 1 using Ia↓ where we see the predicted scaling in τc α−1 for large values; by comparison with the middle panel of Fig. 9, we also note that the observations point to a-values between 2 and some relatively " large value rather than α-values per se smaller than 2. (d) Same as (c) but for E(L2 )/E(L) where we conﬁrm the recent observation that the RMS path scales like the mean independently of a or α, cf. Fig. 9 Numerical Details: The MFP = a/(a − 1)σ was set to unity and 11 physical thicknesses were examined: H = 2−2,...,8 . This was done in a single run for a given value of a, by ﬂagging the history as to which of the 11 media it had not yet escaped. The mean optical thickness τc = σH was computed after the fact from a and H. For each run, 5×106 histories were generated. This resulted in a maximum of 23249, 26317, 24997, 31306, and 44953 scatterings respectively for a = 1.2, 1.4, 1.6, 1.8, and ∞ in the case where H = 256 ×

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On purpose, Fig. 10c for E(L)/H versus (1 − g)τc mimics the display of observations in the middle panel of Fig. 9. We see that, although they are not plotted, the data points populate a region where we will ﬁnd transport models with 2 < a < ∞. Such models have the same asymptotic scaling as the α = 2 case, but its onset as optical depth increases is delayed beyond an already quite large value estimated by Davis and Marshak [58] to be ≈10 for (1 − g)τc . Consequently, it is now clear that 1. the observations of Scholl et al. are mostly not in the asymptotic transport regime, and 2. a model such as the present 1D integral transport equation that accounts for pre-asymptotic behavior is required to further exploit atmospheric A-band data. Our ﬁnal Fig. 10d shows the same RMS-to-mean ratio as in the lower panel of Fig. 9 on the same vertical and horizontal scales. We see that the new model explains well the weak dependence on optical depth with furthermore the right trends in both a and τc . However, we must emphasize here that the choice of upper BC in (132) that expresses diﬀuse isotropic illumination (equivalently, the source terms (137) and (141)) is important to obtain the observed trend at small optical depths. This makes physical sense because the upper level of “cloud” is, by deﬁnition, a strong diﬀuser.23 This last ﬁnding sheds new light on the phenomenology of particle transport with power-law FPDs presented in Sect. 4.3. That approach was entirely predicated on a judicious truncation of the universal law that describes the distribution along orders-of-scattering of particles reﬂected from a semiinﬁnite isotropically"scattering medium. The tight connection we found numerically between E(L2 ) and E(L) tells is that this truncation has to be a very sharp one indeed, irrespective of the ﬁnite optical thickness of the medium. Otherwise, signiﬁcant numbers of straggling particles would aﬀect the higher-order moments.

6 Concluding Remarks We have investigated the impact of unresolved spatial variability on linear particle transport systems at the most fundamental level (i.e., incoherent geometric optics in the case of our primary application to atmospheric radiative transfer). The elementary process of interest here is propagation between an emission or scattering event and the next scattering event or an absorption/escape. In uniform regions —a tempting assumption to make when 23 An improved upper BC would capture the fact that some of the sunlight is transmitted quasi-ballistically through the top layer, but this would call for at least two new parameters (solar zenith angle and the partition between diﬀuse and collimated illumination).

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structure is unknown— this propagation is controlled by an exponential decay in particle ﬂux along a given beam (the well-known Beer’s law in radiative transfer). Concurring with other authors, I showed that on average this decay is in fact sub-exponential in random but spatially correlated media. This is established using both transport-theoretical methods and a somewhat deeper point-process approach pioneered by A. Kostinski and coworkers. Furthermore, the actual mean-free-path is systematically larger than predicted with a uniform-medium assumption based on mass conservation. Finally, for the predicted extension of the mean-free-path and systematic deviation from exponential decay to matter, there needs to be spatial correlations at scales commensurate with the (actual) mean-free-path. Two important consequences are: • Generally speaking, homogenization (or “eﬀective medium”) theory can capture the extended mean-free-path eﬀect but, being limited to exponential kernels, it will fail to capture other eﬀects at scales much larger (and also much smaller) than the mean-free-path. • For applications to the Earth’s cloudy atmosphere, the observed optical variability, which indeed has long-range correlations shaped by small- and large-scale turbulence, leads to propagation kernels with power-law tails. I also critically re-examined the impact of sub-exponential free-path distributions on multiple scattering in ﬁnite media. Guided by early successes of a phenomenological model for solar photon propagation through the cloudy atmosphere based on L´evy (rather than Gaussian) random walks, I proposed here a new 1D transport model with propagation kernels having power-law tails as well as the proper near-ﬁeld behavior. These kernels are particularly well adapted to problems of large-scale radiative transfer in the cloudy atmosphere since they follow directly from the observed Gamma-type distributions of optical variability. This new transport equation is solved numerically using a modiﬁed Monte Carlo scheme. The new model explains all the features of recent observations of the mean and RMS pathlengths of solar photons eventually transmitted to the ground in a narrow ﬁeld-of-view around the zenith direction. In particular, we conclude that large-scale radiation transport in the cloudy atmosphere is in a pre-asymptotic regime where the thickness of the radiative boundary-layer (a couple of actual mean-freepaths) is commensurate with the outer thickness of the medium (from ground level to the top of the highest cloud). In general, the new transport model based on eﬀective propagation kernels oﬀers a ﬂexible intermediate approach to the perennial problem of unresolved variability. It is not as simple as homogenization, but it does avoid its main limitation. Nor is it as complicated as the coupled mean-ﬁeld transport equations for random Markovian media, even in the two-state case. So one hopes to see applications in a wide variety of transport problems in both engineering and natural sciences.

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Acknowledgments and Dedication Sustained ﬁnancial support is acknowledged from the U.S. DOE’s Atmospheric Radiation Measurement (ARM) Program over the long period during which this research was conducted. The author thanks Howard Barker, Brian Cairns, H´el`ene Frisch, Lee Harrison, Andrew Heidinger, Yong-Xiang Hu, Yuri Knyazikhin, Alexander Kostinski, Edward Larsen, Shaun Lovejoy, Alexander Marshak, Qi-Long Min, Jim Morel, Klaus Pfeilsticker, Sydney Redner, Graeme Stephens, and Warren Wiscombe for stimulating discussions around the topics of this paper both in person and in across the internet. Special thanks go to Igor Polonsky for the timely help in reﬁning the numerics in Fig. 10 and much of the LaTeX source material. I wish to thank the volume editor, Frank Graziani, for his help, his patience, and the passion he put into bringing to life the series of interdisciplinary Computational Transport Workshops. Finally, I dedicate this paper to the loving memory of my late father, John F. Davis, M.Eng., M.D., who passed away during the time of its writing. By boldly shifting careers multiple times, moving family-and-all to a new “old” country, and traveling far and wide beyond there, he gave meaning to the proverbial path less-traveled. More to the point of this paper, he demonstrated that the occasional large jump enables the traveler to sample more eﬀectively the vast possibilities that inhabit geographical- and/or career-spaces, given the unknown but ﬁnite time of travel. I’ve tried to follow his example in taking those same kinds of steps/jumps both in real space and in “interest space” and, sure enough, many of my moves proved (so far) to be at once risky and wise. At least I’ve learned to look forward to the next destination and to welcome the next challenge.

References 1. D. Mihalas. Stellar Atmospheres. Freeman, San Francisco (CA), 2nd edition, 1979. 2. O.A. Avaste and G.M. Vainikko. Solar radiative transfer in broken clouds. Izv. Acad. Sci. USSR Atmos. Oceanic Phys., 10:1054–1061, 1974. 3. G.C. Pomraning. Linear Kinetic Theory and Particle Transport in Stochastic Mixtures. World Scientiﬁc Publishing, Singapore, 1991. 4. G.L. Stephens. Radiative transfer through arbitrary shaped optical media, II: Group theory and simple closures. J. Atmos. Sci., 45:1837–1848, 1988. 5. H.W. Barker. A parameterization for computing grid-averaged solar ﬂuxes for inhomogeneous marine boundary layer clouds - Part 1, Methodology and homogeneous biases. J. Atmos. Sci., 53:2289–2303, 1996. 6. H.W. Barker and A.B. Davis. Approximation methods in atmospheric 3D radiative transfer, Part 2: Unresolved variability and climate applications. In 3D Radiative Transfer in Cloudy Atmospheres. A. Marshak and A.B. Davis (eds.). Springer-Verlag, Heidelberg, pages 343–383, 2005.

Eﬀective Propagation Kernels in Structured Media

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7. F. Graziani and D. Slone. Radiation transport in 3D heterogeneous materials: Direct numerical simulation. In Proceedings of the American Nuclear Society Winter Meeting, New Orleans, LA. ANS, LaGrange Park (IL), 2003. 8. R.F. Cahalan. Bounded cascade clouds: Albedo and eﬀective thickness. Nonlinear Proc. Geophys., 1:156–167, 1994. 9. B. Cairns, A.A. Lacis, and B.E. Carlson. Absorption within inhomogeneous clouds and its parameterization in general circulation models. J. Atmos. Sci., 57:700–714, 2000. 10. P.M. Gabriel and K.F. Evans. Simple radiative transfer methods for calculating domain-averaged solar ﬂuxes in inhomogeneous clouds. J. Atmos. Sci., 53:858– 877, 1996. 11. G.W. Petty. Area-average solar radiative transfer in three-dimensionally inhomogeneous clouds: The independently scattering cloudlets model. J. Atmos. Sci., 59:2910–2929, 2002. 12. M.F. Shlesinger, G.M. Zaslavsky, and U. Frisch (eds.). L´evy Flights and Related Topics in Physics. Springer-Verlag, New York (NY), 1995. 13. G. Samorodnitsky and M.S. Taqqu. Stable Non-Gaussian Random Processes. Chapman and Hall, New York (NY), 1994. 14. K. Pfeilsticker. First geometrical pathlengths probability density function derivation of the skylight from spectroscopically highly resolving oxygen Aband observations. 2. Derivation of the L´evy-index for the skylight transmitted by mid-latitude clouds. J. Geophys. Res., 104:4101–4116, 1999. 15. T. Scholl, K. Pfeilsticker, A.B. Davis, H. Klein Baltink, S. Crewell, U. L¨ohnert, C. Simmer, J. Meywerk, and M. Quante. Path length distributions for solar photons under cloudy skies: Comparison of measured ﬁrst and second moments with predictions from classical and anomalous diﬃusion theories. J. Geophys. Res., 2005 (in press). 16. A.B. Davis and Yu. Knyazikhin. A Primer in 3D Radiative Transfer. In 3D Radiative Transfer in Cloudy Atmospheres. A. Marshak and A.B. Davis (eds.). Springer-Verlag, Heidelberg, pages 153–242, 2005. 17. D. Deirmendjian. Electromagnetic Scattering on Spherical Polydispersions. Elsevier, New York (NY), 1969. 18. Yu. Knyazikhin, A. Marshak, W.J. Wiscombe, J. Martonchik, and R.B. Myneni. A missing solution to the transport equation and its eﬀect on estimation of cloud absorptive properties. J. Atmos. Sci., 59:3572–3585, 2002. 19. M. Abramowitz and I.A. Stegun (eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. US Govt. Printing Oﬃce, Washington (DC), 1964. 20. L.C. Henyey and J.L. Greenstein. Diﬃuse radiation in the galaxy. Astrophys. J., 93:70–83, 1941. 21. A. Davis, S. Lovejoy, and D. Schertzer. Supercomputer simulation of radiative transfer in multifractal cloud models.In IRS’92 : Current Problems in Atmospheric Radiation. S. Keevallik and O. K¨ arner (eds.). A. Deepak Publishing, Hampton (VA), pages 112–115, 1993. 22. A. Schuster. Radiation through a foggy atmosphere. Astrophys. J., 21:1–22, 1905. 23. W.E. Meador and W.R.Weaver. Two-stream approximations to radiative transfer in planetary atmospheres: A uniﬁed description of existing methods and a new improvement. J. Atmos. Sci., 37:630–643, 1980.

138

A.B. Davis

24. N.G. van Kampen. Stochastic diﬀerential equations. Physics Reports (Phys. Lett. C), 24:171–228, 1976. 25. F. Reif. Fundamentals of Statistical and Thermal Physics. McGraw-Hill, New York (NY), 1965. 26. A.B. Kostinski. On the extinction of radiation by a homogeneous but spatially correlated random medium. J. Opt. Soc. Amer. A, 18:1929–1933, 2001. 27. A.B. Davis and A. Marshak. Photon propagation in heterogeneous optical media with spatial correlations: Enhanced mean-free-paths and wider-thanexponential free-path distributions. J. Quant. Spectrosc. Radiat. Transfer, 84:3– 34, 2004. 28. W. Feller. An Introduction to Probability Theory and its Applications. Wiley, New York (NY), 1971. 29. U. Frisch and G. Parisi. A multifractal model of intermittency. In Turbulence and Predictability in Geophysical Fluid Dynamics, M. Ghil, R. Benzi, and G. Parisi (eds.), North Holland, Amsterdam (The Netherlands), pages 84–88, 1985. 30. A. Marshak, A. Davis, W.J. Wiscombe, and R.F. Cahalan. Scale-invariance of liquid water distributions in marine stratocumulus, Part 2 - Multifractal properties and intermittency issues. J. Atmos. Sci., 54:1423–1444, 1997. 31. H.-O. Peitgen and D. Saupe. The Science of Fractal Images. Springer-Verlag, New York (NY), 1988. 32. R.F. Cahalan, W. Ridgway, W.J. Wiscombe, T.L. Bell, and J.B. Snider. The albedo of fractal stratocumulus clouds. J. Atmos. Sci., 51:2434–2455, 1994. 33. H.W. Barker, J.-J. Morcrette, and G.D. Alexander. Broadband solar ﬂuxes and heating rates for atmospheres with 3D broken clouds. Quart. J. Roy. Meteor. Soc., 124:1245–1271, 1998. 34. J.W. Goodman. Statistical Optics. Wiley, New York (NY), 1985. 35. L.D. Landau and E.M. Lifshitz. Statistical Physics. Pergamon, New York (NY), 3rd edition, 1980. 36. L. Mandel and E. Wolf. Optical Coherence and Quantum Optics. Cambridge U. Press, New York (NY), 1995. 37. H.W. Barker, B.A. Wielicki, and L. Parker. A parameterization for computing grid-averaged solar ﬂuxes for inhomogeneous marine boundary layer clouds Part 2, Validation using satellite data. J. Atmos. Sci., 53:2304–2316, 1996. 38. A. Marshak, A. Davis, W.J. Wiscombe, and R.F. Cahalan. Radiative smoothing in fractal clouds. J. Geophys. Res., 100:26,247–26,261, 1995. 39. R.A. Shaw, A.B. Kostinski, and D.D. Lanterman. Super-exponential extinction of radiation in a negatively-correlated random medium. J. Quant. Spectrosc. Radiat. Transfer, 75:13–20, 2002. 40. A. Davis and A. Marshak. L´evy kinetics in slab geometry: Scaling of transmission probability. In Fractal Frontiers. M.M. Novak and T.G. Dewey (eds.). World Scientiﬁc, Singapore, pages 63–72, 1997. 41. A.G. Borovoi. On the extinction of radiation by a homogeneous but spatially correlated random medium: Comment. J. Opt. Soc. Amer. A, 19:2517–2520, 2002. 42. A.G Borovoi. Radiative transfer in inhomogeneous media. Dokl. Akad. Nauk SSSR, 276:1374–1378, 1984 (in Russian). 43. A.B. Kostinski. On the extinction of radiation by a homogeneous but spatially correlated random medium: Reply to comment. J. Opt. Soc. Amer. A, 19:2521– 2525, 2002.

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139

44. E.P. Zege, A.P. Ivanov, and I.L. Katsev. Image Transfer Through a Scattering Medium. Springer-Verlag, New York (NY), 1991. 45. K.M. Case and P.F. Zweifel. Linear Transport Theory. Addison-Wesley, Reading (MA), 1967. 46. E. Sparre Anderson. On the ﬂuctuations of sums of random variables. Math Scand., 1:236–285, 1953. 47. U. Frisch and H. Frisch. Universality in escape from half space of symmetrical random walks. In L´evy Flights and Related Topics in Physics, M.F. Shlesinger, G.M. Zaslavsky, and U. Frisch (eds.), Springer-Verlag, New York (NY), pages 262–268, 1995. 48. G.L. Stephens. Reply (to Harshvardhan and Randall). Mon. Wea. Rev., 113:1834–1835, 1985. 49. G.L. Stephens, A.K. Heidinger, and P.M. Gabriel. Photon paths and cloud heterogeneity: An observational strategy to assess eﬀects of 3D geometry on radiative transfer. In 3D Radiative Transfer in Cloudy Atmospheres. A. Marshak and A.B. Davis (eds.). Springer-Verlag, Heidelberg, pages 587–616, 2005. 50. W.M. Irvine. The formation of absorption bands and the distribution of photon optical paths in a scattering atmosphere. Bull. Astron. Inst. Neth., 17:226–279, 1964. 51. V.V. Ivanov and Sh. A. Sabashvili. Transfer of resonance radiation and photon random walks. Astrophysics and Space Science, 17:13–22, 1972. 52. K. Pfeilsticker, F. Erle, O. Funk, H. Veitel, and U. Platt. First geometrical pathlengths probability density function derivation of the skylight from spectroscopically highly resolving oxygen A-band observations: 1. Measurement technique, atmospheric observations, and model calculations. J. Geophys. Res., 103:11,483– 11,504, 1998. 53. Q.-L. Min and L.C. Harrison. Joint statistics of photon pathlength and cloud optical depth. Geophys. Res. Lett., 26:1425–1428, 1999. 54. Q.-L. Min, L.C. Harrison, P. Kiedron, J. Berndt, and E. Joseph. A highresolution oxygen A-band and water vapor band spectrometer. J. Geophys. Res., 109:D02202, doi:10.1029/2003JD003540, 2004. 55. Q.-L. Min, L.C. Harrison, and E.E. Clothiaux. Joint statistics of photon pathlength and cloud optical depth: Case studies. J. Geophys. Res., 106:7375–7385, 2001. 56. R.W. Portmann, S. Solomon, R.W. Sanders, J.S. Daniel, and E. Dutton. Cloud modulation of zenith sky oxygen path lengths over Boulder, Colorado: Measurement versus model. J. Geophys. Res., 106:1139–1155, 2001. 57. M. Quante, H. Lemke, H. Flentje, P. Francis, and J. Pelon. Boundaries an internal structure of mixed phased clouds as deduced from ground-based 95Ghz radar and airborne lidar measurements. Phys. Chem. Earth, 25:889–895, 2000. 58. A.B. Davis and A. Marshak. Space-time characteristics of light transmitted through dense clouds: A Green function analysis. J. Atmos. Sci., 59:2714–2728, 2002. 59. W.J. Wiscombe and G.W. Grams. The backscattered fraction in two-stream approximations. J. Atmos. Sci., 33:2440–2451, 1976. 60. J. Lenoble (ed.). Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures. A. Deepak Publishing, Hampton (VA), 1985.

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61. S.V. Buldyrev, M. Gitterman, S. Havlin, A.Ya. Kazakov, M.G.E. da Luz, E.P. Raposo, H.E. Stanley, and G.M. Viswanathan. Properties of L´evy ﬂights on an interval with absorbing boundaries. Physica A, 302:148–161, 2001. 62. G. Marchuk, G. Mikhailov, M. Nazaraliev, R. Darbinjan, B. Kargin, and B. Elepov. The Monte Carlo Methods in Atmospheric Optics. Springer-Verlag, New York (NY), 1980. 63. H.C. van de Hulst. Multiple Light Scattering: Tables, Formulae and Applications. Academic Press, San Diego (CA), 1980.

Mathematical Simulation of the Radiative Transfer in Statistically Inhomogeneous Clouds Evgueni I. Kassianov Paciﬁc Northwest National Laboratory, 902 Battelle Boulevard, P.O. Box 999, Richland, WA [email protected] Abstract. The solar radiation transport through broken clouds can be treated as photon transport through stochastic media. The stochastic radiative transfer equation and a new statistically inhomogeneous Markovian model are used to derive analytical equations for the ensemble-averaged intensity. The computational method for solving these equations is introduced, and their accuracy and robustness are discussed. Validation tests show good predictive performance for the mean radiative properties.

1 Introduction Transport theory, also called radiative transfer (RT) theory, has been extensively developed over a century. The equation of RT (or linear transport equation) applies successfully to numerous areas, such as atmospheric RT, neutron diﬀusion and sound propagation [e.g., CZ67, Lio02]. The current book provides some new and important applications of this equation. The latter has parameters that are deterministic functions of space and time, thus the equation is deterministic. Sometimes the lack of a complete description of these parameters renders their statistical treatment: although one does not know their values at each space point and time instant, one does know the values they can assume and probabilities of these values. Thus, the RT theory can be formulated in a statistical setting and the corresponding RT equation can be viewed as stochastic [e.g., Pom91]. The stochastic RT equation is applicable for diﬀerent problems: for example, the neutron transport in a boiling water reactor, where liquid water and vapor are considered as stochastic mixture. Broken clouds are an other example of such a potential application. Since the location, geometry, and optical properties of each individual cloud are not known in the deterministic fashion, the cloud properties can be considered as random variables. A radiation ﬁeld transformed by such a cloud ﬁeld also becomes random. This fact dictates the necessity to use statistical methods to study the cloudradiation interaction. The ultimate goal of a statistical approach is to suggest a relatively simple and practical useful treatment of the stochastic radiative transfer problem and to establish the relationship between the statistical parameters of clouds and radiation.

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Here we introduce the statistical treatment of solar radiation in multilayer broken clouds. This treatment is based on the stochastic RT equation (Sect. 2) and a new statistically inhomogeneous Markovian model (Sect. 3). The latter has been used to derive a closed system of equations for the mean solar radiance (Sect. 4). The computational method for their solution is also described in Sect. 4. The accuracy and robustness of the obtained equations are discussed in Sect. 5.

2 Stochastic RT Equation We will assume that broken clouds occupy a certain layer Λ: hb ≤ z ≤ ht. Within Λ, the random monochromatic radiance (or intensity) I (r, ω) at the point r = (x, y, z) in direction ω = (a, b, c) satisﬁes the three-dimensional (3D) stochastic transfer equation ω ∇I (r, ω) + σ (r) κ (r) I (r, ω) = ω0 (r) σ (r) κ (r) g (r, ω, ω ) I (r, ω, ω ) dω ,

(1)

4π

with the boundary conditions I(r0 , ω) = I(x, y, hb, ω) = I ↑ (r0 , ω), c > 0 I(rM , ω) = I(x, y, ht, ω) = I ↓ (rM , ω), c < 0

(2)

where a, b, c are the projections of the unit vector ω on the axes X,Y ,Z (Cartesian coordinate system), respectively, I ↑ (r0 , ω) and I ↓ (rM , ω) are the radiances of the external sources at the boundary of layer Λ, z0 = hb (the cloud base altitude) and zM = ht (the cloud top altitude) σ(r) is the extinction coeﬃcient, ω 0 (r) is the single scattering albedo, and g(r, ω, ω ) is the scattering phase function. Here σ(r), ω 0 (r) and g(r, ω, ω ) are nonrandom functions. In contrast to the deterministic transfer equation, the stochastic one contains the random indicator ﬁeld κ(r), κ(r) = 1 inside clouds and κ(r) = 0 outside the clouds. In other words, the broken cloud ﬁeld may be considered a mixture of two atmospheric components (clouds and clear air), with ﬂuctuating geometrical parameters. In the limiting case, where randomness vanishes (κ(r) is a nonrandom function), (1) describes the 3D deterministic radiative transfer. The statistics of the optical parameters of broken clouds are determined completely by the statistical characteristics of the cloud ﬁeld, κ(r). Construction of a physical model of κ(r) is a complex problem, the solution of which can be based on results of a cloud-resolving model and/or a large eddy simulation model. These models provide 3D cloud ﬁelds which mimic real clouds, thus, the simulated cloud ﬁelds can be a good source for cloud statistics [e.g., OK03; KAM03]. The construction of a physical model

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has not yet been solved, and therefore researchers are forced to use some mathematical models of κ(r). The latter should be selected from the following circumstances: (i) the possibility for determining the input parameters of the model from observations, and (ii) the possibility for deriving relatively simple analytical equations, which links the ensemble-averaged intensity with statistics of κ(r). Below we consider one such model.

3 Statistically Inhomogeneous Model The broken clouds can be represented as a Markovian mixture of cloudy and noncloudy segments [e.g., Tit90; MBP93]. In this case, random indicator ﬁeld κ(r) is described completely by unconditional κ(r) = P {k(r) = 1} and conditional V (r2 , r1 ) = P {κ(r2 ) = 1|κ(r1 ) = 1} probabilities of the cloud presence. Here and in the following, the angular brackets will be used for ensemble averages over κ(r) realizations. Recall that the conditional probability of the event {κ(r2 ) = 1}, given that the event {κ(r1 ) = 1} has occurred, is P {κ(r2 ) = 1|κ(r1 ) = 1} =

P {κ(r2 ) = 1, κ(r1 ) = 1} , P {κ(r1 ) = 1}

(3)

where P {κ(r2 ) = 1, κ(r1 ) = 1} is the joint probability that both events {κ(r2 ) = 1} and {κ(r1 ) = 1} have occurred, and P {k(r1 ) = 1} > 0. For the statistically homogeneous ﬁelds, the unconditional probability κ(r) does not depend on the vertical coordinate, and the conditional one V (r2 , r1 ) depends only on the distance between points r1 and r2 for a ﬁxed direction ω = (r2 − r1 )/|r2 − r1 | [e.g., Tit90, ZT95; MBP93]. Real clouds have strong horizontal and vertical variability. To extend the Markovian approach [e.g., Tit90; MBP93] to multi-layer (or statistically inhomogeneous) broken clouds, one has to specify a statistical relationship between cloud layers. It was demonstrated that (i) the overlap of clouds at two levels tends to fall rapidly as their vertical separation is increased and (ii) the degree of overlap as a function of level separation can be described by a simple inverse exponential expression [e.g., HI00; OK03]. In our approach, we assume that the statistical relationship between two adjusted layers is described by inverse-exponential expression. We represent broken clouds as a set of correlated cloud layers: each layer is homogeneous in vertical but inhomogeneous in horizontal dimensions. Such a representation allows one to use methods previously developed for a single layer of broken clouds. To generalize the Markovian approach [e.g., Tit90; MBP93] to multilayer broken clouds, we suggested a statistically inhomogeneous model of κ(r) [Kas03]. For this model, κ(r) can vary strongly with altitude and the conditional probability is a function of the relative positions of points r1 and r2 . As an illustration, some examples are given below: if the point r1 and r2 belong to the same kth layer, then

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Vk (r2 , r1 ) = (1 − pk ) exp (−Ak |r2 − r1 |) + pk

(4)

where pk = κ(r1 ) = κ(r2 ) is cloud fraction in the kth layer, and Ak is parameter; if the point r1 and r2 belong to diﬀerent adjacent layers, namely kth and mth layers, then Vk,m (r2 , r1 ) = exp (−A∗ |r∗ − r2 |) {Vk (r1 , r∗ ) − pm } + pm ,

(5)

where r∗ = r1 + ω(z∗ − z1 )/c, ω = (a, b, c) = (r2 − r1 )/|r2 − r1 |. For the upward direction (c > 0), m = k + 1, A∗ = Aup k and z∗ equals the top altitude of the kth layer, z∗ = zk+1 ; for the downward direction (c < 0), m = k − 1, A∗ = Adw k and z∗ equals the base altitude of the kth layer dw determine the statistical relationship z∗ = zk−1 . Parameters Aup k and Ak between kth layer and its upper (k + 1) and lower (k − 1) adjacent layers, dw depend on both respectively. Note, all these parameters Ak , Aup k and Ak the 3D cloud structure and the positions of points r1 and r2 . By changing the values of these parameters, one can describe diﬀerent combinations of maximum and random cloud overlaps [Kas03]. This sketched ﬂexibility of the inhomogeneous model is its appealing feature. Also, all input parameters of this model can be estimated from observations [KAM03, KAK05].

4 Ensemble Averaged Radiance The statistically inhomogeneous model of broken clouds and the stochastic transfer equation were used to derive approximated equations for the mean solar radiance [Kas03]. It was assumed that for each kth layer the domain-averaged optical properties are constant (piecewise constant approximation), e.g., the extinction coeﬃcient σ(r) = σ(z) = σk , the single scattering albedo ω 0 (r) = ω 0 (z) = ω 0,k , and the scattering phase function g(r, ω, ω ) = g(z, ω, ω ) = gk (ω, ω ). Also it was assumed that a parallel unit ﬂux of solar radiation is incident on the upper boundary of a given cloud ﬁeld in direction ω⊕ . To get the absolute values, one should multiply the calculated radiative properties by the spectral solar constant weighted with cos(ξ⊕ ), where ξ⊕ is the solar zenith angle. The equations were obtained without considering the aerosol-molecular atmosphere and underlying surface; the latter can be added quite easily. The equation for the mean solar radiance has the form 1 ω 0 (ξ)φ(z, ξ)dξ g(ξ, ω, ω )f (ξ, ω )dω I(z, ω) = |c| Ez

4π

+ j(z, ω) δ(ω − ω⊕ )

(6)

where Ez = (hb, z) if c > 0, and Ez = (z, ht) if c < 0, ht and hb are the top height and the base height of cloud ﬁeld, respectively; j(z, ω) is the mean

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145

direct (unscattered) radiance, and f (z, ω) = σ(r)κ(r)I(r, ω) is the mean collision density, δ(.) is Dirac’s delta function. An integral equation for the mean collision density can be written as (7) f (x) = k(x, x )f (x )dx + Ψ(x) X

k(x, x ) =

ω 0 (z)g(z, ω, ω )η(r, r ) δ 2π|r − r |2

r − r − ω |r − r |

Ψ(x) = σ(z)p(z)v(z, ω)δ(ω − ω⊕ )

(8) (9)

where X is the phase space of coordinates and directions, x = (r, ω). All functions φ(r, r ), η(r, r ) and v(z, ω) = κ(r)j(r, ω) /p(z) are deﬁned by the recurrent expressions [Kas03]. The closed system of (6)–(7) can be solved by using any appropriate numerical methods or analytic techniques (e.g., the spherical harmonic method). Also, the Monte Carlo method can be applied. For kernel (8) in the space L1 we have K ≤ ω 0 ≤ 1, while for a bounded medium K 2 < 1, which ensures convergence of the von Neumann series of (7) f (x) =

∞

K nΨ ,

(10)

n=0

where the operator K n is deﬁned by the formula n n [K Ψ](x) = · · · Ψ(x0 )k(x0 , x1 ) · · · k(xn−1 , xn )dx0 · · · dxn−1 . X

(11)

X

The Monte Carlo tracking of source photons that have not undergone a collision provides an estimation of Ψ0 (the initial collision density), the continued tracking of photons that have undergone one collision provides an estimation of K 1 Ψ0 , and so on. Equation (10) signiﬁes the existence, uniqueness, and positiveness of the solution of (7), and the possibility for using the Monte Carlo method to estimate the linear functionals Jh = X f (x)h(x)dx. According to (6), I(z, ω) is such a functional. The latter can be estimated from the general theory of solution of integral equations by the Monte Carlo method [e.g., MMN80]. To calculate radiative ﬂuxes, another Monte Carlo method can be used. In this simulation technique, the photon trajectory modeling is made in correspondence with the initial Ψ(x) and transitional k(x, x )/ω 0 densities of integral equation (7), while the radiative properties were estimated in accordance with their physical contents [e.g., MMN80]. For instance, the upward ﬂux at the altitude z = zi is estimated from the mean number of photon crossings of the plane z = zi in directions satisfying the inequality c > 0. We used the second Monte Carlo algorithm for solving the obtained equations (6), (7) and for evaluating their accuracy and robustness [KAM03, KAK05].

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5 Validation As was outlined in the previous sections, the statistically inhomogeneous model has relatively few input parameters, which describe only the bulk geometrical statistics of the 3D broken cloud ﬁeld. Thus the question arises: how accurately can equations derived on the basis of the statistically inhomogeneous model represent the mean radiative properties (e.g., mean ﬂuxes, mean absorption) of the 3D broken cloud ﬁelds? We started the validation with the direct radiation and demonstrated that the general analytical equations for the direct radiance (derived by us) are equivalent to those obtained by others for the three limiting cases: broken ﬁeld with maximum overlap, broken ﬁeld with random overlap, and overcast vertically inhomogeneous ﬁeld [KAM03]. To evaluate the accuracy and robustness of equations for the diﬀuse radiation, the following steps were taken [KAM03, KAK05]. First, we obtained the mean radiative properties exactly by applying 3D radiative transfer calculations directly to a given full 3D cloud structure. The calculated radiative properties were considered as a reference. Note that an exhaustive survey of the large number of methods for solving the 1D deterministic transfer equation is given in several monographs and textbooks [e.g., Lio02]. For 3D RT, several approaches were suggested [e.g., Cah05 and bibliography therein]. Second, for a given 3D cloud ﬁeld, we calculated the bulk cloud statistics, which were served as input data for the statistically inhomogeneous model. Next, we estimated the mean radiative properties by applying the approximated equations (6), (7). The obtained radiative properties were considered as approximations of true ones. Finally, we compared the mean radiative properties obtained by using the independent exact method (references) with ones obtained for the statistically inhomogeneous model (approximations). The accuracy of the obtained equations was evaluated by comparing the ensemble-averaged radiative properties that were calculated for 3D cloud Markovian ﬁelds. These ﬁelds were provided by the stochastic Boolean model [KAM03]. The robustness of these equations was estimated by comparing the domain-averaged radiative properties obtained for diﬀerent 2D/3D cloud ﬁelds: the 3D cloud ﬁelds were provided by large-eddy simulation model and satellite and surface retrievals [KAM03], whereas 2D cloud ﬁelds were obtained from radar observations [KAK05]. Figure 1 shows an example of such cloud ﬁelds. It was shown that the approximated equations could provide reasonable accuracy (∼15%) for both the ensemble-averaged and domainaveraged radiative properties. Also, it was demonstrated that the angular distribution histograms and the photon path length distributions of the mean albedo and transmittance, which have been obtained by exact and approximated methods, agree qualitatively and quantitatively.

Radiative Transfer in Statistically Inhomogeneous Clouds

(a) 1.8

∆x0 = 10 m ∆z0 = 30 m

(b)

∆x = 200 m ∆z0 = 30 m

(c)

∆x0 = 10 m ∆z = 150 m

(d)

147

∆x = 200 m ∆z = 150 m

z, km

24 -- 30 18 -- 24 13 -- 18 6.8 -- 13 1.0 -- 6.8

0.9

24.0

25.5

x, km

24.0

25.5

x, km

24.0

25.5

x, km

24.0

25.5

x, km

Fig. 1. Cross section (the horizontal and vertical dimensions) of the extinction coeﬃcient at (a) high resolution and (b,c,d) degraded resolution in (b) the xdirection, (c) the z-direction, and (d) both x- and z-directions. These 2D cloud (x- and z-directions) ﬁelds were obtained from ground-based cloud radar observations [KAK05]

6 Summary The deterministic RT theory has a long history, during which the simplest 1D radiation models have evolved to multi-dimensional ones. However, many physical phenomena can be treated in a statistical fashion. For example, the RT transfer through broken clouds can be considered as photon ﬂow through the stochastic mixture of clouds and clear atmosphere. The ensembleaveraged radiative properties (e.g., mean radiance and ﬂuxes) can be obtained after numerical or analytical averaging of the stochastic RT equation over the ensemble of realizations of clouds. Since the numerical averaging requires signiﬁcant computer time, it is advisable to derive approximate solutions of the problem. Such solutions may be obtained by using the analytical averaging of the stochastic RT equation and an appropriate statistical cloud model. The number of such models is quite limited. A statistically homogeneous Markovian model was successfully used to derive ensemble-averaging solutions analytically [e.g., Pom91; Tit90]. For the statistically homogeneous ﬁelds, the unconditional probability of the cloud presence does not depend on the vertical coordinate, and the conditional probability depends only on the distance between two separated space points. By using the assumption that broken clouds can be represented as a binary mixture with Markovian statistics, mathematically rigorous methods

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for radiative calculations have been developed. However, all of these methods are limited for a single layer of broken clouds. We extended the Markovian approach to multi-layer broken clouds [Kas03, KAM03]. In our approach, we assume that the statistical relationship between two adjusted layers is described by inverse exponential expression. Such assumption has some observation background [e.g., HI00; OK03]. To describe such a statistical relationship, the statistically inhomogeneous Markovian model was suggested [Kas03]. For statistically inhomogeneous ﬁelds, the unconditional probability of the cloud presence can vary strongly with altitude, and the conditional probability is a function of the relative positions of two separated space points. The new statistically inhomogeneous Markovian model and the stochastic RT equation were used to derive analytically the ensemble-averaged solutions for the direct and diﬀuse intensity of solar radiation. It was demonstrated that in the limiting cases the equations for the unscattered radiation yield mathematically correct results. We applied the Monte Carlo method to solve equations for the diﬀuse radiation. The accuracy and robustness of these equations has been estimated using a numerical simulation method and different cloud ﬁelds. Validation tests show good predictive performance for the mean radiative properties [KAM03, KAK05].

Acknowledgments This work was supported by the U.S. Department of Energy’s Oﬃce of Biological and Environmental Research as part of the Atmospheric Radiation Measurement (ARM) Program. It is my pleasure to express gratitude to Professor Georgii Titov for having introduced me to the exciting ﬁeld of stochastic RT and for his continued support through many years.

References [Cah05]

Cahalan, R.F., and Coauthors, 2005: The International Intercomparison of 3D Radiation Codes (I3RC): Bringing together the most advanced radiative transfer tools for cloudy atmospheres, Bull. Amer. Meteor. Soc., (in press). [CZ67] Case, K.M., and Zweifel, P.F. 1967: Linear transport theory, AddisonWesley Publishing Company, 342 pp. [HI00] Hogan, R.J., and Illingworth, A.J. 2000: Deriving cloud overlap statistics from radar, Q.J.R. Meteorol. Soc., 126, 2903–2909. [Kas03] Kassianov, E.I., 2003: Stochastic radiative transfer in multilayer broken clouds. Part I: Markovian approach, J. Quant. Spectrosc. Radiat. Transfer., 77, 373-393. [KAM03] Kassianov, E.I., Ackerman, T.P., Marchand, R.T. and Ovtchinnikov, M., 2003: Stochastic radiative transfer in multilayer broken clouds. Part II: Validation tests, J. Quant. Spectrosc. Radiat. Transfer, 77, 395–416.

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Kassianov, E.I., Ackerman, T.P., and Kollias, P., 2005: The role of cloud-scale resolution on radiative properties of oceanic cumulus clouds, J. Quant. Spectrosc. Radiat. Transfer 91, 211–226. [Lio02] Liou, Kuo-Nan, 2002: An introduction to atmospheric radiation. New York: Academic Press, 583 pp. [MBP93] Malvagi, F., Byrne, R.N., Pomraning, G.C. and Somerville, R.C.J., 1993: Stochastic radiative transfer in a partially cloudy atmosphere. J. Atmos. Sci., 50, 2146–2158. [MMN80] Marchuk, G., Mikhailov, G., Nazaraliev, M., Darbinjan, R., Kargin, B., and Elepov, B., 1980: The Monte Carlo methods in atmospheric optics, Springer-Verlag, Berlin, 208 pp. [OK03] Oreopoulos, L., and Khairoutdinov, M., 2003: Overlap properties of clouds generated by a cloud-resolving model, J. Geophys. Res., 108, doi:10.1029/2002JD003329. [Pom91] Pomraning, G.C., 1991: Linear kinetic theory and particle transport in stochastic mixtures, World Scientiﬁc, 235 pp. [Tit90] Titov, G. A., 1990: Statistical description of radiation transfer in clouds. J. Atmos. Sci., 47, 24–38. [ZT95] Zuev, V.E., and Titov, G., 1995: Radiative transfer in cloud ﬁelds with random geometry, J. Atmos. Sci., 52, 176–190.

Transport Theory for Optical Oceanography N.J. McCormick University of Washington, Mechanical Engineering Department, Seattle, Washington 98195-2600∗∗

Abstract. A general introduction to the ﬁeld of ocean optics is presented, including features that make optical oceanography problems similar to and diﬀerent from other transport problems. Methods for solving time-independent, one-dimensional problems are discussed that are appropriate for passive illumination conditions.

1 Introduction 1.1 Background on Optical Oceanography Non-military applications of ocean optics are a sub-discipline of biological oceanography because photon absorption drives photosynthesis which drives phytoplankton (“primary”) production at the bottom of the food chain. The correlation of optical radiance measurements from satellites with other ocean physics measurements (wave characteristics, temperature, salinity, etc.) is becoming more commonplace with recent enhanced eﬀorts in ecological monitoring. Such applications include both remote and in situ sensing of ocean waters, typically done for a passive time-independent surface illumination, which is the emphasis of the discussion here. The solution of the time-dependent transport equation is needed, on the other hand, for active illumination imaging applications such as mine detection or communication by pulsed optical signal propagation. The types of in-water problems being solved with the classic radiative transfer equation include a) forward problems for determining the angledependent and angle-integrated light ﬁeld, and b) inverse problems for determining, for example, the scattering and absorption properties in order to monitor the biological primary production of water or the ecology of coral reefs. Good introductions to radiative transfer for oceanographic forward problem applications are available. The text by Kirk [Kir94] is for people more interested in the biological aspects of radiative transfer. Mobley [Mob94] gives details of how ocean optical transfer calculations can be done, as well as a physicist’s approach to biological oceanography. The text by Thomas and ∗∗

[email protected]; http://faculty.washington.edu/mccor/

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Stamnes [TS99] and the monograph by Walker [Wal94] each contain an introduction to atmospheric and oceanic optics, with the former giving more details about atmospheric line-by-line radiative transfer calculations and the latter containing an extensive coverage of sea-surface and refracted light statistics. The monograph by Bukata et al. [BJK95] pertains to water optical properties. For oceanographic inverse problem applications, an excellent review article is that by Gordon [Gor02]. The key features of optical transport in seawater are: 1. Light in the 400–700 nm range is of primary interest, where most biological activity occurs and the optical transmission is the greatest. 2. Ocean waters are diﬃcult to characterize because • they consist of water plus dissolved organic and inorganic matter not well-characterized as to type and location, • the “microscopic cross sections,” from which the eﬀective absorption and scattering properties are determined, can only be idealized (e.g., with spherical or other regular shapes), • the air-water interface condition is diﬃcult to simulate because of refractive eﬀects and surface waves. 3. Ocean waters usually are modeled with the plane parallel approximation for natural light illuminations because • the water column is assumed to be layered (i.e., there are minimal horizontal variations of the water constituents), • the bottom is assumed ﬂat and of uniform composition or very deep, • the incident radiation from the atmosphere is assumed to be uniform over the sea surface. The reasons ocean waters are diﬃcult to characterize are: 1. The wavelength-dependent scattering of phytoplankton depends on the composition and particle size that can vary from approximately 0.7 µm to ∼100 µm. An approximate model for the number size distribution n(x) of biological particles with an equivalent diameter x, per unit volume in open ocean waters, is [Mob94] n(x) ∝ x−s for a slope of ln n(x) = −s with s ≈ 4. 2. The suspended sediments are predominately scatterers, rather than absorbers, although strong absorption features have been observed in ironrich sediment minerals. 3. Colored dissolved organic matter (CDOM) absorbs most strongly in ultraviolet wavelengths with various absorption peaks. The similarities to other transport problems include: 1. The linear Boltzmann equation for neutral particles is the governing equation.

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153

2. Active illumination problems (e.g., with a laser beam) usually are threedimensional. 3. The coeﬃcients of individual, optically-active constituents of seawater are assumed additive so, for example, for the absorption coeﬃcient a (in units of m−1 ) is atotal = aseawater + aphytoplankton + aother

particles

+ aCDOM + · · · .

4. Phytoplankton and CDOM have the ability to ﬂuoresce by re-emitting light within distinct wavebands that are somewhat longer than the absorbed light, which means a coupled wavelength analysis may be necessary. 5. Raman (inelastic) scattering events cause a wavelength increase (i.e., downscattering in energy). 6. Bioluminescence1 is analogous to an external source from neutron-gamma reactions in gamma transport or spontaneous ﬁssion in neutron transport analyses. The diﬀerences from most other linear transport problems include: 1. “Inverse problems” (e.g., to characterize properties of medium) are very important, more so than for analyses of designed systems (e.g., nuclear engineering applications). 2. A plane-parallel approximation usually is suﬃcient for waters illuminated by natural light so the radiance L (i.e., the angular dependence of the radiation ﬁeld) depends only on the water depth z and the polar angle θ = cos−1 µ and azimuthal angle ϕ. 3. The optical sensors most often used enable a simpliﬁed plane-parallel approximation so the azimuthal angular dependence need not be determined. 4. The scattering is very strongly forward peaked (e.g., more like the scattering of atmospheric aerosols than neutrons), with peak forward-tobackward scattering ratios of O(105 –106 ) [Pet72,Mob94]. 5. The refractive index mis-match at the air-water interface causes angledependent internal reﬂection for some surface emerging directions. 6. Air-water interface waves are hard to characterize. 7. Problems with polarization eﬀects have four radiance components for each wavelength. It is worth emphasizing that ocean waters are dynamically changing systems that are hard to characterize, and hence the challenge of obtaining good input data for computations means the old adage “garbage in, garbage out” is a big concern. Furthermore, the environment for performing optical experiments in the ﬁeld is often diﬃcult, which leads to appreciable measurement 1

Bioluminescence results from an exothermic internal chemical reaction releasing energy as light, either spontaneously or in response to mechanical stimulation, and occurs with some species of bacteria, dinoﬂagellates, zooplankton, and ﬁsh.

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uncertainties. For these reasons computational results are not needed with more precision than measurement uncertainties. 1.2 The Transfer Equation Used Because the water optical properties typically vary little in the horizontal direction, and because the surface illumination often is nearly uniform over the sea surface, ocean optics problems often can be treated spatially with only the depth variable provided the eﬀects of surface waves are averaged out. In this case, in the customary notation of the ocean optics literature, the governing Boltzmann integrodiﬀerential for the radiance (i.e., “angular ﬂux”) L(z, µ, ϕ, λ) (in Wm−2 sr−1 nm−1 ) is ∂L(z, µ, ϕ, λ) + c(z, λ)L(z, µ, ϕ, λ) ∂z 2π 1 ˜ µ , µ, ϕ , ϕ, λ)L(z, µ, ϕ, λ) + S(z, λ) , = b(z, λ) dϕ dµ β(z, µ

(1)

−1

0

where µ is the cosine of the polar angle between the downward distance z from the sea surface and the direction of interest and ϕ is the azimuthal angle in the horizontal plane. The sum of the absorption and scattering coefﬁcients is the so-called beam attenuation coeﬃcient c(z, λ) = a(z, λ) + b(z, λ) (in m−1 ). These coeﬃcients are “inherent optical properties” (IOP). The ˜ µ , µ, ϕ , ϕ, λ) (in sr−1 ), also is an IOP. The scattering phase function is β(z, source from Raman scattering or ﬂuorescence is eﬀectively isotropic, S(z, λ) = 0

∞

dλ

0

2π

dϕ

1

dµ k(z, µ , ϕ , λ , λ)L(z, µ , ϕ , λ )

(2)

−1

for a conversion kernel k(z, µ , ϕ , λ , λ) (in m−1 sr−1 nm−1 ). Often it is suﬃcient to use only a simpliﬁed transport equation that does not depend on the source S(z, λ) because bioluminescent sources are negligibly small during the daytime hours for most ocean waters and Raman scattering and ﬂuorescence eﬀects also are small in comparison to the sea surface illumination except for deeper depths (τ > ∼ 30) and longer visible wavelengths (e.g., λ > ∼ 520 nm). In such cases the radiative transfer equation can be solved independently for each wavelength λ. Another simpliﬁcation that often can be made when attempting to predict subsurface measurements is to use a transport equation that depends only on µ and not ϕ, even though the sea surface illumination depends on both angle variables. The reason the single-angle equation can be used is that the sensors for passive (i.e., ambient) light typically measure only the “apparent optical properties” of the downward and upward “planar irradiances” (i.e., “partial currents”)

Transport Theory for Optical Oceanography

2π

Ed (z, λ) =

1

dϕ 0

2π

Eu (z, λ) =

dµ µL(z, µ, ϕ, λ) ,

(3)

dµ µL(z, −µ, ϕ, λ) ,

(4)

0 1

dϕ 0

155

0

and the vertically upward radiance L(z, −1, λ) =

2π

L(z, −1, ϕ, λ)dϕ .

(5)

0

Such sensors depend on sunlight and they are restricted to measurements down to a depth z ∼ 150 m, depending on the water properties. Polarization eﬀects for the radiation can be important near the air-water interface, in which case the source-free form of (1) must be replaced by ∂I(z, µ, ϕ, λ) + c(z, λ)I(z, µ, ϕ, λ) µ ∂z 2π 1 ˜ = b(z, λ) dϕ dµ Σ(z, µ , µ, ϕ , ϕ, λ)I(z, µ , ϕ , λ) + S(z, λ) 0

(6)

−1

for the Stokes vector I = [I, Q, U, V ]T , a diagonal interaction matrix c(z, λ), ˜ and the Stokes scattering phase matrix Σ(z, µ , µ, ϕ , ϕ, λ). The eﬀects of polarization for ocean optics applications has been investigated by Kattawar and collaborators [AK93, RK98]. 1.3 Fourier Uncoupling of the Transfer Equation Many ocean optics computational programs take advantage of the relatively simple geometry to convert (1) that depends on µ and ϕ into a set of uncoupled equations that depend only on µ, especially when only the downward and upward planar irradiances and the vertically upward radiance are needed. Consider the equation 2π 1 ∂L(τ, µ, ϕ) ˜ µ , ϕ, ϕ )L(τ, µ , ϕ ) (7) + L(τ, µ, ϕ) = ω dϕ dµ β(µ, µ ∂τ 0 −1 in terms of the optical distance variable τ = c(z)dz, the single-scattering albedo ω = b/c, and an implicit single wavelength λ. An expansion of the phase function as ˜ µ , ϕ, ϕ ) = (2π)−1 β(µ,

M

(2 − δm,0 )β˜m (µ, µ ) cos m(ϕ − ϕ )

(8)

m=0

gives the set of uncoupled equations [Cha60,Mob94] 1 ∂Lm (τ, µ) m µ + L (τ, µ) = ω β˜m (µ , µ)Lm (τ, µ )dµ ∂τ −1

(9)

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N.J. McCormick

for m = 0 to M , with L(τ, µ, ϕ) =

M

(2 − δm,0 )Lm (τ, µ) cos mϕ + Lu (τ, µ) .

(10)

m=0

Here Lu (τ, µ) accounts for that portion of the radiance not included in the expansion (e.g., that may arise from a Dirac-delta type surface illumination). The reduced phase function can be expanded in terms of the associated Legendre functions Pnm (µ) as β˜m (µ , µ) =

N

fn αnm Pnm (µ )Pnm (µ) ,

(11)

n=m

with f0 = 1 and αnm =

(2n + 1) (n − m)! fn , 2 (n + m)!

n≥m.

As mentioned previously, often just the simplest form of the transport equation for m = 0 is needed, 1 N ∂L(τ, µ) + L(τ, µ) = (ω/2) (2n + 1)fn Pn (µ) Pn (µ )L(z, µ )dµ , ∂τ −1 n=0 (12) as when only Ed (τ ), Eu (τ ), L(τ, −1), and the “scalar irradiance” (i.e., “total ﬂux”) 2π 1 dϕ dµL(τ, µ, ϕ) E0 (τ, µ) = µ

0

−1

are desired.

2 Aspects Requiring Special Computational Attention 2.1 Optical Properties “Case 1” ocean waters are those for which the inherent optical properties are determined primarily by phytoplankton and co-varying CDOM and detritus. (It should be noted that the label is not a synonym for open ocean waters, although many such waters are of the case 1 type). Examples for correlations of the absorption and scattering coeﬃcients with the non-water “particle” concentration C in mg m−3 are [PS81] ap (λ, C) = 0.06Ap (λ)C 0.602 , 550 bp (λ, C) ≈ 0.5Bp (550) C 0.62 , λ

(13) C < 10 .

(14)

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(Here Ap (λ) is a tabulated coeﬃcient normalized to the particle absorption at 440 nm and Bp (550) ranges from 0.12 to 0.45 m−1 .) “Case 2” water optical properties are signiﬁcantly inﬂuenced by CDOM, detritus, mineral particles, bubbles, and other substances whose concentrations do not co-vary with the phytoplankton concentration. These waters, not necessarily coastal, are a signiﬁcant complication for ocean optics computations compared to the many other ﬁelds of transport where the input material parameters follow regular laws of physics. A diﬃculty often also arises when attempting to describe the strongly anisotropic scattering phase function. The classic experimental scattering data of Petzold [Pet72,Mob94] traditionally has been used to model case 1 waters. Another approach is the “Fournier-Forand model” [FF94] that is for an ensemble of particles that have a hyperbolic particle distribution given by F (r) ∝ r−k for radius r and slope parameter k. For a phase function scattering angle ψ, the phase function P (ψ) of that model is [MSB02] 1 [ν(1 − η) − (1 − η ν ) 4π(1 − η)2 η ν ψ −2 ν + [η(1 − η ) − ν(1 − η)] sin 2 ν 1 − η180 2 + ν (3 cos ψ − 1) , 16π(η180 − 1)η180

P (ψ) =

where ν=

3−k , 2

η=

4 sin2 3(n − 1)2

ψ . 2

(15)

(16)

Here n is the real index of refraction of the particles and η180 is η evaluated at ψ = 180◦ . Knowledge of the shape of the phase function for a scattering angle of ≤ 15◦ is not necessary in many applications, however [Gor93]. An examination of the eﬀects of diﬀerent scattering models on the computed radiance has shown that [MSB02]: • Use of the correct phase function is just as necessary as use of the correct absorption and scattering coeﬃcients. • An accurate shape of the phase function is not so important if the overall shape does not diﬀer greatly from the correct shape, but the backscattering fraction must be nearly correct. • The Fourier-Forand phase function works well. 2.2 Index of Refraction Eﬀects Light rays change direction when passing through the air-water interface because of Snell’s law, sin θair = nW sin θwater ,

(17)

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for a relative index of refraction of water with respect to air, nW ≈ 4/3. Such an interface phenomenon does not occur for most other transport problems (e.g., those involving neutrons). To analyze those light rays passing from air to water through a ﬂat interface, only a simple change of direction is required to obtain the direction beneath the surface. But for those within the water that reach the interface, only those with a polar angle of θwater ≤ ∼ 48.6◦ with respect to the downward normal can pass through the interface while those with ∼ 48.6◦ ≤ θwater ≤ 90◦ are totally reﬂected back into the water. This means that numerical techniques that do not discretize the polar angle variable, such as the spherical harmonics method, are cumbersome to implement. The mis-match of the diﬀerent indices of refraction for air and water tends to aﬀect the radiance mostly near the surface. At a few optical depths beneath the surface, where almost all photons have undergone one or more scattering interactions, the directional dependence of the radiance is dominated by the optical properties more than the directional dependence of the surface illumination. For deep, spatially uniform waters containing no sources (e.g., inelastic scattering) the radiance approaches an asymptotic regime where the directional dependence does not change with increasing depth, although the magnitude of the radiance decreases exponentially [McC92]. 2.3 Sea Surface Waves and Wave Focusing Eﬀects If the radiance is to be accurately computed near the surface, then the effects of waves also must be considered. This requires incorporating the index of refraction direction changes at the point on the wave surface where the ray strikes. The Monte Carlo technique, the logical method of choice, has been implemented for this purpose [Mob94]. But a major problem is how to describe the shape of the waves. Traditionally the classic Cox-Munk model [CM54a,CM54b,Mob94,Wal94] has been used. It is a wave-slope wind-speed correlation for capillary (i.e., small surface) waves. The dimensionless horizontal wave slopes in the alongwind and crosswind directions vary independently and are normally distributed with zero means, with variances proportional to the wind speed U in ms−1 as σu2 = au U, σc2 = ac U,

where au = 3.16 × 10−3 m−1 s, where ac = 1.92 × 10−3 m−1 s .

The wave elevation also is distributed normally. Wind produces a frictional drag at the interface that transmits energy to the sea surface, so when a steady wind starts then small capillary waves develop ﬁrst. Energy then is transferred to the longer-wavelength gravity waves and the spectrum energy grows until equilibrium is reached such that the input wind energy balances the energy dissipation. For a radial spectrum S(k) and an angular speading function Φ(k, ϕ), the “Elfouhaily” statistical

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model gives a sea surface roughness spectrum Ψ (k, ϕ) that is the Fourier transform of the autocorrelation function of the surface height [ECK96], Ψ (k, ϕ) = k −1 S(k)Φ(k, ϕ) ≈ (2πk)−1 S(k)[1 + a2 cos(2ϕ)] .

(18)

Another consequence of a wind-roughened surface is that wave focusing occurs near the surface and the plane geometry approximation is no longer valid [ZBB01]. There may be signiﬁcant implications of this for remote sensing applications, but for in-water measurements the eﬀects are damped out with increasing optical depth, like those due to the air-water index of refraction mis-match.

3 Computational Programs An excellent comparison of diﬀerent computational methods used for oceanographic analyses has been given by Mobley et al. [MGG94]. A recent but more limited comparison also is available [CCG03]. R 3.1 HYDROLIGHT

This is a commercially available program [MS00] in Fortran for solving the sub-surface light ﬁeld given an incident illumination on the sea surface. For the unit sphere of directions oriented with a downward vertical axis, it separates out the azimuthal angle dependence with the Fourier decomposition of Sect. 1.3. The polar angle range 0 ≤ θ ≤ 180◦ is divided into equally-spaced segments of ∆θ, and equally-spaced segments also are used for ∆ϕ. The result is a set of surface areas or “quads”. The program uses the invariant imbedding procedure to solve (9) for each azimuthal component of the light ﬁeld [Mob94]. This procedure has an advantage for calculations involving many layers of distinct optical properties. A key feature of the program is that it contains a Monte Carlo ray-tracing capability for simulating waves at the surface to generate the in-water boundary condition just beneath the sea surface. The program has an extensive collection of built-in data sets to model a wide variety of water optical properties that are mixed according to user-speciﬁed constituent concentrations in the water. The PC-computer version has a graphical user interface with many input options. The program also provides a means of coupling atmospheric radiative transport with the in-water light ﬁeld because the inherent optical properties for both water and atmosphere are available to cover the wavelength range 300 nm to 1000 nm. A nice feature is that the Fortran source code of the program is available for personal modiﬁcation (e.g., a spherically-symmetric problem has even been solved with it [MK00]). Because the program has been developed commercially, it is not inexpensive; the current price is $10K that includes updated versions from the authors.

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3.2 DISORT This program is based on the classic discrete ordinates method [STW88,JS94, TS99] for solving the set of azimuthally-independent transfer equations (9), with extra ordinates to allow for the change in direction across the air-water interface. The program uses an “iteration of the source function method” (i.e., “post processing”) to compute the radiance in directions other than the ordinates speciﬁed in the program input. The newest version for ocean applications is the Coupled Atmosphere-Ocean (CAO-DISORT) program [GSH03,YK03]. A vector version, VDISORT, also is available [SS00] for investigating polarized radiation problems. 3.3 Analytic Discrete Ordinates Method This is a discretized version of the analytic eigenmode expansion of the homogeneous radiative transfer equation [Sie00]. The advantage of preserving that functional form of the solution is that highly accurate benchmarkquality results can be obtained for the radiance. With this method the variables τ and µ of (9) for the phase function of (11) are separated with Lm (τ, µ) = φm (ν m , µ)exp(−τ /ν m ). With a suppressed superscript m for φ and ν, the eigenfunctions φ(ν, µi ) satisfy

1−

N K ω µi fn αnm Pnm (µi ) wk Pnm (µk )[φ(ν, µk ) φ(ν, µi ) = ν 2 n=m k=1

+(−1) φ(ν, −µk )] , (19) N K µi ω 1+ φ(ν, −µi ) = fn αnm Pnm (µi ) wk Pnm (µk )[(−1)n−m φ(ν, µk ) ν 2 n=m n−m

k=1

+φ(ν, −µk )]

(20)

for quadrature nodes µi on [0, 1], i = 1 to K, and integration weights wi . From these equations the eigenvalues ±νj , j = 1 to J, can be computed so that the solution in vector form for a region 0 ≤ τ ≤ τ0 can be expanded with L± (τ ) =

J

[Aj Φ± (νj )e−τ /νj + Bj Φ∓ (νj )e−(τ0 −τ )/νj ] + Lp,± (τ )

(21)

j=1

for L± (τ ) = [L(τ, ±µ1 ), L(τ, ±µ2 ), . . . L(τ, ±µK ]T , Φ± (νj ) = [φ(ν, ±µ1 ), φ(ν, ±µ2 ), . . . , φ(ν, ±µimax )]T ,

(22)

where the Aj and Bj are expansion coeﬃcients that depend on the surface illumination and Lp,± (τ ) is the particular solution [BGS00] incorporating

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the eﬀects of any Dirac-delta-like boundary conditions and internal sources. Analytically-based post processing is used to compute the radiance in directions other than at the speciﬁed collocation points. To date the method has been used only for idealized atmospheric problems and no eﬀort has been made to incorporate refractive eﬀects at an air-water interface.

4 Computing Challenges Because of the relatively simple geometry for most optical oceanography problems, computations to date largely have been performed with the onedimensional transfer equation, which makes these problems relatively simpler than the complicated ones arising in other transfer applications. However, three-dimensional optical oceanography problems arise if the: • Incident sunlight is not uniform over the each pixel of the water surface (e.g., caused by a partial cloud cover) • Bottom is not parallel to the water surface • Water constituents are not horizontally uniform within a layer Such problems can be analyzed with Monte Carlo techniques [MS03], although deterministic multi-dimensional techniques developed for solving transport problems for other applications also can be used provided adjustments are made to incorporate the eﬀects of the index of refraction at the air-water interface and the statistical nature of surface waves. To better simulate ocean optical radiances, a better characterization of the water inherent optical properties is needed. The computational techniques discussed here for solving forward problems, when combined with methods for solving inverse problems, presently are being investigated for use in obtaining water properties [FSS98,FES01,LC02,LCA02]. The inate variability of any case 2 ocean water, however, introduces uncertainties when attempting to extrapolate the results to other such waters. Depending on where the measurements are taken, possible non-plane-parallel geometry (sea surface waves and wave focusing) eﬀects also can complicate the analysis. Perhaps for this reason the recent emphasis in funded optical oceanography research has shifted from the development of more sophisticated computer programs to more extensive experimental activities.

References [AK93]

Adams, C.N., Kattawar, G.W.: Eﬀect of volume-scattering function on the errors induced when polarization is neglected in radiance calculations in an atmosphere-ocean system. Appl. Opt. 32, 4610–4617 (1993) [BJK95] Bukata, R.P., Jerome, J.H., Kondratyev, K.Ya., and Pozdnyakov, D.V.: Optical Properties and Remote Sensing of Inland and Coastal Waters. CRC Press, Boca Raton (1995)

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[BGS00] Barichello, L.B., Garcia, R.D.M., Siewert, C.E.: Particular solutions for the discrete-ordinates method. J. Quant. Spectrosc. Rad. Transfer 64, 19–226 (2000) [CCG03] Chalhoub, E.S., Campos Velho, H.F., Garcia, R.D.M., Vilhena, M.T.: A comparison of radiances generated by selected methods of solving the radiative-transfer equation. Transport Th. Stat. Phys. 32, 473–503 (2003) [Cha60] Chandrasekhar, S.: Radiative Transfer. Dover, New York (1960) [CM54a] Cox, C., Munk, W.: The measurement of the roughness of the sea surface from ptographs of the sun’s glitter. J. Opt. Soc. Am. 44, 838–850 (1954) [CM54b] Cox, C., Munk, W.: Statistics of the sea surface derived from sun glitter. J. Mar. Res. 13, 198–227 (1954) [ECK96] Elfouhaily, T., Chapron, B., Katsaros, K., Vandemark, D.: A uniﬁed directional spectrum for long and short wind-driven waves, J. Geophys. Res 102, No. C7, 15781–15796 (1996) [FSS98] Frette, Ø., Stamnes, J.J., Stamnes, K.: Optical remote sensing of marine constituents in coastal waters: a feasibility study. Appl. Opt. 37, 8318– 8326 (1998) [FES01] Frette, Ø., Erga, S.R., Stamnes, J.J., Stamnes, K.: Optical remote sensing of waters with vertical structure. Appl. Opt. 40, 1478–1487 (2001) [FF94] Fournier, G.R., Forand, J.L.: Analytic phase function for ocean water. In: Jaﬀe, J.S. (ed) Ocean Optics XII. Proc. SPIE 2258, 194–201 (1994) [Gor93] Gordon, H.R.: Sensitivity of radiative transfer to small-angle scattering in the ocean: Quantitative assessment. Appl. Opt. 32, 7505–7511 (1993) [Gor02] Gordon, H.R.: Inverse methods in hydrologic optics. Oceanologia 44, 9– 58 (2002) [GSH03] Gjerstad, K.I., Stamnes, J.J., Hamre, B., Lotsberg, J.K., Yan, B., Stamnes, K.: Monte Carlo and discrete-ordinate simulations of irradiances in the coupled atmosphere-ocean system. Appl. Opt. 42, 2609–2622 (2003) [JS94] Jin, Z., Stamnes, K.: Radiative transfer in nonuniformly refracting layered media: atmosphere–ocean system. Appl. Opt. 33, 431–442 (1994) [Kir94] Kirk, J.T.O.: Light & Phototsynthesis in Aquatic Ecosystems, 2nd ed. Cambridge, Cambridge (1994) [LC02] Lee, Z., Carder, K.L.: Eﬀect of spectral band numbers on the retrieval of water column and bottom properties from ocean color data. Appl. Opt. 41, 2191–2201 (2002) [LCA02] Lee, Z., Carder, K.L., Arnone, R.A.: Deriving inherent optical properties from water color: a multband quasi-analytical algorithm for optically deep waters. Appl. Opt. 41, 5755–5772 (2002). [McC92] McCormick, N.J.: Asymptotic optical attenuation. Limnol. Oceanogr. 37, 1570–1578 (1992) [MK00] McCormick, N.J., Ka¸ska¸s, A.: Isotropic spherical source analysis for ocean optics. Appl. Opt. 39, 4902–4910 (2000) [Mob94] Mobley, C.D.: Light and Water Radiative Transfer in Natural Waters. Academic, New York (1994) [MGG94] Mobley, C.D., Gentili, B., Gordon, H.R., Jin, Z., Kattawar, G.W., Morel, A., Reinersman, P., Stamnes, K., Stavn, R.H.: Comparison of numerical models for computing underwater light ﬁelds. Appl. Opt. 32, 7484–7504 (1994)

Transport Theory for Optical Oceanography [MS00]

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Mobley, C.D., Sundman, L.K.: Hydrolight 4.1. Sequoia Scientiﬁc, Inc., Redmond, Wash. (2000) [MS03] Mobley, C.D., Sundman, L.K.: Eﬀects of optically shallow bottoms on upwelling radiances: Inhomogeneous and sloping bottoms. Limnol. Oceanogr. 48, 329–336 (2003) [MSB02] Mobley, C.D., Sundman, L.K., Boss, E.: Phase function eﬀects on oceanic light ﬁelds. Appl. Opt. 41, 1035–1050 (2002) [Pet72] Petzold, T.J.: Volume scattering functions for selected ocean waters, SIO Ref 71-78, Scripps Institution of Oceanography, San Diego, Calif. (1972) [PS81] Prieur, L., Sathyendranath, S.: An optical classiﬁcation of coastal and oceanic waters based on the speciﬁc absorption of phytoplankton pigments, dissolved organic matter, and other particlate materials. Limnol. Oceanogr. 26, 671–689 (1981) [RK98] Rakovi´c, M.J., Kattawar, G.W.: Theoretical analysis of polarization patterns from incoherent backscattering of light. Appl. Opt. 37, 3333–3338 (1998) [SS00] Schulz, F.M., Stamnes, K.S.: Angular distribution of the Stokes vector in a plane-parallel, vertically inhomogeneous medium in the vector discrete ordingate radiative transfer (VDISORT) model. J. Quant. Spectrosc. Rad. Transfer 65, 609–620 (2000) [Sie00] Siewert, C.E.: A concise and accurate solution to Chandrasekhar’s basic problem in radiative transfer. J. Quant. Spectrosc. Rad. Transfer 64, 109–130 (2000) [STW88] Stamnes, K.S., Tsay, S-C., Wiscombe, W., Jayaweera, K.: Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media. Appl. Opt. 27, 2502–2509 (1988) [TS99] Thomas, G.E., Stamnes, K.: Radiative Transfer in the Atmosphere and Ocean. Cambridge Univ. Press, Cambridge (1999) [Wal94] Walker, R.E.: Marine Light Field Statistics. Wiley, New York (1994) [YS03] Yan, B., Stamnes, K.: Fast yet accurate computation of the complete radiance distribution in the coupled atmosphere-ocean system. J. Quant. Spectrosc. Rad. Transfer 76, 207–223 (2003) [ZBB01] Zaneveld, J.R.V., Boss, E., Barnard, A.: Inﬂuence of surface waves on measured and modeled irradiance proﬁles, Appl. Opt. 40, 1442–1449 (2001)

Perturbation Technique in 3D Cloud Optics: Theory and Results Igor N. Polonsky1 , Anthony B. Davis1 and Michael A. Box2 1

2

Los Alamos National Laboratory, Space and Remote Sensing Sciences Group (ISR-2), Los Alamos, NM 87545, USA [email protected] School of Physics, Univ of New South Wales, Sydney, NSW, 2052 Australia [email protected]

1 Introduction It is well known that generally to simulate accurately radiative transfer through a realistic cloudy atmosphere one should use numerical approaches such as Monte Carlo [12], or SHDOM [3]. However, it is usually required too much time to make a simulation which is inconvenient when just we need an answer on a simple question like how signiﬁcant the 3D eﬀects are for a given problem. The perturbation method is what comes to mind ﬁrst if we need to go further into modelling of the radiative transfer through cloud atmosphere starting from the simplest framework of one dimensional radiative transfer [2, 8]. Recently, two perturbation approaches have been used. One is somewhat orthodox [14] and based on assuming that some term in the radiative transfer equation is small enough to be considered as a small parameter to construct a perturbation series [4,5,9–11]. A diﬀerent type of perturbation approach may be formulated on the basis of the joint consideration of both the direct and corresponding adjoint problem [1, 7, 13, 15, 18]. This approach can be derived using a variational principle [17] allowing one to obtain a required solution practically without eﬀort. The goal of this paper is to formulate a variational principle to derive the perturbation approach speciﬁcally for the problem of cloud optics. We will also demonstrate how it can be used to explain some eﬀects in cloud optics.

2 Deﬁnition of the Problem Let us consider radiation propagation through a cloud which has a shape of a slab and is illuminated by a steady, uniform, and collimated beam (e.g., sunlight). We introduce a Cartesian coordinate system with the origin on the upper slab surface with the z-axis directed toward the inner normal. The direction is deﬁned by the Euler polar, θ and azimuth φ angles. The radiance I(r, vn) at the point r in the direction n can be calculated using the framework of the radiative transfer equation (RTE) [e.g, 6]

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n · I(r, n) + σe (r)I(r, n) =

σs (r) 4π

P (r, n, n )I(r, n )dn + S(r, n) . (1)

4π

Here n and r are the vectors which deﬁne the direction and position, respectively, I(r, n) is the radiance at the point r in the direction n, σe (r) and σs (r) are the extinction and scattering coeﬃcients, respectively, and P (r, n, n ) is the phase function normalized as 1 P (r, n, n )dn = 1 4π 4π

S(r, n) is the source function which in the case of the sunbeam has the form S(r, n) = S0 δ(n − n0 )δ(z − ε) . Here S0 is the Sun ﬂux at the top of the cloud, and ε is inﬁnitesimal small and has been introduced for convenience. Certainly, (1) has to be complimented with the boundary conditions I(r, n)|z=0 = 0, µ > 0 , 1 I(r, n)|z=H = A(r, n, n )|z=H I(r, n )|z=H |dn , µ < 0 . π

(2) (3)

µ>0

Here A(r, n, n ) describes the reﬂection properties of the underlying surface if one exists.

3 Variational Principe to Derive the Radiative Transfer Equation We may assume that the most measurements in cloud optics can be describe by integration of the radiance with some receiver function R(r, n) (4) P = R(r, n)I(r, n)dndr Ξ

where Ξ denotes the region of interest in the position-direction space. According [17] we may introduce a functional ˜ ˜ n)S(r, n) dndr R(r, n)I(r, n) + I(r, J = I(r, n)LI(r, n)dndr − Ξ

Ξ

+

˜ n)I(r, n)dndr I(r,

z=0,µ>0

− z=H,µ<0

⎡

˜ n) ⎣I(r, n) − 1 I(r, π

µ>0

⎤ A(r, n, n )I(r, n )dn ⎦ dndr .

(5)

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Here to simplify notation the operator L = n · + σe (r) −

σs (r) 4π

dn P (r, n, n )⊗ ,

(6)

4π

has been introduced. Note that the notation ⊗ is used to indicate that the ﬁnal term is an integral operator, not merely a deﬁnite integral. This func˜ n) results in RTE tional has a clear property that the variation of it over I(r, (1) with boundary condition (2). The variation of the functional over I(r, n) ˜ n) results in the equation for I(r, ˜ n) = σs (r) P (r, n , n )I(r, ˜ n )dn + R(r, n) (7) ˜ n) + σe (r)I(r, −n · I(r, 4π 4π

with boundary conditions ˜ n)|z=0 = 0, µ < 0, I(r, 1 ˜ n )|z=H |dn , µ > 0 . ˜ I(r, n)|z=H = A(r, n , n)|z=H I(r, π

(8) (9)

µ<0

This equation constitutes the adjoint formulation of our problem. The adjoint solution has a strict physical meaning. As can be noticed if we reverse direction in (7–8) we obtain a RTE which is similar to (1–2). This means that the adjoint formulation of a given problem tracks photon backward starting from the receiver and ending up at the source.

4 Perturbation If we substitute either (1–2) or (7–8) into (5) shows that at the minimum the functional value is the same as our measured signal (4) ˜ n)dndr = −P . J|min = − R(r, n)I(r, n)dndr = − S(r, n)I(r, Ξ

Ξ

This fact provides us with possibility to estimate how small variations of the problem parameters aﬀect the signal. Expanding our functional into series over small variations of all functions in the neighborhood of its minimum, keeping only the ﬁrst order terms, and taking into account (1–2, 7–8), we obtain that

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˜ n)∆LI(r, n)dndr I(r,

∆J = −∆P = Ξ

−

˜ n)∆S(r, n) dndr ∆R(r, n)I(r, n) + I(r,

Ξ

+

1 π

˜ n) I(r, µ>0

z=H,µ<0

where ∆σs (r) ∆L = ∆σe (r) − 4π

∆A(r, n, n )I(r, n )dn dndr . (10)

σs (r) dn P (r, n, n ) ⊗ − 4π

4π

dn ∆P (r, n, n ) ⊗ .

4π

(11) Equation (10) is the major formula of the perturbation approach. It says that if both the direct and the adjoint solutions of the same problem are known the eﬀect of the variation can be simply estimated by integration. To reveal the physical sense of (10) let us consider a special point perturbation of L, ∆L = ∆σe (r)δ(r − rp ) We have that the eﬀect has the form ˜ p , n)∆σe (r)I(rp , n)dn . ∆J = I(r 4π

Recalling the physical sense of the adjoint solution, this formula shows that the eﬀect generated by this perturbation is the same as an eﬀective source ∆σe (r)I(rp , n)δ(r − rp ) has been added at point rp . Thus, in the most general case we can treat variations of the media parameters as addition some distribution of the eﬀective sources.

5 A Toy Example As an example let us consider how addition of the periodic in x extinction coeﬃcient aﬀect the reﬂected radiance. This problem in more details was considered in [9, 16]. The formulation of the problem assumes that our base problem parameters are constant over the medium and ∆L = εσ e sin(k0 x) The variation of the reﬂected radiance has the form

Perturbation Technique in 3D Cloud Optics: Theory and Results 0

0.2 0.7

−20

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0.16

0.14

−40

0.9

0.12

0.5

0.1

0.95 −80

0.7

ρ,[a.u]

φ (k),[deg]

−60

0.08

−100

0.2 0.06 0.9

−120

0.99 0.95

0.04

−140

0.99 0.999 0.02

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1.0

0.9999 −180

0

1

2

3

4

5_ k0/σe

6

7

8

9

1.0

0

10

0

0.5

1

1.5

_ 2 k0/σe

2.5

3

3.5

Fig. 1. Left panel. The phase shift, φ(k0 ) as a function of k0 /σ e . Right panel. Amplitude coeﬃcient, φ(k0 ), as a function of k0 /σ e . Isotropic phase function. The ﬁgures at the curves show the solar zenith cosine µ0 −3

2

x 10

1.5

1

δE,[a.u]

0.5

0

−0.5

−1

−1.5

−2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

k0 x

Fig. 2. The upwelling zenith radiance proﬁle as a function of k0 x simulated using the SHDOM. Isotropic phase function. Wavenumber, k0 , is 1.0σ e (dotted line), 2.0σ e (solid line), and 3.0σ e (dashed line)

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∆J(x) = ερ(k0 ) sin [k0 x + φ(k0 )] . where [16] determined analytical expression for ρ(k0 ) and φ(k0 ). Figure 1 shows how parameters ρ(k0 ) and φ(k0 ) depend on k0 . For the simulation a slab of optical thickness 10.0 has been chosen. For the sake of simplicity the medium is assumed to be non absorbing with an isotropic phase function. The ﬁgure shows that for the case of normal illumination at k0 /σ e ≈ 2.0 the amplitude coeﬃcient, ρ, is close to zero which means that for this kind of the extinction coeﬃcient perturbation the reﬂected radiance does not change and the radiance measurement cannot reveal periodicity of the extinction coeﬃcient proﬁle. We tested this prediction performing the corresponding simulation employing SHDOM [3]. The results of the simulation are shown on Fig. 2 which clearly shows that there is quite a weak variation of the reﬂected radiance in the case of k0 /σ e ≈ 2.0 which shape does not resemble the sin-wave shape of the extinction coeﬃcient variation.

References 1. M. A. Box, B. Croke, S. A. W. Gerstl, and C. Simmer. Application of the perturbation theory for atmospheric radiative eﬀects: Aerosol scattering atmospheres. Beitr. Phys. Atmosph, 62:200–211, 1989. 2. S. Chandrasekhar. Radiative Transfer. Oxford University Press, 1950 (Reprinted by Dover, New York, 1960). 3. K. F. Evans. The spherical harmonics discrete ordinate method for threedemensional atmospheric radiative transfer. J. Atmos. Sci., 55:429–446, 1998. 4. V. L. Galinsky and V. Ramanathan. 3D radiative transfer in weakly inhomogeneous medium. Part I: Diﬀusive approximation. J. Atmos. Sci., 55:2946–2959, 1998. 5. V. L. Galinsky. 3D radiative transfer in weakly inhomogeneous medium. Part II: Discrete ordinate method and eﬀective algorithm for its inversion. J. Atmos. Sci., 57:1635–1645, 2000. 6. A. Ishimaru. Wave Propagation and Scattering in Random Media. Academic Press, New York, 1978. 7. J. Landgraf, O. P. Hasekamp, M.A. Box, and T. Trautmann. A linearized radiative transfer model for ozone proﬁle retrieval using the analytical forwardadjoint perturbation theory approach. J. Geophys. Res., 106:27291–27305, 2001. 8. J. Lenoble, editor. Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures. A.Deepak, Publishing, Hampton, Va., USA, 1985. 9. J. Li, J. W. Geldart, and P. Ch´ ylek. Perturbation solution for 3D radiative transfer in a horizontally periodic inhomogeneous cloud ﬁeld. J. Atmos. Sci., 51:2110–2122, 1994. 10. J. Li, J. W. Geldart, and P. Ch´ ylek. Second order perturbation solution for radiative transfer in clouds with a horizontally arbitrary periodic inhomogeneity. J. Quant. Spectrosc. Radiat. Transfer, 53:445–456, 1995.

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11. A. I. Lyapustin and Y. Knyazikhin. Green’s function method in the radiative transfer roblem. II. spatially heterogeneous anisotropic surface. Appl. Opt., 41 (27):5600–5606, 2002. 12. G. Marchuk, G. Mikhailov, M. Nazarliev, R. Darbinjan, B. Kargin, and B. Elepov. The Monte-Carlo Methods in Atmospheric Optics. Springer-Verlag, Heidelberg, 1980. 13. G. I. Marchuk. Equation for the value of information from weather satellite and formulation of inverse problems. Kosmicheskie Issledovaniya, 2:462–477, 1964. 14. P. M. Morse and H. Feshbach. Methods of Theoretical Physics. McGraw-Hill book Company, New York, Toronto, London, 1952. 15. I. N. Polonsky and M. A. Box. Perturbation technique to retrieve scattering medium stratiﬁcation. J. Atmos. Sci, 59:758–768, 2002. 16. I. N. Polonsky, M. A. Box, and A. Davis. Radiative transfer through inhomogeneous turbid media: Implementation of the adjoint perturbation approach at the ﬁrst order. J. Quant. Spect. Rad. Tran., 78(1):85–98, 2003. 17. G.C. Pomraning. A variational principle for linear systems. J. Soc. Indust. Appl. Math., 13(2):511–519, 1965. 18. E. A. Ustinov. The inverse problem of the photometry of solar radiation reﬂected by an optically thick planetary atmosphere. Mathematical methods and weighting functions of linearized inverse problem. Kosmicheskie Issledovaniya, 29:604–620, 1991.

Vegetation Canopy Reﬂectance Modeling with Turbid Medium Radiative Transfer Barry D. Ganapol Departments of Hydrology and Water Resources and Aerospace and Mechanical Engineering University of Arizona [email protected] Abstract. Biophysical considerations for vegetation canopy reﬂectance modeling are presented. Included is a brief overview outlining strengths and weaknesses of four possible canopy reﬂectance models. The overview is followed by the description of the LCM2 coupled leaf/canopy turbid medium reﬂectance model based on natural averaging. The model follows conventional radiative transfer theory with modiﬁcation for canopy architecture as characterized by leaf orientation. The presentation concludes with a demonstration of LCM2 in a multiple pixel mode to estimate the amount of ripe coﬀee cherries at harvest in the ﬁelds of the Kauai Coﬀee Company and to detect targets hidden beneath canopies.

1 Introduction Vegetation plays a signiﬁcant role in sustaining life on planet Earth. In particular, vegetation is responsible for the exchange of O2 and CO2 to maintain an oxygen rich atmosphere and the conversion of the sun’s energy into photosynthetic activity for nutrient biogeochemical recycling. In addition, the global climate is strongly linked to vegetation’s existence and the ocean’s activity. Thus humankind’s very survival depends on the health of Earth’s vegetation canopies. For this reason, as part of NASA’s Earth Science Enterprise, a vegetation canopy research initiative has been active for the past 20 years. The goal of this eﬀort has been to gain understanding of how radiant energy interacts with vegetation. Such understanding is essential if governments are to make sensible decisions concerning future development of Earth’s resources while maintaining a commitment to the environment. In addition, a detailed knowledge of how these interactions lead to the canopy reﬂectance increases our understanding of nature’s processes. For example, in the investigation of life sustaining photosynthesis and leaf evapo-transpiration, plant physiologists are primarily concerned with the complex biochemical interactions driven by radiant energy in the visible part of the sun’s energy spectrum. The agronomist, on the other hand, is concerned with how the morphology of crop canopies inﬂuence leaf biochemistry to promote photosynthesis and subsequently yield. In these applications, the relative amounts of biochemical agents as well as the local environmental conditions and canopy architecture are primary factors in predicting canopy health and intra- and

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inter-annual photosynthetic gain. Fortunately, for both the plant physiologist and agronomist, information concerning biochemical agents in addition to canopy structural (photometric) characteristics can be inferred from the spectral variation of photons reﬂected from vegetation. The remote sensing of the reﬂected energy, therefore, can provide the opportunity to infer chemical content, canopy yield and in general overall canopy health. This information is encoded in the canopy spectral response to passive sunlight in the form of the canopy reﬂectance or, more generally, in the bi-directional reﬂectance distribution function (BRDF). The key to decoding this information is a fundamental consideration in how a canopy inﬂuences that response from microscopic interactions of photons within leaves to the distribution of radiant energy across leaf aggregates. This is where radiative transfer enters into the investigation. 1.1 Primary Science Issue The fundamental science issue to be addressed here is “Can reﬂected information be reliably interpreted through the vegetation canopy reﬂectance?” The magnitude of the undertaking can be assessed from Fig. 1. Sunlight penetrates the atmosphere, a portion of which is then reﬂected from the foliage, re-enters the atmosphere and is detected by an airborne or satellite sensor. The reﬂected photons contain information about the elements supported by

Fig. 1. Passive photons reﬂected from a canopy contain information about the canopy

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the canopy such as fruits, branches, stems and leaves, as well as information concerning the soil and atmosphere. In order to distill the speciﬁc chemical information concerning the foliage contained in a detected signal, the undesired physical distortions introduced by the soil, atmosphere and canopy architecture must be accounted for. In principle, distortions can be removed through modeling the canopy reﬂectance (CR) with radiative transfer. Thus, the role of a canopy reﬂectance model is to enable the interpretation of remotely sensed observations of a canopy by removing that part of the signal that distorts the desired information whether that is concerning chemical content, species or target identiﬁcation. 1.2 Social Implications of CR Models The social implications of a reliable canopy reﬂectance model are many. For instance in basic science, the understanding of the plant photo-systems can unlock the secrets of photosynthesis. At a more comprehensive level, CR investigations can lead to the establishment of ecological principles which could, in turn, provide improved forest management strategies. In the area of precision agriculture and crop management, properly interpreted remotely sensed information can improve crop yield and production eﬃciency thus beneﬁting humankind. Another signiﬁcant application of CR modeling is to global climate models (GCMs) where canopy reﬂectance becomes the terrestrial boundary condition. While presently not as important as cloud forcing at this time, a representative boundary condition will, in future, progressively become more important as GCMs mature. Finally, in the military arena, reliable optical foliage reﬂectance models along with synthetic aperture radar (SAR) are an important component of precision battleﬁeld engagement (PBE). They can provide warﬁghter asset management and estimation of adversary asset strength and location by enabling the detection of relocatable targets under foliage. In addition, such models can be used to design camouﬂage, concealment and detection (CC&D) to protect assets or counter CC&D systems to more eﬃciently ﬁnd targets. Hopefully, in this way, collateral damage can be limited with the goal of reducing unnecessary human causalities and property loss. 1.3 General CR Modeling Considerations In considering a CR model, the ultimate goal is to uncover signatures. To do so, the following ﬁve prominent vegetation signatures play a signiﬁcant role: + Spectral λ: + + + +

Wavelength response of canopy reﬂectance and transmittance indicating speciﬁc chemical absorption → Arrangement of scattering objects within the canopy Spatial ( r ): Temporal (t): Intra- and inter-annual variability Direction (Ω): Anisotropy from the canopy surface roughness Polarization (Q): Polarized state of reﬂected photons.

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Variation with respect to the wavelength of the response is the most important signature. Indeed, CR investigations are sometimes called hyperspectral investigations for this reason. In the canopy, the photons will selectively be scattered and absorbed at particular wavelengths. For example, as shown in Fig. 2, at water bands where the absorption is particularly high (∼1100, 1450, 1920 nm), the signal will indicate photon depletion. At 550 nm, which is called the green peak, the reﬂectance will have a local maximum since the blue and red wavelength photons are strongly absorbed by chlorophyll on either side. The wavelength spectrum considered in the optical regime extends from ultra violet UV (400 nm) to the mid infrared MIR (2500 nm). Spatial signatures come from objects which are larger than the incident wavelengths of light and reﬂect light macroscopically. Such objects include hidden targets. The temporal signature is a result of changes in the canopy whether of anthropogenic (human made) or biogenic (naturally occurring) origin. Remote sensing addressing temporal variation is called change detection. Directional eﬀects are caused by the canopy surface roughness leading to an anisotropic non-Lambertian response. In addition, the “hot spot” resulting from viewing in the retro direction (in the direction of the sun) where minimal shadowing is observed is a directional eﬀect. Finally, polarization can be a relatively strong signature which can eﬀectively be used to detect hidden human made targets. This is a result of the leaf surface being a weak natural linear polarizer and the leaf’s interior essentially a non polarizer.

Canopy Reflectance 0.40

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Fig. 2. A typical canopy reﬂectance spectrum

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Canopy response signatures through the canopy reﬂectance are inﬂuenced by many factors. Some of these are the + size, shape and distribution of objects within the canopy + biophysical parameters such as − Leaf Area Index (LAI = canopy optical depth) − leaf optical properties − the sun (zenith) angle − Leaf Angle Distribution (LAD) − fAPAR–fraction of absorbed phytosynthetically active radiation. 1.4 A Brief Description of Canopy Reﬂectance Models The interaction of radiant energy with plant canopies can be broadly characterized by the following four approaches. Empirical Models Probably the most simple of all modeling strategies is the empirical model. At the same time empirical models are the most inﬂexible and lead to the least amount of information. In this strategy, a surface is ﬁt to the reﬂectance data. The simplest of all surfaces is the Lambertian surface empirical model for which the canopy reﬂectance is represented by [1] Rf =

cos (θi ) Ei π

where Ei is the incident radiance and θi is the sun zenith angle. While no surface is ever truly a Lambertian surface, this idealized assumption is used quite often because of its simplicity. A more complex representation is given by Minnaert’s model [2] k

Rf = c

[cos (θi ) cos (θv )] Ei cos (θi )

and the Walthall model [3] Rf = a + bθv θi cos (φv − φi ) + cθv2 θi2 + d θv2 + θi2 where a, b, c, d and k are adjustable parameters and the subscripts i and v indicate incident and view directions respectively. One of the major limitations of empirical models is the lack of physical meaning of the parameters and thus the inability to adjust them for diﬀerent physical situations. A second limitation is their use in directions for which they were not intended.

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First Principle Physical Models In this class of models, the turbid medium assumption ﬁgures prominently. This assumption enables the use of a modiﬁed form of conventional participating media radiative transfer theory as represented by Fig. 3. Speciﬁcally, the canopy is assumed to interact with photons as a (green) gas would. At this level of application, the photons do not sense the canopy structure and interact with the individual atoms assumed to be points. This assumption is also commonly called the atomistic assumption for obvious reasons. Contained in this supposition are the continuous media and far ﬁeld assumptions. The medium is also assumed to be microscopically homogeneous. The discrete nature of the scattering centers and shadowing is therefore not part of the model which can represent a signiﬁcant limitation. These assumptions therefore best ﬁt a dense canopy. Canopy architecture is introduced via the leaf angle distribution (LAD) accounting for the distribution of leaf orientation. In what follows, the LCM2 model will be featured as an example of a ﬁrst principles turbid medium model. The reference to ﬁrst principles comes from the intent of these models to be primarily physically based with a minimal of adjustable parameters. Turbid medium modeling in it present form originated at the Estonian School of Actinimetry headed by J. Ross in the mid 70’s [4].

Fig. 3. Conventional radiative transfer participating medium

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Geometric-Optic (GO) Models In this formulation, the canopy is treated as an assemblage of vegetation ﬁlled 3D objects as shown in Fig. 4. Radiative transfer theory is assumed within the assemblages and shadowing is assumed between assemblages. An advantage of the GO model is the inclusion of shadowing. A disadvantage is that a particular canopy realization must be speciﬁed and the statistical nature of a scene is lost.

Fig. 4. Geometric-Optic Model allows for shadows

Computer Simulation Models The ﬁnal model category presents the most faithful depiction of vegetation canopies. Here ray tracing as shown in Fig. 5 is used to render a scene. Particles are tracked as they interact with the canopy leaves and other elements and are tallied as they cross speciﬁed areas. As one can imagine, such a procedure is quite computationally intensive making inversion for canopy properties virtually impossible. In this category, radiosity is a commonly applied method of simulation. Here, leaves are considered as diﬀusely scattering surfaces. Each leaf can “communicate” with every other leaf through view factors as shown in Fig. 6. Radiosity models are also computationally intensive. Figure 7 shows a rendering from the Botanical Plant Modeling System (BPMS) which is one of the most eﬀective rendering models currently available.

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Fig. 5. Ray tracing

Fig. 6. Leaf communication in a radiosity models through view factors

2 Description of the LCM2 Coupled Leaf/Canopy Radiative Transfer (RT) Model 2.1 General Turbid Medium Canopy Modeling Considerations/Modeling Approach By any estimation, the vegetation canopy (see Fig. 8) represents an extremely complex biosystem that would seem to defy any consistent mechanistic analysis. In general, a canopy consists of stems, branches, ﬂowering or non-ﬂowering buds and leaves. In most CR modeling eﬀorts, leaves are considered to be

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Example BPMS Simulation

Fig. 7. Rendering from BPMS program

Fig. 8. A canopy is a very complicated biosystem

the primary scattering and absorbing element. Leaves are classiﬁed as either broad-leaf as found in deciduous vegetation or needle-leaf as found in ﬁr trees. The complexity of a canopy makes the radiative transfer characterization ill posed. For example, questions arise such as – How are the medium biophysical properties to be appropriately deﬁned? – or How can the radiative transfer equation be written in a fractal-like setting? – or How can radiative transfer

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be expected to provide a measure of the radiance when the medium is so biologically diverse? In a word, we are virtually “ignorant” when it comes to deﬁning the radiative transfer setting. The approach taken in the development of the LCM2 model [5] is to acknowledge our ignorance and to rely heavily on natural averaging. So rather than become consumed in modeling the detailed canopy structure or the detailed scattering of photons, we consider biophysical interactions in an average sense since our ignorance allows for nothing more. In conjunction with this modeling approach, we also do not seek answers to diﬃcult questions either. We only address the most basic of issues such as the determination of canopy reﬂectance on a mixed pixel by pixel basis or what is fAPAR – not the determination of speciﬁc species within a canopy or the amounts of each of the hundreds of biochemical agents within a leaf. Another element of this modeling approach is that we can determine the minimum amount of detail required for an adequate description. In other words – what we can get away with. In this regard, we will construct the simplest of models and compare to experiments to determine adequacy or not. If not, then additional eﬀort is put into the model detail. 2.2 The Nested RT Model The radiative transfer modeling approach most often taken in treating such a complex system is to consider intra- and inter-leaf models separately. In the following, each sub-model in LCM2 will be individually described. The collective model will include three sub-models – for the leaf, the canopy and the bridge between them through the bi-Lambertian leaf scattering assumption. Within Leaf Scattering: The LEAFMOD Module Observing Fig. 9 it comes as no surprise that the leaf is a complex multibodied structure of wax layers, elongated palisade parenchyma cells situated on top of a chaotically conﬁgured spongy mesophyll composed of vacuoles. Each element within the leaf has a speciﬁc function. For example, the light harvesting elements, called grana, are contained in the palisade parenchyma and are the initiators of photosynthesis. The function of the spongy mesophyll is to serve as a back scattering medium to optimize the capture of photons in the parenchyma by reducing non productive leakage of photons through the abaxial (bottom) leaf surface. The air-epicuticular wax interfaces called the upper and lower epidermis are deﬁned by the outer leaf adaxial (top) and abaxial extent. The epidermal layers are composed of multilayered membranes of pectin, cellulose, cutin and wax and as will be seen are responsible for leaf polarization. The Leaf Scattering Phase Function The deﬁning feature of the leaf radiative transfer model is the scattering phase function. The scattering phase function used in the LEAFMOD [6]

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Cuticular wax

Palisade Parenchyma

Chlorophylls

Vein

Carotenoid Pigments Cuticular wax Spongy Mesophyll

Anthocyain Pigments

Fig. 9. Typical anatomical structure of a leaf

module is the simplest possible in keeping with our ignorance. Therefore, we assume isotropic scattering. This choice is justiﬁed on the basis that photon deﬂection results from the change of the index of refraction at cellular walls. Since leaf cells, to a ﬁrst approximation, are nearly circular, photons coming from all directions would, on average, experience uniform deﬂection in angle averaged over a cell as depicted in Fig. 10.

dL

Fig. 10. Isotropic within leaf scattering is assumed out of ignorance

Radiative Transfer (RT) Equation In order to avoid the complicating detail of the leaf conﬁguration, an atomistic approximation of the leaf’s interior is assumed. This enables the use of the 1D radiative transfer equation describing the within leaf angular radiance

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ω ∂ + 1 I (τ, Ω; Ω0 ) = dΩ I (τ, Ω ; Ω0 ) Ω ∂τ 4π 4π

I (0, Ω; Ω0 ) = δ (Ω − Ω0 ) I (∆L , Ω (−µ, φ) ; Ω0 ) = 0 valid at each wavelength. The longitudinal direction is measured from the adaxial surface as shown in Fig. 11. A light beam or diﬀuse light is assumed to illuminate the top or bottom surface of the leaf resulting in light transmission through the leaf and reﬂectance from the leaf. Again a 1D model is preferred since nothing can be said about the 3D nature of the leaf with any conﬁdence. In the RT model, the basic physical parameters are the scattering and absorption coeﬃcients (or proﬁles) Σs , Σa , which are wavelength λ dependent, and the leaf thickness dL (see Fig. 10). Ω

ΩL

Ω00

Ω ‘ ′

τ Ω′

Ω

Σ ω≡ s Σ

Σss+ Σaa = =Σ Σ

∆L = dLΣ Fig. 11. Adaxial leaf surface illuminated by light beam or diﬀuse light

Within Leaf Radiance The solution of the one angle (inclination) form of the equations is accomplished using Siewert’ s FN method [7]. The solution strategy is to form two singular integral equations coupling the boundary radiances. These equations are solved by expanding the outgoing existences in a spectral approximation involving Legendre polynomial basis functions and ﬁnding the unknown (blending) coeﬃcients by collocation and matrix inversion. The desired reﬂectance and transmittance outputs are shown in Fig. 12. The above transport formulation presupposes knowledge of the scattering Σs and absorption Σa coeﬃcients respectively at each wavelength. Unfortunately, this information is not available like in the case of neutrons. For this reason, an additional calibration procedure must be performed.

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L 1

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Fig. 12. Solution by the FN method (D = Direct, H = Hemispherical, R = reﬂectance, T = Transmittance)

Calibration of the Leaf Scattering Coeﬃcient To describe the scattering coeﬃcient one could postulate a model of leaf scattering and determine an estimate of this coeﬃcient as is done elsewhere. Here however, we choose to calibrate the scattering coeﬃcient using experimental leaf reﬂectance and transmittance data. The procedure begins with the LOPEX/Leaf Data Set [8]. This is a dataset containing experimental leaf reﬂectance and transmittance measurements for about 70 broad leaf species over the wavelengths 400 nm to 2500 nm. The measurements are speciﬁed as the average of ﬁve repetitive measurements per species. In addition, the leaf average thickness and chemical assays are given. From this extensive dataset, it will be possible to calibrate the scattering proﬁle Σs . The procedure is as follows. Say one is interested in determining the reﬂectance from a broad leaf maple canopy whose average leaf thickness is known and the amounts of chlorophyll, protein, cellulose and lignin and moisture are speciﬁed to simulate a particular environmental condition. This is called the leaf of interest (LoI-See Fig. 13). Next, a reference maple leaf is identiﬁed in the LOPEX library. The reference leaf (RF) will, of course, have a diﬀerent thickness and chemical makeup relative to the LoI to be investigated; however, there will be several similarities. First, the primary biochemical agents will most likely be the same; and second, the scattering coeﬃcients will be similar. The latter similarity is argued on the basis of how scattering comes about. Since scattering is a result of the variation of index of refraction across cell walls and

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Leaf of Interest

LOPEX Leaf DataSet Reference Leaf Database of leaf HH ρ and τ

Fig. 13. Scattering within the reference leaf (RF) and leaf of interest (LoI) is similar

from the similarity of the leaf’s anatomical structure over species, one would logically conclude that the scattering coeﬃcients are similar. For this reason, the scattering within the RF is assumed to be the same as the LoI. Thus, the RF is used for the scattering coeﬃcient of the LoI. This is accomplished by equating the experimental reﬂectance and transmittance measurements of LOPEX and the analytical expressions from the FN solution: ρL (Σa , Σs , dL ; λ) = ρEXP τL (Σa , Σs , dL ; λ) = τEXP where

(1)

L−1 ω = cα dµµφα (−µ) 2 α=0

(2a)

L−1 ω = dα dµµφα (µ) . 2 α=0

(2b)

1

ρL (Σa , Σs , dL )

0

1

τL (Σa , Σs , dL )

0

The solution (inversion) to this set of nonlinear equations gives the absorption and scattering proﬁles for the RF: Σa (λ) , Σs (λ) .

(3)

The scattering coeﬃcient is then assumed for the LoI, but the absorption coeﬃcient is discarded since it is appropriate only for the RF. The absorption proﬁle is reconstructed from the speciﬁc absorptivities σj associated with each major biochemical component: Σa ≡

J j=1

ρj σj

(4)

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where ρj is the density of component j. The speciﬁc absorptivities are obtained from the literature [8]. The ﬁnal step is to run LEAFMOD in the forward mode to determine the leaf reﬂectance ρL (Σa , Σs , dL ) and transmittance τL (Σa , Σs , dL ) for the LoI. As an example how the calibration applies, consider a nominal maple leaf (LoI) with dL = 1.34 mm 3 ρw = 0.723 gm/cm ρch = 38.8 µg/cm2 . A representative maple leaf from LOPEX (RF), whose experimental reﬂectance and transmittance proﬁles will be used in (1), has dL = 0.9 mm. Now consider two cases of canopy stress. The ﬁrst case is for a chlorotic maple leaf where the chlorophyll concentration has been reduced to half of its nominal value ρch = 19.4 µg/cm2 . For the second case, we consider a water stressed maple where the moisture contents is reduced by half ρw = 0.367 gm/cm2 . Figure 14 shows the resulting reﬂectances obtained from the calibration for both cases. Note that the chlorosis only eﬀects the visible (400–680 nm) and the water stress only the NIR (700–2500 nm). Figure 15 shows the variation over wavelength of the canopy reﬂectance for the two cases indicating the expected diﬀerence in reﬂectance from the nominal reﬂectance that should be observed. Thus, with this model, canopy reﬂectance can be leaf property speciﬁc which is the unique feature of LCM2. 2.3 The Leaf Area Scattering Phase Function: The Leaf/Canopy Connection In this module, leaf optical properties are appropriately formulated for the canopy radiative transfer model. For this purpose, the leaf within the canopy is assumed to act as an idealized bi-Lambertian diﬀusely reﬂecting surface. Energy is assumed to be isotropically emitted from the leaf surfaces as shown in Fig. 16. The appropriate form for the leaf phase function is ⎧ 1 ⎪ ⎨ ρL |Ω · ΩL | , (Ω · ΩL ) (Ω · ΩL ) < 0 π (5) γD (Ω , Ω; ΩL ) = ⎪ ⎩ 1 τ |Ω · Ω | , (Ω · Ω ) (Ω · Ω ) > 0 . L L L L π

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Leaf Reflectance 0.7

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Fig. 14. Leaf reﬂectance for chlorotic and water stressed example

The area scattering phase function is then deﬁned as 1 ΓD (Ω , Ω) = dΩL |Ω • ΩL | gL (ΩL ) γD (Ω , Ω; ΩL ) π

(6)

2π

where Ω , Ω are the incoming direction and outgoing directions respectively and gL (ΩL ) is the leaf angle distribution (LAD). The LAD represents the leaf orientation within the canopy and is the characterizing feature of canopy architecture. The bi-Lambertian assumption, therefore, allows the LoI leaf reﬂectance and transmittance, ρL , τL as determined by LEAFMOD, to be

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Canopy Reflectance 0.40

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Fig. 15. Canopy reﬂectance for Chlorotic and water stressed canopies

Reflection from Adaxial Surface

macro Transmission from Adaxial Surface Fig. 16. Bi-Lambertian leaf scattering assumed within canopy

used directly – hence the connection between the LoI and the canopy. The one-angle version of the diﬀuse area scattering phase function to be input into CANMOD is obtained by integration over the azimuthal angle

1

ΓD (µ , µ) = 2

dωgL (ω) a (µ , ω) b (µ, ω)

(7)

−1

where the integrand is a relatively complicated function of both µ, µ and µL . The area scattering function also contains a term for specular reﬂection from the leaf surface as depicted in Fig. 17. The contribution can be expressed as

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Ω′

γ

Ω

γ

Air-leaf cuticular layer interfacial surface

Fig. 17. Specular reﬂection at the leaf surface

1 Γsp (µ , µ) = 4π

π

dωgL (µ∗L [cos (ω)]) K (κ, γ [cos (ω)]) Fs (n, γ [cos (ω)])

0

(8) where the Fresnel coeﬃcient is 1 sin2 (γ − γ˜ ) tan2 (γ − γ˜ ) Fs (n, Ω • ΩL ) = + 2 sin2 (γ + γ˜ ) tan2 (γ + γ˜ )

γ˜ ≡ sin

−1

sin (γ) n

.

The specular component also introduces a linear polarizing component. There is experimental evidence [9] that leaves polarize the signal as a result of specular reﬂection from the leaf surface. The linear polarization component of the area scattering phase function is expressed as ΓQ (µ , µ) =

1 4π

π

dωgL (µ∗L (cos (ω))) K (κ, γ cos (ω)) FQ (n, γ cos (ω)) (9)

0

where FQ (n, γ) = and r|| =

1 ρ|| − ρ⊥ 2

1 sin2 (γ − γ˜ ) 2 sin2 (γ + γ˜ )

1 tan2 (γ − γ˜ ) r⊥ = . 2 tan2 (γ + γ˜ ) With polarization, the following area scattering phase matrix can be constructed: π 1 dωgL (µ∗L ) L (−λ) • Γ (µ , µ) = 4π (10) 0 • K (κ, γ (Ω • Ω)) T (n, γ (Ω • Ω)) L (λ ) where

Reﬂectance Modeling with Turbid Medium Radiative Transfer

T (n, γ) =

Fs (n, γ)

FQ (n, γ)

FQ (n, γ)

Fs (n, γ)

and Fs (n, γ) =

191

1 r|| + r⊥ 2

1 r|| − r⊥ . 2 The area scattering phase matrix Γ (µ , µ) is then the “bridge” between leaf scattering and the canopy phase function as demonstrated in the following section. FQ (n, γ) =

Within Canopy Scattering: The CANMOD Module The Canopy Radiative Transfer Algorithm With the various scattering components now deﬁned, a vector transport equation can be constructed for the photon intensity and linear polarization components of the radiance. Only the simplest of polarization models (linear) will be considered since there is little information regarding circular or partial elliptical polarization states. With a transport equation set, the numerical solution methodology will then be described. Here we veer from the tried and true semi-analytical methods of LCM2 used in the past to the standard discrete ordinates scheme however still with an emphasis on accuracy. Surprisingly, as will be shown, a very accurate method is developed by mining the solution via several convergence accelerators. The Vector Transport Equation Since we will only be concerned with the intensity and linear vertical polarization components, the appropriate transport equation will be for a 2-vector and not the usual 4-vector equation of elliptical polarization. The simpliﬁed Stokes vector is now deﬁned as → I (τ, µ) . (11) (τ, µ) ≡ I Q (τ, µ) The ﬁrst component is the intensity which contains no information about the state of polarization unlike the second component which contains information concerning the linearly polarized state. The vector canopy transport equation that characterizes the variation of the Stokes vector as photons are scattered and absorbed in a vegetative medium can be written generally as

∂ + G (µ) I µI ∂τ

→

1

I (τ, µ) = −1

→

dµ Γ (µ , µ) I (τ, µ )

(12)

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Barry D. Ganapol

The second term represents a total loss of photons in a beam of direction Ω necessitating that a photon be lost either by a scattering or an absorbing event. The intercept function G is given by 1 dΩL |Ω • ΩL | gL (ΩL ) (13) G (Ω) = 2π 2π +

and represents all the leaf area presented perpendicular to direction Ω. The boundary conditions at the upper canopy boundary (τ = 0) and the lower boundary (τ = ∆) are → 1 (0, µ) = δ (µ − µ0 ) I 0 1 (14) → 1 I (∆, −µ) = 2ρT dµ µ I (∆, µ ) p . T 0

Unpolarized sunlight enters the canopy, is scattered and absorbed with a fraction linearly polarized and both unpolarized and polarized components can then be scattered back into the canopy by a partially Lambertian reﬂecting (target or soil) surface situated underneath the canopy. The strength of the reﬂected polarized signal from the surface is determined by an assigned degree of polarization pT . The solution is more conveniently obtained if the Stokes vector is decomposed into its uncollided and collided components →

→

→

I (τ, µ) = I 0 (τ, µ) + I c (τ, µ)

giving two transport equations → ∂ + G (µ) I I 0 (τ, µ) = 0 µI ∂τ 1 → → ∂ + G (µ) I I c (τ, µ) = dµ Γ (µ , µ) I (τ, µ ) . µI ∂τ

(15a)

(15b)

−1

The uncollided contribution is easily solved → 1 −τ /ξ δ (µ − µ0 ) Θ (τ /ξ) I 0 (τ, µ) = 0 e

(16)

where ξ ≡ µ/G (µ). Substituting the total vector into the second transport equation for the collided component gives

Reﬂectance Modeling with Turbid Medium Radiative Transfer

→ ∂ + G (µ) I I c (τ, µ) µI ∂τ 1 → 1 −τ /ξ0 e = dµ Γ (µ , µ) I c (τ, µ ) + Γ (µ0 , µ) 0 → Ic → Ic

−1

193

(17)

(0, µ) = 0

1 1 −τ /ξ0 (∆, −µ) = 2ρT µ0 e + dµ µ Ic (∆, µ ) pT 0

Once (17) is solved numerically for the angular intensity, the desired quantities are the vector reﬂectance

Rf I Rf Q

1 = µ0

1

→

dµ µ I (0, −µ )

(18a)

0

as well as the degree of polarization at the top of canopy (ToC) 1 Dp0 ≡

1

dµ µ Q (0, −µ ) / 0

dµ µ I (0, −µ ) .

(18b)

0

The Converged SN (CSN) Algorithm Simplicity and versatility are the hallmarks of the discrete ordinates (SN) numerical algorithm developed for neutron transport calculations but widely applied in all particle transport ﬁelds. The method amounts to a convenient bookkeeping scheme for a particle population as one sweeps in a speciﬁed direction across the spatial domain. Of course, inherent in the method are numerical errors resulting from the discretization of the spatial and angular domains; and as a result, the SN method has always been considered an approximate numerical scheme. In this section, a variation of the SN algorithm will be devised for radiative transfer in canopies. The methodology couples a Romberg iterative strategy with a Wynn-Epsilon (Wε) acceleration to generate nearly 4-place accuracy for the canopy reﬂectance. SN/Romberg/Wε Theory The method will be developed for the radiative transfer equation (17) for a canopy slab of optical depth (LAI) ∆ and an impinging (plane) beam source in direction µ0 at ToC. At the bottom canopy boundary, a partially reﬂecting condition is imposed possibly representing a target or soil. After inclusion of the impinging source as a volume source, introducing the SN approximation and integrating over a spatial interval h, as shown in Fig. 18, we arrive at

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τj

h

τj+1

Fig. 18. Discretized spatial domain

the following SN equations without spatial approximation for the collided → intensity I c :

→ → → µm I c,j+1,m − I c,j,m + Gm dτ I c,m (τ ) =

N m =1

ωm Γ (µm , µm )

h →

dτ I c,m (τ )

(19)

h

1 + ξ0 e−τj /ξ0 − e−τj+1 /ξ0 Γ0,m , 0 where Γ (µ m , µm ) =

Lmc

ωj gj aj,m bj,m .

j=1

The m-subscript represents the angular discretization, while the j-subscript represents the spatial discretization at the interval edges. The angular quadrature points are chosen to be the zeros of the Legendre polynomial of degree N/2 over the half ranges [–1,0] and [0,1] PN/2 (±µm ) = 0, m = 1, N/2 . The quadrature weights ωm are for the corresponding Gauss/Legendre quadrature. Spatial discretization is uniform over [0,∆] with h ≡ ∆/Nh . In (19), the following quantities have been deﬁned: → I c,j,m

→

≡ I c (τj , µm )

Gm ≡ G (µm ) Γm ,m ≡ Γ (µm , µm ) .

Reﬂectance Modeling with Turbid Medium Radiative Transfer

195

If we were to follow the common practice of relating the integral of the intensity over interval h to the edge ﬂuxes, we would call the average intensity (represented by the integrals) the average of the interval edge ﬂuxes. Here, we adopt a more general view where the integration is interpreted as a quadrature approximation of a given order K K dxg (x) = αk gk + O hK+1 k=1

h

If a trapezoidal rule is assumed, then K is 3 and the α’s are 1/2 and we have the usual “diamond diﬀerence” approximation. More importantly, we know the order of the error and the form of the error tail of the solution [10] →Exact → I c,j,m = I c,j,m

+

∞ → β k h2k .

(20)

k=1

With this knowledge, a Romberg iterative scheme [11] can be applied to (19) in the fully discretized form →

→

→

− + Tm I c,j+1,m − Tm I c,j,m = q j,m

h + 2

h + 2 with − Tm

+ Tm

⎡→ ⎤ I c,j+1,m + ⎦ ωm Γm ,m ⎣ → + I c,j,m m =1,=m

N/2

(21)

⎡→ ⎤ I c,j+1,m + ⎦ ωm Γm ,m ⎣ → + I c,j,m m =N/2+1,=m N

h h ≡ µm + Gm I − ωm Γm,m 2 2 h h ≡ µm + Gm I + ωm Γm,m 2 2

−x /ξ 1 −xj+1 /ξ0 j 0 Γ0,m ≡ ξ0 e −e 0 in order to successively eliminate the higher order error terms in the error tail of (20). This is an extension of Richardson’s extrapolation as applied previously to the transport equation [12]. → q j,m

Iteration Strategy The SN algorithm of order N is implemented in the standard way with sweep iteration in the positive and negative angular directions. The convergence of the sweeps is accelerated using the Wynn-Epsilon algorithm. In this algorithm,

196

Barry D. Ganapol (n)

(n)

(n)

(n+1)

ε−1 = 0, ε0 εk+1 = εk

= Sn

−1 (n+1) (n) + εk − εk , (n)

Sn represents the sweep iterates and ε2k the subsequent improved approximation forming a tableau of approximations. The tableau diagonal is then interrogated for convergence of the canopy reﬂectance. Each Sn calculation contributes the ﬁrst element of a Romberg scheme for convergence in spatial discretization. Finally, convergence in the quadrature order N is performed again through a Wε acceleration. Thus, three convergence accelerations are used to mine the SN solution for high accuracy.

3 LCM2 Demonstration The three modules described above have been combined into the LCM2 nested radiative transfer CR code. The code can be run in two distinct modes – Single Pixel and Independent Pixel Approximations. In this section, both modes will be demonstrated. 3.1 Single Pixel Approximation (SPA) Figure 19 shows the variation of the reﬂectance of the intensity component for the visible and NIR wavebands for a canopy of with an LAI of 2 and erectophile (upright) LAD as the surface index of refraction varies. A soil background reﬂectance of 0.2 is assumed. The behavior is similar at all wavelengths. In particular, as the index of refraction increases from 1 (air) to 1.5, the reﬂectance increases. With increasing specular reﬂection from the leaf’s adaxial surface, more photons avoid the leaf’s interior and therefore there is less absorption allowing more to exit the canopy. This is in light of the tradeoﬀ of less photons being diﬀusely scattered from the leaf’s interior. The magnitude of the overall eﬀect will no doubt depend on canopy properties, but the tendency to increase reﬂection with increasing refractive index seems readily apparent. Figure 20 displays the change in the reﬂection of the Q-component with increasing index of refraction n · Rf Q also increases with n as does the degree of polarization at ToC. This is to be expected since an index of refraction diﬀerent from 1 is the origin of linear polarization. It should also be noted that Rf Q does not exhibit the usual pronounced variation at the green peak associated with the leaf’s interior. While there is some evidence of a green peak, there is no distinct chlorophyll well eﬀect. This result is a direct consequence of the fact that polarization arises from the leaf surface as modeled. The slight rise at green (factor of 0.5 compared to a factor of 2 for Rf ), is a result of multiple scattering of the intensity component providing the

Reﬂectance Modeling with Turbid Medium Radiative Transfer

197

0.09

n=1 0.08

n = 1.01 n = 1.1

0.07

n= 1.2

RfI

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n = 1.4 n = 1.5

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RfI

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700

750

800

850

900

950

1000

1050

λ(nm) Fig. 19. Canopy reﬂectance with specular leaf reﬂection for an unpolarized target

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Barry D. Ganapol 0.006

LAI = 2 0.005

n=1 n = 1.25

0.004

n = 1.5

RfQ

0.003

0.002

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800

900

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1100

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Dp0

0.03

0.02

0.01

0.00

300

λ(nm) Fig. 20. Linearly polarized reﬂectance RfQ and degree of polarization (DP0)

Reﬂectance Modeling with Turbid Medium Radiative Transfer

199

source of polarization. The diﬀerence between the two components is more clearly evident in Fig. 21 where the two components are shown for increasing canopy over-story (increasing canopy LAI). Saturation is observed in both components when, as the canopy become denser, there is little change in the reﬂectances. One diﬀerence to be noted is that with increasing LAI, Rf decreases in the visible and increases in the NIR. This is a result of the highly absorbing nature of the leaf in the visible–allowing for increased absorption in a dense canopy and its highly scattering nature in the NIR–allowing for increased scattering out of the canopy. Also note that the increase in the NIR reﬂectance over the visible is a factor of two for the polarized component while it is a factor of 10 for the intensity component – again a consequence of leaf surface scattering being responsible for polarization. 3.2 Independent Pixel Approximation (IPA) Application to Precision Agriculture Currently the transport methods development group at the University of Arizona is a part of a demonstration of the use of Un-piloted Aerial Vehicles (UAVs) by NASA in precision agriculture. In particular, the eﬀort is focused on using a UAV to provide a synoptic view of the Kawai Coﬀee Company coﬀee ﬁelds. The Pathﬁnder UAV, carrying several cameras to record the visible and NIR reﬂectance (shown in Fig. 22), was ﬂown over the coﬀee ﬁelds. The intent of the campaign was to explore the possibility of transferring NASA technology to the agricultural community. LCM2 in the IPA mode was the basis of a predictive Neural Net (NN) to distinguish the amount of yellow coﬀee cherries (ripe crop) from green (under ripe) and red cherries (over ripe) in the ﬁelds. LCM2 was used to train the NN to predict the three cherry classiﬁcations in a scene given reﬂectance estimates. The reﬂectances from the UAV ﬂyover was then introduced as input and a prediction made based on the LCM2 model as shown in Fig. 23. The prediction of yellow cherries agreed to within 10% of the ground truth which is truly remarkable agreement. Linearly polarized Targets Now consider a linearly polarizing target beneath the canopy. To test LCM2 in a more realistic manner, a 64 (8 × 8) pixel scene was constructed. The LAI and soil reﬂectance were ﬁxed at 2 and 0.2 for all pixels respectively and random amounts of 5 LAD distributions were assumed to represent a random LAD. Figure 25 shows the reﬂectances for the scene at three wavelengths 550 nm, 680 nm and 800 nm. For the same wavelengths, Fig. 26, shows a T72 tank in the clear, which is subsequently to be hidden under a canopy of various LAIs. The vehicle surface is assumed to be fully linearly polarizing and reﬂects at 0.3. The surrounding soil is assumed to be reﬂecting at 0.1.

200

Barry D. Ganapol 1

LAI = 1:Rf LAI = 2 LAI = 4 LAI = 8 LAI = 1:RfQ

Rf

0.1

LAI = 2 LAI = 4 LAI = 8 0.01

0.001 300

400

500

600

700

800

900

1000

300

400

500

600

700

800

900

1000

0.007

0.006

RfQ

0.005

0.004

0.003

0.002

λ(nm) Fig. 21. Variation of LAI over unpolarized target (LAD = Planophile)

Reﬂectance Modeling with Turbid Medium Radiative Transfer

201

Fig. 22. Helios and Pathﬁnder UAVs

Figures 27a,b show RfI and Dp0 for the T-72 under canopies of LAIs of 4 and 6 at the three wavelengths. With increasing LAI, the T-72 becomes increasingly obscured in both measures and all wavelengths as expected. An alternative way of viewing the scene information is to plot the degree of polarization (Dp0) against RfI . When this is done for the canopy without a target we have Fig. 28a indicating no correlation between the degree of polarization and the RfI . Doing the same for the target under a canopy of LAI = 6 and recalling that the target could not be identiﬁed at all from the scene variations of RfI , and Dp0 individually, we observe the result that the T-72 target becomes clearly deﬁned in all wavelengths with the NIR yielding the clearest deﬁnition. This result can be explained by noting that the T-72 is a much brighter object than the surrounding background and it polarizes while soil and canopy do not. In the Dp0/RfI plot this places the T-72 target in the upper right corner and the soil variation is conﬁned to the bottom. The soil variation however still remains uncorrelated as in Fig. 28a on a very narrow scale. We believe this to be a very signiﬁcant result indeed in the ﬁeld of target identiﬁcation. Future Computational Challenges Facing CR Modeling Several serious computational challenges face canopy modelers in the future. One major challenge is the inverse problem of which the coﬀee cherry estimation in Kauai is an example. Can canopy state variables, important for ecological prediction and precision agriculture, be reliably obtained by inversion?

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Fig. 23. Yellow coﬀee cherry amounts predicted in indicated ﬁelds (Field 408 – topmost ﬁeld)

This challenge involves not only more powerful computational architectures, but also the development of fast running and accurate inversion algorithms. NNs are an example of the wave of new methods, but more powerful optimization concepts need to be developed if CR models are expected to see routine use. Of course, this must go hand in hand with the development of larger memories and faster parallel methodologies. In addition, programming languages must keep pace with the new architectures to make their use readily accessible. Finally, the establishment of reliable canopy optical properties must also be part of the mix. Experimental resources should be dedicated to providing more representative leaf and canopy properties as the CR models become more sophisticated.

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203

Fig. 24. Predicted green/yellow/red cherry distributions. (Field 408 Block 4)

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Barry D. Ganapol 550nm Degree of Linear Polarization (Dp0)

7

0.025 0.026 0.027 0.028 0.029 0.030 0.031 0.032

6

Y Data

5

4

at TOC

LAI = 2

3

2

680nm

X Data 1

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X Data

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Fig. 25. Canopy reﬂectance for scene with a random LAD

6

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Reﬂectance Modeling with Turbid Medium Radiative Transfer

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Contour Graph 1 6

5

Y Data

ρΤ = 0.1 Soil 4

T

ρΤ= 0.3 T-72 PT = 1

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Fig. 26. T-72 tank in the clear

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Fig. 27a. T-72 under foiliage of LAI = 4

Contour Graph 7

Reﬂectance Modeling with Turbid Medium Radiative Transfer

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Fig. 27b. T-72 under foiliage of LAI = 6

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

B. Hapke and E. Wells, J. Geophys Res., 86(B4), 3055–3060, (1981). M. Minnaert, Astrophys. Jour., 93, 403–410 (1941). C. Walthal, et. al., Appl. Opt., 24(4), 383–387 (1985). J. Ross, The Radiation Regime and Architecture of Plant Stands, W. Junk, The Hague, Netherlands (1981). B. Ganapol ,et. al., Rem. Sens. Environ. 70:153–166 (1999). B. Ganapol, et. al., Rem. Sens. Environ. 63:182–193 (1998). C. Siewert, Nucl. Sci. & Eng., 69, 156–160 (1979). B. Hosgood, et. al. European Commission, Joint Research Centre, Institute for Remote Sensing Applications, Report EUR 16095 EN (1995). L. Grant, C. Daughtry., and V. Vanderbilt., Envir. Exp. Bot. 27,139–145 (1987). E. Larsen and W.F. Miller, Nucl. Sci. & Eng., 73, 76–83 (1980). F. Press, Numerical Recipes, Cambridge Press, (2001). W. Abbott and E. Allen, Nucl. Sci. & Eng., 108, 278–288 (1991).

Rayspread: A Virtual Laboratory for Rapid BRF Simulations Over 3-D Plant Canopies Jean-Luc Widlowski, Thomas Lavergne, Bernard Pinty, Michel Verstraete and Nadine Gobron Institute for Environment and Sustainability, Joint Research Centre, TP. 440, Via E. Fermi 1, 21020 Ispra (VA), Italy [email protected]

Accurate knowledge of the spatial (and temporal) variability of the biosphere’s characteristics is useful not only to address critical scientiﬁc issues (climate change, environmental degradation, biodiversity preservation, etc.) but also to provide appropriate initial state and boundary conditions for general circulation or landscape succession models. In particular, the 3-D structure of vegetation emerged as a crucial player in processes aﬀecting carbon sequestration, landscape dynamics and the exchanges of energy, water and trace gases with the atmosphere e.g., [BWG04]. The growth and development of plant architecture, in turn, are primarily conditioned by eﬀective interception of solar radiation that provides the necessary energy for photosynthesis and other physiological processes [VB86]. In this context, space borne, optical remote sensing provides a convenient, eﬃcient and cost-eﬀective way to acquire information on the state of terrestrial vegetation, over large areas and at spatial resolutions adequate to address many key ecological and climate change related issues. The physical interpretation of such remote sensing measurements, however, can provide reliable quantitative information only on the relevant state variables that control the interactions of the radiation ﬁeld with all intervening media from the light source to the detector e.g., [VPM96]. The simulation of such processes, using physically based radiative transfer (RT) models, thus allows to estimate the most probable value of a remote sensing measurement, given that the values of all state variables in the model, the conditions of observation and the nature and role of all relevant radiative processes in the system are speciﬁed in advance. This modeling approach is known as the direct or forward mode, and can be used, for example, to determine which state variable in a given model is primarily responsible for the observed signal variability under speciﬁc condition of observation and illumination. It also provides ample testing ground for the intercomparison of diﬀerent radiation transfer models [PWT04]. The interpretation of remote sensing data requires applying the same model in inverse mode, or more speciﬁcally inverting the model against the data set, in order to retrieve the state variables of interest [GS83, KKP00]. At the Earth’s surface, the spectral, directional and polarization signatures

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of the various scattering elements, together with their number, geometry and spatial distribution, are responsible for the radiative characteristics of the reﬂected electromagnetic signal in the optical part of the solar spectrum. Furthermore, the conditions of observation and illumination, as well as the state of the atmosphere at the time of measurement, may signiﬁcantly alter the reﬂectance values of terrestrial surfaces. State of the art inversion techniques involve the pre-computing of reﬂectance ﬁelds (and/or related quantities) for a pre-selected ensemble of likely canopy structures and under a wide variety of possible scenarios of illumination and observation geometries. These look-up-tables (LUT) can then be searched to identify the best-matching candidate capable of explaining the entire string of available (multi-spectral and/or -directional) remote sensing data. Such approaches reduce the amount of retrievable solutions to the inversion but require (1) the availability of detailed knowledge regarding typical canopy architectures (in particular the size, shape, and number of leaves, branches and crowns) as well as their temporal evolutions, (2) a RT model capable to actually represent such complex 3-D structures, and (3) the accurate and reliable simulation of the reﬂectance ﬁelds for these canopy representations under all envisaged spectral, viewing and illumination conditions. This contribution will provide a concise description of the major RT characteristics in plant canopies. Following this, the Rayspread model will be presented, which addresses the need for rapid, yet accurate, simulations of the reﬂectance ﬁeld over arbitrarily complex plant environments. Validation and veriﬁcation of this model are followed by a small case study on the impact of woody structures on the directional signatures of vegetation canopies.

1 Canopy Radiation Transfer Fundamentals The radiative transfer equation is a statement of the law of energy conservation [Cha60]. Under steady state conditions and neglecting any polarization, diﬀraction and emission eﬀects, its integro-diﬀerential form describes the change in the intensity I [W m−2 sr−1 ] of monochromatic radiation travelling in direction Ω at some location r: σ ˜s (r, Ω → Ω) I(r, Ω ) dΩ (1) Ω • ∇I(r, Ω) + σ ˜e (r, Ω, Ω0 ) I(r, Ω) = 4π

Variations in intensity are thus either due to radiation – originally travelling in direction Ω – being intercepted by the foliage (second left hand term), or else, due to radiation being scattered from some direction Ω into the direction of interest Ω at location r (right hand term). More speciﬁcally, σ ˜e [m−1 ] is the extinction coeﬃcient, which indicates the loss in intensity due to absorption and scattering of radiation away from Ω, and σ ˜s [sr−1 ] is the diﬀerential scattering coeﬃcient for photons scattered from some direction Ω into a unit solid angle about direction Ω at point r. For the purpose of

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highlighting RT features that are speciﬁc to vegetation canopies, however, a one-dimensional approach may be more suitable at ﬁrst since the spatial ˜s will only add additional complexity to the dependency of both σ ˜e and σ RT formalism. In the following we thus consider a structurally homogeneous plane-parallel leaf canopy of depth H, that is being illuminated from direction Ω0 in a spatially uniform manner from the top (z = H). 1.1 Foliage Properties Aﬀecting the Scattering Coeﬃcient, σ ˜s The probability for radiation, travelling along some arbitrary direction Ω , to be scattered into a particular direction Ω clearly depends both on the amount of foliage area present within the canopy volume, i.e., the leaf area density Λ(z) [m2 /m3 ], the distribution of the foliage orientation in the canopy, i.e., gL (z, ΩL ), and the leaf scattering phase function f (Ω → Ω; ΩL , z). Assuming that all leaf normals ΩL (θL , φL ) – where θL is the zenith angle and φL is the azimuthal angle – point into the upper hemisphere (2π + ), the diﬀerential scattering coeﬃcient, σ ˜s at some level z in the canopy can be written as: Λ(z) σ ˜s (z, Ω → Ω) = gL (z, ΩL ) |Ω · ΩL | f (Ω → Ω, ΩL , z) dΩL (2) 2π 2π+ where, f (Ω → Ω; ΩL ) describes the fraction of intercepted energy (from photons that initially travelled in the direction Ω ) that is scattered by a leaf with outward normal ΩL into a unit solid angle about the direction Ω. Note that the integral of f (Ω → Ω; ΩL ) over all exiting directions yields the single scattering albedo, ωo , which may depend on both the initial photon direction Ω and the leaf normal orientation ΩL [SM88]. Probability distributions of the leaf orientation also have a signiﬁcant impact on the interception probability for radiation travelling within a canopy. More speciﬁcally, the leaf-normal distribution (LND) function gL (z, ΩL ), denotes the fraction of total leaf area in the horizontal layer of unit thickness at height z whose normals fall within a unit solid angle around the direction ΩL , and must satisfy the following normalization criterion [Ros81]: 1 2π

gL (ΩL ) dΩL = 2π +

1 2π

2π

0

π/2

gL (θL , φL ) sin θL dθL ≡ 1

dφL 0

Although some plants are known to change their orientations in response to solar illumination conditions, heliotropism (and other two-dimensional LND functions) are primarily treated in theoretical studies [SGR85, Ver87] because the assumption of azimuthally random leaf distributions signiﬁcantly reduces the mathematical complexity of the function gL (θL , φL ). Among the most commonly used azimuthally invariant leaf normal distribution functions are 1) the trigonometric functions of [Bun78]:

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(3)

and (2) the beta functions of [GS84]: ∗ gL (θL ) =

2 xξ−1 (1 − x)ν−1 π B(ξ, ν)

0<x<1

(4)

where B(ξ, ν) = Γ (ξ)Γ (ν)/Γ (ξ + ν) is the beta function, Γ the gamma func∗ tion, x = 2θL /π and gL (θL ) = gL (θL ) sin θL . Table 1 provides the parameter values for (3) and (4) to generate six common LNDs, including an erectophile (planophile) LND that features predominantly vertical (horizontal) foliage orientations typical for most grasses (watercress). The impact of diﬀerent LNDs on the surface-leaving radiation ﬁeld is such as to alter the observed BRF1 values anywhere from −10 % to +40 % with respect to a 1-D canopy ∗ (θL ) = sin θL ). On with directionally invariant interception probability (gL the other hand, for a homogeneous foliage canopy with a given LND, the differences in BRF values arising due to the divergence of (3) and (4) amounts to only ∼ 0.5 % on average in both the red and near-infrared wavelengths. Table 1. Parameter values for the trigonometric (a, b, c) [Bun78] and beta (ξ, ν) ∗ (θL ) function [GS84] representation of 6 diﬀerent leaf-normal distributions, gL

a b c ξ ν

Uniform

Spherical

Planophile

Erectophile

Plagiophile

Extremophile

2/π 0 0 1.000 1.000

sin θL 0 0 1.101 1.930

2/π 2/π 0 2.770 1.172

2/π –2/π 0 1.172 2.770

2/π 0 –2/π 3.326 3.326

2/π 0 2/π 0.433 0.433

1.2 Canopy Properties Aﬀecting the Extinction Coeﬃcient, σ ˜e Due to the physical size of foliage elements the extinction of solar radiation within a given canopy volume is wavelength independent but highly sensitive to the direction of propagation. More speciﬁcally, the probability, per unit pathlength of travel, that photons may hit a ﬁnite-sized leaf depends not only on the amount and orientation of the scatterers but also on the direction of travel of the photons. As a consequence, the probability that single-scattered solar radiation exits a “cloud of oriented ﬁnite-sized scatterers” is signiﬁcantly 1 The BRF [-] is the radiant ﬂux density (Φtarget /Ar ) reﬂected into some inﬁnitesimal solid angle divided by the radiant ﬂux density reﬂected by an ideal, perfectly diﬀusing Lambertian reﬂector (Φlambert /Ar ) into the same inﬁnitesimal solid angle under identical conditions of illumination [NRH77].

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enhanced if its new direction lies along (or is close to) the retro-reﬂection direction [Hap81, Ger88, Kuu91]. Thus a viewer looking at a canopy target along the solar illumination direction will not see any shadows but a local peak in the surface-leaving reﬂectance ﬁeld, commonly known as the hot spot (or, opposition, or, Heiligenschein) eﬀect. To account for this retro-reﬂection peak [PV97] have proposed to write the extinction coeﬃcient as: ˜ Ω, Ω0 ) = Λ(z) G(Ω) O(z, ˜ Ω, Ω0 ) σ ˜e (z, Ω, Ω0 ) = σe (z, Ω) O(z,

(5)

where σe (z, Ω) is the extinction coeﬃcient for a purely turbid medium, i.e., an inﬁnite number of oriented point scatterers. Λ(z) [m2 /m3 ] is the leaf area density (LAD) at level z, and G(Ω) is the well known function of [Ros81] accounting for the projection of the leaf normal orientations in a particular direction of interest. G(Ω) is an indicator of the likelihood that radiation may be intercepted by the canopy if travelling in direction Ω, since it is the mean projection of a unit foliage area, in the direction Ω, per unit volume of canopy: 1 gL (ΩL ) ΩL · Ω dΩL G(Ω) = 2π 2π+ where for azimuthally independent leaf normal distributions G(Ω) simpliﬁes to G(θ) and the probability distribution function of the leaf normal orientations gL (ΩL ) reduces to gL (θL ). Note that the probability of interception remains directionally constant (G(θ) = 0.5) if the LND is perfectly spherical, ∗ (θL ) = sin θL . i.e., gL ˜ Ω, Ω0 ) in (5) is a geometric correction that aims Finally, the term O(z, at accounting for the reduced extinction coeﬃcient along the retro-reﬂection direction in a canopy composed of ﬁnite sized leaves. [VPD90], who studied the physical mechanism behind the hot-spot eﬀect – among others, e.g., [Hap81, Ger88, Kuu91] – idealised this free space between scatterers by considering two cylindrical volumes V1 and V2 drawn along the incoming Ω0 and outgoing Ω directions, respectively. In analogy to an incident and reﬂected beam of light, V1 and V2 have a common base deﬁned by a circular sun ﬂeck of radius r, and therefore share a common volume free of scatterers. With the assumption that the leaf area density Λ and the Ross function G(Ω) are constant along the coordinate z, the correction function ˜ Ω, Ω0 ) is given by ∂( V• / V2 )/∂z for the extinction coeﬃcient O(z, which is equal to [PV97]: ! 2 −1 2 1/2 ˜ O(z, Ω, Ω0 ) = 1 − cos ζ∗ − ζ∗ [1 − ζ∗ ] π and unity if ζ∗ = min (1, ζT ) = 1, where ζT = zGf /2r > 1. [GPV97] describes how to retrieve an estimate of r, and the geometric factor Gf is deﬁned as a combination of illumination (observation) zenith θ0 (θv ) and azimuth φ0 (φv ) angles:

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Gf =

1/2 tan2 θ0 + tan2 θv − 2 tan θ0 tan θv cos(φ0 − φv )

A direct consequence of this formulation is that the smaller the radius r of the circular sun ﬂecks at some depth z in the canopy the narrower the ˜ Ω, Ω0 ) will provide signiﬁcant contributions. For angular range to which O(z, homogeneous leaf canopies, one can thus expect the hot spot contribution due to radiation that has only collided once with the underlying soil (i.e., the black canopy contribution [PGW04]) to be slightly narrower than that due to radiation having collided once only with the foliage elements of the canopy overstory. This is illustrated in the main left panel of Fig. 1 which shows the BRF contributions due to the single collided by the leaves (dashed) and the single collided by the soil (solid) radiation components for a homogeneous leaf canopy with an LAI of 3 in the red spectral band.

0.030

0.025

0.020

total BRF in X−plane 0.015 -60

0.03

0.02

Bidirectional Reflectance Factor

0.04

0.030

0.05

-40

-20 0 20 40 Viewing angle [degree]

60

single-collided by leaves

0.01

Bidirectional Reflectance Factor

Bidirectional Reflectance Factor

Bidirectional Reflectance Factor

0.05

0.04

total BRF in X−plane -60

-20 0 20 40 Viewing angle [degree]

60

-20 0 20 40 Viewing angle [degree]

60

single-collided by leaves multiple-collided

0.00 -40

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single-collided by soil

0.02

single-collided by soil -60

0.020

0.015

0.03

0.01

0.00

0.025

-60

-40

-20 0 20 40 Viewing angle [degree]

60

Fig. 1. Main panels: The BRF contributions in the principal plane (θ0 = 50◦ ) due to the single-collided by the soil (solid ), single-collided by the leaves (dashed), and multiple collided (dotted ) radiation components for structurally homogeneous (left panel ) and heterogeneous (right panel ) leaf canopies in the red spectral band. Inlaid graphs: The total BRFs in the orthogonal (or cross) plane for the same scenes

Heterogeneous leaf canopies are characterized by a conglomerate of differently sized (and shaped) gaps among their foliage elements, that all have their own hot spot contribution (of diﬀerent angular widths). Consider, for example the simple case where all foliage elements are uniformly distributed within spherical volumes – of identical size and LAD – that are randomly located above a perfectly ﬂat background (compare with the lower right panel of Fig. 2). In this case, only two typical scales of voids exist among the scatterers in the scene: 1) gaps of the order of a few centimeters that occur amid the foliage elements in the spherical volumes, and 2) gaps of the order of a

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Structurally Homogeneous Vegetation Canopy Stochastic Representation

Deterministic Representation

Structurally Heterogeneous Vegetation Canopy Stochastic Representation

Deterministic Representation

Fig. 2. Top panels: structurally homogeneous vegetation canopies, where the foliage is uniformly distributed throughout the available volume. Bottom panels: structurally heterogeneous vegetation architectures, where the scatterers are conﬁned within the volumes of several non overlapping spheres. In the right panels, the discrete foliage elements are represented in a deterministic manner, whereas in the left panels, they are displayed in a stochastic manner using pseudo-turbid objects

few meters that exist between the spherical foliage clusters. As a consequence the black canopy BRF component will feature both a narrow hot spot contribution, due to radiation exiting the canopy via the spherical foliage volumes, and another much broader contribution due to radiation exiting the canopy from within the gaps in between the spheres. Both of these signatures are readily discernible in the black canopy BRF contribution (solid line) in the right panel of Fig. 1. At the same time the angular width of the hot spot contribution due to the single collided by the leaves BRF component (dashed line), is – just as in the homogeneous case – primarily driven by the gap sizes among the scatterers in the foliage clusters (spheres). Note also, that the multiple-scattering component does not feature any discernible hot spot contributions. Remotely acquired hot spot observations thus consist of several nested contributions of increasing angular widths, that arise from mutual shading conﬁgurations amid target constituents occupying diﬀerent spatial scales (e.g., leaf, tree, terrain) [Mer89, QX94] and [GQW97]. The black canopy component is of particular interest here since it is conditioned by the shapes, sizes and spatial organisation of the overlying canopy structure without, however, being aﬀected by the spectral properties of that medium. In addition, the shape of its BRF ﬁeld is known to exhibit a

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bell-shaped pattern – ignoring the immediate hot spot region [PWG02] – because the presence of gaps in between the foliage structures reduces the domain averaged extinction coeﬃcient for radiation exiting the heterogeneous canopy at small θv (unlike at large θv where the optical depth increases dramatically). Given that the RT in vegetation canopies is essentially driven by single-scattering interactions in the red, the bell-shaped black canopy BRF component is thus in direct competition with the remaining, predominantly bowl-shaped BRF components to form the overall “shape” of the surface leaving radiation ﬁeld (compare with the inlaid graphs in Fig. 1). Since the reﬂectance values of soils tend to be larger than those of foliage in the red, the magnitude and shape of the overall BRF ﬁeld is largely driven by the amount and spatial organisation of the gaps, in addition to the opacity and shapes of the foliage clusters in the canopy. In fact, there is now ample observational evidence to link the reﬂectance anisotropy of 3-D vegetation targets (in the red spectral band) to the heterogeneity of the canopy [WPG01,LG02,CRH03] and [WPG04]. Furthermore, [DM04] have recently proposed a germane contribution which highlights and helps identifying geophysical conditions where 3-D eﬀects have a very signiﬁcant contribution to the establishment of the radiative transfer regimes. Their approach and results derive from statistical analyses of various types of internal variability of the extinction coeﬃcient. The interpretation of multi-directional remote sensing observations could thus, in principle, provide additional information on the degree of heterogeneity that exists within terrestrial environments. This statement is, however, subject to the dimensionality of the RT model at hand, since in inversion mode only those vegetation structures can be retrieved that the model is actually capable of simulating (in forward mode). Obviously the more free parameters a model possesses, the larger the number of possible combinations (and hence structural and spectral scenarios) that may be simulated. Fractal or L-system based approaches may facilitate the generation of visually impressive plant representations. Nevertheless, the construction of ecologically (and bio-mechanically) accurate canopy architectures does require access to dozens (or more) structural parameters that may be diﬃcult to come by. In eﬀect, detailed ﬁeld measurements of structural and spectral parameters suitable for inclusion in 3-D RT model simulations are still lacking for most species and geographical locations. Recently, [WPG03] provided a compilation of allometric relationships that allow the faithful reconstruction of ﬁve major European tree species for 3-D canopy reﬂectance modelling purposes at medium spatial resolution. For any such forest structure, variable boundary conditions may have to be accounted for, that is, variable proportions of diﬀuse radiation at the top of the canopy, variable soil scattering properties and albedo values beneath the canopy, and – depending on the spatial resolution of the observing sensor – variable horizontal radiation ﬂuxes due to the discrete nature of adjacent canopy structures. Luckily the net impact of the lateral boundary conditions is certainly negligible for medium spatial

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resolution sensors like, for example, MODIS, MERIS, and MISR. Similarly the directional distribution of the incident radiation may be described with formulae like those of [Ste77] for clear sky radiation. Recent developments now also allow for a decoupling of the soil optical properties from the BRF simulations of 3-D plant canopies [PGW04]. As such, for each one of the 3-D canopy structure scenarios that are generated in the context of a LUT-based inversion approach, canopy reﬂectance simulations are required to account for the natural variability among leaf and bark optical properties, their changes as a function of wavelength, and diﬀerent positions of the sun in the sky. The Rayspread model has been developed speciﬁcally to perform accurate and computationally eﬀective BRF simulations over arbitrarily complex 3-D plant environments.

2 The Rayspread Model Monte Carlo (MC) ray-tracing techniques lend themselves nicely to the simulation of reﬂectance ﬁelds over complex canopy representations in that they stochastically sample the various interactions of the incident radiation within the scene until a certain level of stability is reached (for a recent overview of MC canopy reﬂectance models see [DLN00]). One of their main assets, however, is the capability to perform radiative transfer simulations in 3-D media of almost arbitrary complexity (subject to the available computer memory). On the other hand, such MC ray-tracing models are – in their simplest form – computationally demanding, which either leads to their parallelisation and integration into multi-processor environments [GV96], or else, to the implementation of variance reduction techniques [TG98] that speed up the simulations. The Rayspread model aims at combining both of these approaches. 2.1 Description of the Model The Rayspread model, which is an extension of the virtual laboratory of [GV98], samples all relevant radiative processes within a 3-D scene by following individual primary rays from their energy source, through all relevant interactions, until an absorption or exit event occurs. In addition, a variance reduction method known as “photon spreading” [TG98] has been implemented to achieve signiﬁcantly faster simulations of BRF ﬁelds. This local estimator technique implies the “spreading” of secondary rays towards a series of detectors from every physical interaction point in the path of the primary ray. Yet, Rayspread maintains backward compatibility with the model syntax of [GV98], to allow for the computation of radiative quantities other than BRFs (i.e., transmission, absorption, albedo, etc.), and relies on the Message Passing Interface (MPI) as a communicating layer within a distributed memory, parallel processor architecture. In its current form, the Rayspread model assumes that 1) light propagation can be described entirely

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with geometrical optics, 2) incident monochromatic radiation can be simulated with a ﬁnite number of mutually non-interacting rays, and 3) whenever a ray-matter interaction occurs along the trajectory of a primary ray, the ray is scattered in one and only one direction under elastic scattering conditions. To set up an experiment with this model, the structural and optical properties of the medium of interest have to be deﬁned prior to the computation of the ray trajectories. A set of 12 geometric primitives (e.g., disc, cone, sphere, cylinder, ellipse), may be combined using Constructive Solid Geometry (CSG) techniques to produce objects of great complexity. Every object within the bounding volume of the scene (known as the world object) is characterized by its position with respect to the world cartesian coordinate system, its spatial extension and an interaction model that speciﬁes the object’s scattering properties if a ray intersects its outer envelope or is to be propagated within the spatially homogeneous media deﬁned inside its (closed) interior. Figure 2 shows some examples of structurally homogeneous (top) and heterogeneous (bottom) vegetation canopies, that are depicted both in a deterministic manner (right), where every primitive is explicitly described, and in a stochastic manner (left), where the statistical foliage properties are represented by volumes of spatially homogeneous media with uniform characteristics. A light source is deﬁned in terms of its location, extent, intensity and directionality. Multiple light sources may be combined to simulate complex illumination conditions. Last but not least, a series of virtual ﬁlters (and logical combinations thereof) can be applied to BRF measurements such that only certain scattering orders, or, object-speciﬁc physical interactions may contribute towards the radiance counter of a given detector. Primary Ray Generation and Event Tracking Upon emission from a light source, a primary ray k is tagged by a set of parameters describing that event: kj = {rj ; Ωj ; Tj ; Ol }, where rj is a position vector indicating the origin of the ray, Ωj is a unit vector describing the travelling direction, Tj marks the type of j-th interaction (emission if j = 0, absorption, reﬂection or transmission if j ≥ 1) and Ol identiﬁes the l-th object within the scene where the interaction occurs. For every new interaction between the primary ray and the scene, another event is added to the proﬁle of the ray path. In order to determine the position of the next point of intersection (rj+1 ) an optimized geometric-sorting algorithm, based on the uniform subdivision of the scene into smaller volumes called ‘voxels’, is applied. Ray Interaction at the Interface Between Two Media If a primary ray falls onto an open surface, or is scattered inside some medium, the type of interaction and the outgoing direction are both determined with respect to the local coordinate system ( ) at the point of the scattering event. Knowing the incoming direction of the ray (Ωj ) as well as

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the bidirectional reﬂectance and transmission distribution functions (from the interaction model of the object), it is possible to compute the hemi spherically integrated probabilities of reﬂection Pρ (Ωj ), transmission Pτ (Ωj ) and absorption Pα (Ωj ) = 1 − Pρ (Ωj ) − Pτ (Ωj ), e.g., [AM90]. The type of interaction Tj+1 is simulated by generating a uniform random number R over the interval [0, 1], such that a reﬂection event occurs if R < Pρ (Ωj ), a transmission if Pρ (Ωj ) ≤ R < Pρ (Ωj ) + Pτ (Ωj ) and an absorption event if Pρ (Ωj ) + Pτ (Ωj ) ≤ R. In the ﬁrst two cases the outgoing direction after reﬂection (transmission) is statistically simulated with respect to the scattering distribution function of the object, normalized by Pρ (Pτ ). This sampling scheme may either use the acceptance/rejection technique [Dev86], or, more direct approaches if available. In any case, scattering directions are ﬁrst determined with respect to the local coordinate system – where the surface normal always points to (0, 0, 1) – and then in the coordinate system of the world object (Ωj+1 ). The soil interface is treated in exactly the same manner, allowing for the deﬁnition of various scattering distribution functions, e.g., Lambertian, Gaussian, Torrance-Sparrow [TS67], or the RPV formulation [RPV93]. Ray Propagation within a Scattering Medium Ray propagation within a spatially homogeneous medium containing a stochastic representation of ﬁnite-sized or point-like scatterers is not only controlled by the simulations of the interaction types and scattering angles as before, but also by the reckoning of the actual distances between successive interactions. Rays are expected to travel over distances of the order of the mean free path of photons in the real world, with a probability of interaction with the pseudo-turbid medium that increases exponentially with the distance travelled since the last interaction [SG69]. The actual path length da (Ωj ) that a ray may travel since its last point of interaction rj inside a pseudo-turbid medium of optical depth τ (Ωj ), is simulated using a uniform random number R (bound in the interval [0, 1]) such that da (Ωj ) = − ln R / τ (Ωj ). In the case of ﬁnite size scatterers, the formulation of the optical depth τ (Ωj ) makes use of the geometric-statistical hot spot model of [VPD90] (see Sect. 1.2) which requires some assumptions as to the shape and organisation of the scatterers within the medium. Nevertheless, a physical interaction will occur only if the next position of interaction rj+1 = rj +Ωj da is actually contained within the medium. Otherwise the ray will be placed just outside the current medium, and (accounting for its direction of propagation Ωj ) the probability of having a physical interaction in this new medium will be computed in a likewise manner. Secondary Rays and BRF Measurements At every physical interaction j that characterises the random walk process by which a primary ray samples a scene, Rayspread traces secondary rays

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towards a pre-deﬁned number of detectors. Given the direction Ωj of the primary ray k prior to its collision with object Ol at location rj , the probability that a secondary ray s actually reaches a detector d (located at rd ) without any physical interactions in between is given by: Ps (Ωj , rj ; Ωd , rd ) = P (Ol ; Ωj → Ωd ) P (rj → rd ) where P (Ol ; Ωj → Ωd ) is the probability of being scattered by object Ol into the direction of a detector Ωd , and P (rj → rd ) accounts for the probability of reaching that detector without further physical interactions. P (Ol ; Ωj → Ωd ) is computed using the optical properties and scattering distribution functions of the object where the interception of the primary ray occurred. If a detector is located at inﬁnity, then all secondary rays travel along identical directions Ωd irrespective of their origin in the scene. P (rj → rd ), on the other hand is evaluated using the geometric-sorting algorithm (mentioned previously) to verify if there are any physical interactions with other objects along the direction of propagation of the secondary ray. If such an event takes place, then the probability that the secondary ray reaches the detector is set to zero. Alternatively, if the secondary ray enters a spatially homogeneous medium containing a stochastic representation of oriented scatterers, then P (rj → rd ) has to be modulated by the probability of traversing that medium without interception since entering and exiting a medium does not constitute a physical interaction per se [TG98]. In the case of a detector d located at inﬁnity its radiant ﬂux density counter, Rd is then updated for all contributions due to secondary rays reaching it from physical interactions (of primary rays) which satisfy the speciﬁed measurement conditions. In the case where all physical interactions of the primary rays are accepted by the measure: Rd =

Nk Nin

Ps (Ωj , rj ; Ωd , rd )

k=1 j=1

where Nk is the number of physical interactions occurring within the trajectory of primary ray k, and Nin is the total number of primary rays entering the world object’s top area Ar . The resulting BRF value for that detector is then obtained by: BRFd =

Φd /Ar Rd Φlambert /Ar Nin cos θd

where the approximate sign arises from the fact that the contributions due to secondary rays with P (Ωj , rj ; Ωd , rd ) ≤ 10−9 are currently ignored. Obviously, it is possible to perform local BRF measurements within a larger 3-D scene, by restricting the number of primary and secondary rays that may participate to only those that enter and exit the scene, respectively, via a predeﬁned reference area Ar located at the top of the world object. These reference areas can be of square, circular or elliptical shapes.

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Lateral Boundary Conditions Within Rayspread, the computations associated with the physical interactions of a primary ray are pursued until 1) an absorption event occurs, 2) the primary ray exits the world object upward, or 3) a user-speciﬁed maximum event number is reached. Contrary to a pure MC scheme, primary rays are used only to sample the scene’s geometric and scattering properties and thus do not contribute to any BRF measurements. To simulate the horizontal exchange of primary (and secondary) rays with neighbouring areas cyclic boundary conditions may be applied, that is, laterally exiting rays are re-ingested into the world object through the side opposite the one of their departure without aﬀecting their direction of travel. This assumes that the simulated scene is part of a larger and similarly structured domain. 2.2 Benchmarking of the Model The 3-D virtual laboratory of [GV98] was evaluated both against actual laboratory BRF measurements and with respect to a panoply of other RT models in the context of the RAdiation transfer Model Intercomparison (RAMI) exercise [PWT04]. Among others, RAMI compares the total, single and multiple collided BRF components generated by its participating RT models over a variety of structurally homogeneous and heterogeneous canopy representations, similar to those presented in Fig. 2. The results displayed on the web site http://rami-benchmark.jrc.it/ indicate that the Raytran model of [GV98] is in close agreement with other 3-D RT models in the heterogeneous cases, and was deemed accurate enough to contribute towards the “most credible” BRF solutions for the various homogeneous test cases. Given these credentials, and since the Rayspread model utilizes many of the geometrical routines and radiation transfer formulations implemented within the model of [GV98], a direct comparison of their various BRF components is a ﬁrst step in the validation process of the Rayspread model. The two panels of Fig. 3 show that the implementation of the photon spreading acceleration technique did not introduce any bias between the Raytran and Rayspread generated BRF values for the discrete homogeneous (left) and heterogeneous (right) test cases of the second phase of RAMI. Included in these graphs are BRF simulations along diﬀerent planes of observation, a wide range of illumination conditions as well as spectral surface properties. Since these simulations do not permit an evaluation of the Rayspread model in absolute terms, the black canopy BRF component for a vertically and spatially homogeneous turbid medium canopy with spherical LND and Lambertian soil was generated (using diﬀerent numbers of primary rays), and subsequently compared against the analytical true solution, which is: −0.5 LAI −0.5 LAI αsoil exp BRF (θv ) = exp |µ0 | |µv |

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RAMI Phase 2 HOM DIS

RAMI Phase 2 HET DIS 0.004

D = -5.47 10-5

0.005

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D = (BRFRaytran - BRFRayspread)

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-0.010 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 BRFRayspread

-0.004 0.0

0.1

0.2 0.3 0.4 BRFRayspread

0.5

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Fig. 3. The diﬀerence, D between BRF simulations of the Raytran and Rayspread models over the entire set of homogeneous (left panel) and heterogeneous “ﬂoating spheres” (right panel) test cases of the RAMI (phase 2) exercise. The mean deviation between the two model outputs, D is also indicated.

where µ = cos θ, LAI = Λ(z) H is the leaf area index, αsoil is the albedo of the soil, and Beer-Bouger’s exponential law is applied to both the downward and the upward transmission probabilities within a canopy with G(θ) = 0.5. The left panel of Fig. 4 shows how the normalized absolute BRF diﬀerence , when averaged over 119 view directions (θv ≤ 75◦ ) and multiple model runs with diﬀerent starting seeds, decreases as a function of the number of primary rays. To assess the eﬃciency of the speed-up, results are shown for two contrasting scenarios: 1) a sparse canopy with bright soil and a small solar zenith angle (thick lines), and 2) a very dense canopy with dark soil and a large solar zenith angle (thin lines). As can be expected for MC √ simulations, the results of both cases improve as a power law: ∝ 1/ Nin for both Rayspread (solid lines) and Raytran (dashed lines) provided that the latter receives suﬃcient numbers of primary rays into its angular BRF bins. This is particularly visible in the dense canopy scenario where the true BRF(θv = 0) ≈ 0.000168, and it takes Raytran more than 106 primary rays to guarantee suﬃcient BRF contributions. For case 1, any primary ray number in the range 103 to 107 yields an improvement in the accuracy of Rayspread simulations that is constant with respect to that of Raytran and given by the oﬀset between the two (thick) lines: rayspread raytran / 50. By the same token, in case 1 the Raytran model will need about 2500 times the number of

104

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LAI=5.0 θ0 = 50o

αsoil = 0.10

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104 105 106 107 Number of primary rays

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Homogeneous turbid canopy, LAI=3, red 102

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Heterogeneous discrete canopy; red 103

104 105 106 107 Number of primary rays

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Fig. 4. The mean value of the normalized absolute diﬀerence, that exists in between a set of reference BRF values and simulations by the Rayspread (solid lines) and Raytran (dashed line) models as a function of the number of primary rays used by these models. Left panel: results for the black canopy BRF contribution over two homogeneous turbid medium canopy scenarios (thick and thin lines) where the reference BRFs are known theoretically. Right panel: results for the total BRF over the homogeneous (thick) and heterogeneous “ﬂoating spheres” (thin) test cases of RAMI. The reference BRFs were given by a Rayspread run with 109 primary rays

primary rays of the Rayspread model to achieve a similar accuracy level (for case 2 and N ≥ 106 the advantage of the Rayspread model is even larger). It can be seen, for example, that the Rayspread model lies within 1% of the true solution (of case 1) after only 10,000 primary rays, whereas Raytran requires more than 107 rays to achieve a similar BRF accuracy. The right panel of Fig. 4 displays the same uncertainty measure, i.e., the mean value of the normalized absolute diﬀerence , existing between a reference set of total BRFs and the corresponding simulations carried out by the Rayspread (solid lines) and Raytran (dashed lines) models over the homogeneous (thick) and heterogeneous “ﬂoating spheres” (thin) test cases of RAMI (these data were also part of Fig. 3). Based on the previous discussion the reference BRFs were taken from a Rayspread reference run with 109 primary rays. Again, the slope of the various lines is close to –0.5, and the oﬀset between Rayspread and Raytran indicates that the latter will need about 4500 times the number of primary rays of the former to achieve a similar accuracy level . Figure 5 shows the quasi-linear speed-up that can be achieved by running Rayspread in parallel mode on a small PC/Linux cluster. [GV96] found that this linearity was maintained for at least 100 processors in the case of the Raytran model. When run on a single processor, it took ∼17 minutes for a 2.4 GHz machine (2 Gb RAM) to compute the total

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10

Speed Up = time (1 proc) / time (N proc)

Speed Up

8

6

4

RAMI "Floating Spheres" scene 750,000 scatterers

2 2

4 6 8 Number of processors, N

10

Fig. 5. Speed-up in computation time achieved when running Rayspread on increasing numbers of processors using the Message Passing Interface (MPI) as a communication layer between the various nodes

BRF using 100,000 primary rays in the near-infrared over the heterogeneous “ﬂoating spheres” scenario with 750,000 individual scatterers. 2.3 Case Study: Impact of Woody Structure on BRF Pattern The woody structure of vegetation canopies is in general disregarded in canopy reﬂectance simulations, despite the fact that land surface-atmosphere interaction models do acknowledge their eﬀect on surface albedo, canopy absorption, evaporation and run-oﬀ [ZSD02]. The central graph of Fig. 6 shows variations of the total BRF, in the red spectral band and along the principal plane, obtained over 3-D canopy representations where ﬁrst the branches within the crowns, and then also the trunks themselves were omitted. Shown are BRF simulations over a mixed coniferous – deciduous forest target in late autumn. Only the Scots Pine trees still feature their crowns, which were modelled to contain 1) a vertical LAI gradient, 2) a foliage free zone (of vertically varying width) that surrounds the main stem, and 3) ﬁve branches per whorl (as well as multiple second order branches) [SKK94] – as displayed in the left panel of Fig. 6. All necessary spectral and structural tree characteristics were compiled from the available scientiﬁc literature [WPG03]. Assuming

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vertical LAI profile

0.20

0.15

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0.381 0.303

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Bidirectional Reflectance Factor Bidirectional Reflectance Factor

Foliage−free volume

Branching levels

Red

Only crowns

0.30

Full canopy No branches Only crowns

projected crown area

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40

60

Fig. 6. The middle panel shows the total BRF simulated in the principal plane for the three forest representations depicted in the right panel: Full tree representation – as shown in the left panel – (solid line), no branches within crowns (dashed line), and no woody elements in canopy (dotted line). Also indicated is the canopy absorption in each case. Note that the woody structures were brightened for better contrast in the right hand colour plates

Lambertian scattering properties, the reﬂectance (transmittance) values for the foliage was set to 0.054 (0.014), whilst woody elements had a reﬂectance of 0.142, and αsoil = 0.125. The density of trees with (without) crowns was 106 (331) stem per hectare and their height was 24.0 (19.8) m, resulting in an LAI of 1.50 for the entire scene (250 × 250 m2 ). As expected, the presence or absence of woody elements does not aﬀect the overall width of the hot spot which is primarily governed by the size (and shape) of the tree crowns. On the other hand, it was found that the omission of woody elements in 3-D canopy representations reduced the canopy absorption (in this particular scenario by 27.4%), and increased the directional hemispherical albedo values of the canopy. More speciﬁcally, canopy representations without the (mainly horizontal) branch structures lead to increases in the surface leaving radiation ﬁeld at predominantly small θv values, whereas the omission of the (mainly vertical) tree trunks showed its largest impact at increased θv values. In the latter case, BRF values in the forward scattering direction (positive θv ) were mostly aﬀected.

3 Conclusion A concise overview of the impact of vegetation structure on the radiative transfer within plant canopies was given. Furthermore, in the context of inverting 3-D RT models against satellite observations the need for speedy, yet

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accurate and reliable, RT simulations over realistic 3-D canopy architectures have been highlighted. To this eﬀect, the Monte Carlo ray-tracing model Rayspread makes use of a variance reduction technique, known as photon spreading, and the capability to operate in a multi-processor environment. Veriﬁcation and benchmarking tests of the Rayspread model, as well as, a small study regarding the impact that 3-D woody structures may have with respect to the surface leaving BRF ﬁeld are also provided.

References [AM90]

[BWG04]

[Bun78]

[Cha60] [CRH03]

[DM04]

[Dev86] [DLN00]

[Ger88]

[GPV97]

[GS83]

Antyufeev, V. S., Marshak, A. L.: Monte Carlo method and transport equation in plant canopies. Remote Sensing of Environment, 31, 183–191, (1990) Baldocchi, D. D., Wilson, K. B., Gu L.: Inﬂuences of structural and functional complexity on carbon, water and energy ﬂuxes of temperate broadleaved deciduous forest. Tree Physiology, 22, 1065-1077, (2002) Bunnik, N. J. J.: The Multispectral Reﬂectance of Shortwave Radiation of Agricultural Crops in Relation With Their Morphological and Optical Properties. Mededelingen Landbouwhogeschool, Wageningen, The Netherlands (1978) Chandrasekhar, S.: Radiative Transfer, Dover Publications, Inc., New York, (1960) Chopping, M. J., Rango, A., Havstad, K. M., Schiebe, F. R., Ritchie, J. C., Schmugge, T. J., French, A. N., Su, L., McKee, L., Davis, R.: Canopy attributes of desert grasslands and transition communities derived from multi-angular airborne imagery. Remote Sensing Environment, 85, 339– 354, (2003) Davis, A. B., Marshak, A.: Photon propagation in heterogeneous optical media with spatial correlations: Enhanced mean free-paths and widerthan-exponential free-path distributions. Journal of Quantitative Spectroscopy and Radiative Trannsfer, 84, 3–34, (2004) Devroye, L.: Non-Uniform Random Variate Generation. Springer-Verlag, New York (1986) Disney, M. I., Lewis, P., North, P. R. J.: Monte Carlo Raytracing in Optical Canopy Reﬂectance Modelling, Remote Sensing Review, 18, 163– 196, (2000) Gerstl, S. A. W.: Angular Reﬂectance Signature of the Canopy Hotspot in the Optical Regime, In: The 4th International Colloquium On Spectral Signatures of Objects in Remote Sensing, Aussois, France, ESA report SP-287, (1988) Gobron, N., Pinty, B., Verstraete, M. M., Govaerts, Y.: A Semi-Discrete Model for the Scattering of Light by Vegetation, Journal of Geophysical Research, 102, 9431–9446, (1997) Goel, N. S., Strebel, D. E.: Inversion of Vegetation Canopy Reﬂectance Models for Estimating Agronomic Variables. I. Problem Deﬁnition and Initial Results Using Suits Model. Remote Sensing of Environment, 13, 487–507, (1983)

The Rayspread Canopy Reﬂectance Model [GS84]

[GQW97]

[GV96]

[GV98]

[Hap81] [KKP00]

[Kuu91] [LG02]

[Mar89]

[Mer89] [NRH77]

[PV97]

[PWG02]

[PWT04]

229

Goel, N. S., Strebel, D. E.: Simple Beta Distribution Representation of Leaf Orientation in Vegetation Canopies. Agronomy Journal, 76, 800– 803, (1984) Goel, N. S., Qin, W., Wang, B.: On the estimation of leaf size and crown geometry for tree canopies from hotspot observations. Journal of Geophysical Research, 102, 29,543–29,554, (1997) Govaerts, Y., Verstraete, M. M.: Devellopment and parallelization of a Monte Carlo ray tracing code for radiative transfer modelling: Initial lessons from using the Cenju-3 machine. Speedup Journal, 10, 62–67, (1996) Govaerts, Y., Verstraete, M. M.: Raytran: A Monte Carlo Ray Tracing Model to Compute Light Scattering in Three-Dimensional Heterogeneous Media. IEEE Transactions on Geoscience and Remote Sensing, 36, 493– 505, (1998) Hapke, B.: Bidirectional Reﬂectance Spectroscopy. I. Theory. Journal of Geophysical Research, 86, 3039–3054, (1981) Kimes, D. S., Knyazikhin, Y., Privette, J. L., Abuelgasim, A. A., Gao, F.: Inversion Methods for Physically-Based Models, Remote Sensing Review, 18, 381–439, (2000) Kuusk, A.: he Hot Spot Eﬀect in Plant Canopy Reﬂectance, In: PhotonVegetation Interactions, Springer-Verlag, New York, (1991) Lovell, J. L., Graetz, R. D.: Analysis of POLDER-ADEOS data for the Australian continent: the relationship between BRDF and vegetation structure. International Journal of Remote Sensing, 23, 2767–2796, (2002) Marshak, A. L.: The Eﬀect of the Hot Spot on the Transport Equation in Plant Canopies. Journal of Quantitative Spectroscopy and Radiation Transfer, 42, 615–630, (1989) Meerkoetter, R.: Reﬂection functions for inhomogeneous land surfaces. Applied Optics, 29, 4192–4198, (1989) Nicodemus, F. E., Richmond, J. C., Hsia, J. J., Ginsberg, I. W., Limperis, T.: Geometrical Considerations and Nomenclature for Reﬂectance, National Bureau of Standards, U.S. Department of Commerce, Washington, DC, NBS Monograph, 160, (1977) Pinty, B., Verstraete, M. M.: Modeling the scattering of light by vegetation in optical remote sensing. Journal of the Atmospheric Sciences, 55, 137–150 (1997) Pinty, B., Widlowski, J.-L., Gobron, N. Verstraete, M. M.: Uniqueness of Multi-angular Information – Part 1: A Surface Heterogeneity Indicator from MISR, IEEE Transactions on Geoscience and Remote Sensing, 40, 1560-1573, (2002) Pinty, B., Widlowski, J.-L., Taberner, M., Gobron, N., Verstraete, M. M., Disney, M., Gascon, F., Gastellu, J.-P., Jiang, L., Kuusk, A., Lewis, P., Li, X., Ni-Meister, W., Nilson, T., North, P., Qin, W., Su, L., Tang, R., Thompson, R., Verhoef, W., Wang, H., Wang, J., Yan, G., Zang, H.: The RAdiation transfer Model Intercomparison (RAMI) Exercise: Results from the Second Phase, Journal of Geophysical Research, 109, D06210, doi:10.1029/2004JD004252

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[PGW04] Pinty, B., Gobron, N., Widlowski, J.-L., Lavergne, T., Verstraete, M. M.: Synergy between 1-D and 3-D radiation transfer models to retrieve vegetation canopy properties from remote sensing data. Journal of Geophysical Research, 109, D21205, doi:10.1029/2004JD005214, (2004) [QX94] Qin, W., Xiang, Y.: On the hotspot eﬀect of leaf canopies: Modeling study and inﬂuence of leaf shape. Remote Sensing of the Environment, 50, 95–106, (1994) [RPV93] Rahman, H., Pinty, B., Verstraete, M. M.: Coupled Surface-Atmosphere Reﬂectance (CSAR) Model. 1. Model Description and Inversion on Synthetic Data. Journal of Geophysical Research, 98, 20,779–20,789, (1993) [Ros81] Ross, J.: The Radiation Regime and Architecture of Plant Stands, W. Junk, Boston (1981) [SM88] Shultis, J. K., Myneni, R. B.: Radiative Transfer in Vegetation Canopies With Anisotropic Scattering. Journal of Quantitative Spectroscopy and Radiation Transfer, 39, 115–129, (1988) [SG69] Spanier, J., Gelbard, E. M.: Monte Carlo principles and Neutron Transport Problems. Addison-Wesley, Reading, (1969) [SKK94] Stenberg, P., Kuuluvainen, T., Kellomaeki, T., Grace, J. C., Jokela, E. J., Gholz, H. L.: Crown structure, light interception and productivity of pine trees and stands, In: A comparative analysis of pine forest productivity, Ecological Bulletins (43), Copenhagen, (1994) [Ste77] Steven, M. D.: Standard distributions of clear sky radiance. Quarterly Journal of the Royal meteorological Society, 103, 457–465. [SGR85] Strebel, D. E., Goel, N. S., Ranson, K. J.: Two-dimensional Leaf Orientation Distributions. IEEE Transactions on Geoscience and Remote Sensing, 23, 640–647, (1985) [TG98] Thompson, R. L., Goel, N. S.: Two Models for Rapidly Calculating Bidirectional Reﬂectance: Photon Spread (PS) Model and Statistical Photon Spread (SPS) Model, Remote Sensing Reviews, 16, 157–207, (1998) [TS67] Torrance, K. E., Sparrow, E. M.: Theory for oﬀ-specular reﬂection from roughened surfaces. Journal of the Optical Society of America, 57, 916– 925, (1967) [Ver87] Verstraete, Michel M.: Radiation Transfer in Plant Canopies: Transmission of Direct Solar Radiation and the Role of Leaf Orientation. Journal of Geophysical Research, 92, 10,985–10,995, (1987) [VPD90] Verstraete, M. M., Pinty, B., Dickinson, R. E.: A Physical Model of the Bidirectional Reﬂectance of Vegetation Canopies. 1. Theory. Journal of Geophysical Research, 95, 10,985–10,995 [VPM96] Verstraete, M. M., Pinty, B., Myneni, R. B.: Potential and Limitations of Information Extraction on the Terrestrial Biosphere From Satellite Remote Sensing, Remote Sensing of Environment, 58, 201–214, (1996) [VB86] Vogelmann, T. C., Bjorn, L. O.: Plants as light traps. Physiol. plant. 68, 704–708, (1986) [WPG01] Widlowski, J-L., Pinty, B., Gobron, N., Verstraete, M. M., Davies, A. B.: Characterization of Surface Heterogeneity Detected at the MISR/TERRA Subpixel Scale, Geophysical Research Letters, 28, 4639– 4642, (2001) [WPG03] Widlowski, J-L., Verstraete, M. M., Pinty, B., Gobron, N.: Allometric Relationships of Selected European Tree Species, EC Joint Research Centre, Technical Report EUR 20855 EN, Ispra, Italy, (2003)

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[WPG04] Widlowski, J-L., Pinty, B., Gobron, N., Verstraete, M. M., Diner, D. J., Davies, A. B.: Canopy Structure Parameters Derived from Multi-angular Remote Sensing Data for Terrestrial Carbon Studies, Climatic Change, 67, 403–415 , (2004) [ZSD02] Zeng, X., Shaikh, M., Dai, Y., Dickinson, R. E., Myneni, R. B.: Coupling of the Common Land Model to the NCAR Community Climate Model, Journal of Climate, 15, 1832-1854, (2003)

Part III

High Energy Density Physics

Use of the Space Adaptive Algorithm to Solve 2D Problems of Photon Transport and Interaction with Medium A. V. Alekseyev, R. M. Shagaliev, I. M. Belyakov, A. V. Gichuk, V. V. Evdokimov, A. N. Moskvin, A. A. Nuzhdin, N. P. Pleteneva, and T. V. Shemyakina

1 Introduction Numerical simulation of multidimensional particle transport processes is among the most diﬃcult problems in applied mathematics with high computational burden. Deterministic methods are widely used at present time for solving transport equations numerically. Further development of such methods opens a prospect for simulating various physical processes of particle and energy transport in more realistic assumptions and with more accurate and profound consideration of details and speciﬁc features of the particular problems. One of the main diﬃculties of solving a multidimensional transport equation numerically is that the number of variables required for adequate simulation of a given system may be large enough. In particular, a 2D timedependent transport equation should be solved in 6D phase space and a 3D equation – in 7D phase space [1]. When developing the particular deterministic numerical methods for multidimensional transport equations, stringent requirements of memory and cost-eﬃciency of the numerical methods for solving large-size grid equations are imposed. Another problem of simulating a transport equation numerically consists in that it is often required to solve the equation in complex geometry with sub-regions essentially diﬀering in their optical properties. In so doing, the dependence of the transport equation factors describing the radiation/medium interaction on the transport equation solution is essentially nonlinear. However, introduction of a ﬁne spatial grid in the computational domain as a whole is not required, moreover, such grid signiﬁcantly increases the problem run time and the requirement of memory resources. The paper describes the results of eﬀorts on development and numerical studies of the space adaptive method for numerically solving multidimensional transport equations under the contract with LANL [2]. The space adaptive sub-grid method for solving a 2D time-dependent multiple-group transport equation is described. The idea of the adaptive method is that every time step some multiply connected, in general, sub-domain is determined by examining grid solutions in mathematical regions of the problem to be solved, where the numerical solution to the transport equation has to

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be found with a higher precision. Within the sub-domain above, the original (reference) grid cells are fragmented into smaller cells according to certain prescribed rules. The transport equation is approximated separately for each small cell to provide the required higher precision of numerical solution. The number of reference grid’s cells contained in the sub-domain above is considerably less, as a rule, than the total number of the reference grid’s cells. This also provides a relative cost-eﬃciency of computations using adaptive grids of such a kind.

2 Statement of a 2D Transport Equation The method is based on the kinetic multiple-group model describing the radiation transport processes using the classic divergent form of a 2D timedependent transport equation. The equations are as follows [1, 3]: 1 ∂εi χai χsi (0) Qi + L εi + χni εi = εip + ε + , c ∂t 2π 2π i 2π i = 1, . . . , i 1 is the group number, L εi = µ

(1)

1 ∂ " ∂εi 1 ∂ " + 1 − µ2 · sin ϕ · εi , (2) r · 1 − µ2 · cos ϕεi − ∂z r ∂r r ∂ϕ i1 i1 ∂E (0) = χai · εi ∆ωi − χai εip ∆ωi ∂t i=1 i=1

(3)

The equation has to be solved in the axially symmetric domain D = {(r, z) ∈ L}, where L is the cross section of the solid of revolution by a plane passing across Z-axis (Fig. 1).

R

0

Fig. 1. Transport problem geometry

Z

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The following table of symbols is used to write the equations: r, z

are cylindrical coordinates of a particle (Fig. 1);

Ω (µ, ϕ)

is a unit vector in the direction of the particle ﬂight (Fig. 1); µ = cos (θ) ;

θ

is an angle between vector Ω and axis of symmetry z (Fig. 1);

ϕ

is an angle between the projection of vector Ω to the plane passing through point (r, z) normally to Z-axis and the vector joining points (0, z) and (r, z) (Fig. 1);

→

→

→

−1 ≤ µ ≤ 1, εi = εi (r, z, µ, ϕ, ωi , t) ωi ∆ωi T = T (r, z) E = E(ρ, T ) χai = χai (ρ, T, ωi ) Qi = Qi (r, z, ωi ) χsi = χsi (ρ, T, ωi ) χni = χai + χsi εip = εip (T, ωi ) Qi = Qi (r, z, ωi )

0 ≤ ϕ ≤ π;

is the radiation intensity function (the desired function), is mean energy of photons in group i, is width of the interval with respect to energy variable ω, is temperature of medium, is internal energy, is absorption cross-section, is an independent source, is scattering cross-section, is full cross-section, is Planck function, is an independent source.

Photon absorption and scattering processes are taken into account. Processes of energy re-emission by a medium are simulated in approximation to local thermodynamic equilibrium (using Planck function). The boundary condition on the outer surface is speciﬁed in the form ε (t, r, z, µ, ϕ)(r,z)∈Γ = φ(t, rΓ , zΓ , µ, ϕ)

(4)

→∗ → → Ω n < 0. Here, Γ is generatrix of the solid of revolution; n is the outer normal to generatrix Γ; φ(t, rΓ , zΓ , µ, ϕ) is the speciﬁed function (a ﬂow of particles entering into the solid of revolution). Besides, the system of (1)–(4) is supplemented with initial conditions in time-dependent case. Note that only one-group transport equation will be considered in the subsequent description. Of course, the transport equation solution scheme in

with

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a one-group case will essentially correspond to the multiple-group one in all variables except for the energy variable.

3 Description of 2D Transport Equation Approximation Methods To approximate the transport equation in angular variables, DSn -type schemes [1, 3–5] are used. In space variables, the transport equation is approximated using the so-called “extended template” [6]. Consider region L covered with the quadrangular spatial grid. Introduce the grid values of function ε at nodes (εPl ), on edges (εPl,l+1 ) and at the centers of quadrangles (εP0 ) ( Fig. 2).

Fig. 2. 2D cell

The algorithm of solving the system of grid equations with the speciﬁed right-hand side of the transport equation is described below. For some ﬁxed value of parameter µm−1/2 , the grid equations are solved successively for each value of ϕ¯q,m (q = 1, 2, . . .) beginning from ϕ = π. Then, the system of balance equations is solved in combination with the additional equations, initial and boundary conditions for the chosen direction µm−1/2 , ϕ¯q−1/2,m−1/2 . If the right-hand side of the transport equation is known, the system can be reduced to the system with the triangle matrix according to the algorithm described in [1], i.e. it is solved using the sweep (point-to-point) computational algorithm.

4 Description of the Space Adaptive Computational Algorithm for Transport Equation The idea of the multiple-grid method considered in the paper is that every time step some sub-domain (a multiply connected one, in general) is selected by examining the grid solutions in computational domains of the problem, where the numerical solution to the transport equation has to be found with a

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higher accuracy. Spatial cells of such sub-domain are partitioned into smaller cells according to the certain rules. In so doing, there emerges a possibility to reﬁne spatial cells belonging to that space sub-domain only, where a higher-accuracy numerical solution is desired. The set of the reference spatial grid cells to be reﬁned is variable. Note the main features of the multiple-grid method under consideration for solving 2D transport problems. (1) During each time step, some sub-domain (subset of spatial cells), which is a multiply connected one, in general, is selected with regard to the particular features of the problem stated by examining grid solutions in computational domains of the problem, where the numerical solution to the X-ray transport equation has to be found with a higher accuracy. The analysis of gradients of the grid solution to the transport equation from the previous time step is performed to estimate the set of cells. (2) Each space cell of the selected sub-domain is fragmented into smaller cells according to the prescribed rules. One and the same reference cell can be fragmented into various numbers of smaller cells, or it can be not fragmented at all. The transport equation is approximated separately in each of the smaller cells to provide a higher-accuracy numerical solution. (3) The system of grid radiation transport equations is solved numerically using the implicit sweep (point-to-point) computational scheme. The order of resolving the small cells comprising the reference cell is determined by the cell exposure. The numerical solution to the transport equation at the reference cell’s central point can be found using the obtained numerical solutions to the transport equation at smaller cells. In so doing, the scheme conservativeness is preserved, i.e. the number of photons in the reference cell remains equal to the total number of photons in all of the smaller cells within the reference one. The energy equation is also solved in smaller cells. The adaptive method under consideration uses the following algorithm of subdividing spatial cells into smaller ones (Fig. 3). Each of the four edges of the reference cell is subdivided into equal segments. The opposite edges have the same number of segments. The following main ideas underlie the development of the adaptive method for solving 2D transport equations: • Each cell of a reference spatial grid can be partitioned into smaller cells of an adaptive reﬁned grid (adaptive cells). • The adaptively reﬁned grid is built by partitioning each space direction into 2N equal intervals, where N is the adaptive grid level. • The adaptive grid level in rows and columns of a cell may change with transition from a previous step to the next one.

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R 4 1

2 3

Z Fig. 3. The reference cell fragmentation geometry

• The adaptive grid is built at the beginning of a time step basing on examination of the spatial distribution of the solution function on the reference spatial grid obtained at the previous time step. • The geometric parameters of the adaptive grid are calculated and the solution functions are interpolated from the older adaptive grid to the new one. • The order of resolving space cells during sweep (point-to-point) computations using the transport equation solution module is determined on the reference grid. If a cell is adaptively reﬁned, the subsystem of equations corresponding to the transport equation approximation on the adaptive grid of the given cell is solved. • All the solution functions are stored in the special dynamic data structure on the adaptive grid, as well as integrated onto the reference grid and stored in standard arrays. As it was mentioned above, a space adaptive grid can vary dynamically (it can be either reﬁned or enlarged) during time steps. Since the values of a set of functions are stored and calculated in the adaptive grid cells, there occurs a need in re-interpolating the values of the corresponding grid functions, which can be determined both at the centers of space cells and on their edges, with each change of the adaptive grid.

5 Results of Computational Investigations of the Adaptive Method Performance 5.1 Problem with Analytical Solution The task is to calculate the process of radiation transport and interaction with plane 1D medium [7]. Computations for the 1D problem above by a

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R 1000.1

1000.0

0

0.5

Z

Fig. 4. The system geometry

2D code using the transport equation written in cylindrical coordinates were carried out for the following geometry (Fig. 4). The incoming radiation ﬂux corresponding to ε(0, µ, t) =

ct , µ > 0, C1 = −0.85903205 . . . c + (cC1 µ + 1)

where ε(z, µ, t) is the radiation intensity multiplied by π, is speciﬁed at the left end {Z = 0, 1000 ≤ R ≤ 1000.1} At the right end {Z = 0.5, 1000 ≤ R ≤ 1000.1}, the incoming radiation ﬂux is set equal to zero. On the upper {0 ≤ Z ≤ 0.5, R = 1000.1} and lower {0 ≤ Z ≤ 0.5, R = 1000} lateral surfaces the boundary condition “mirror reﬂection” is speciﬁed. As it was mentioned above, the dependences of energy on temperature and absorption cross-section are considered in the form E = 2058 · T 4 , χa = 1 2058·T 4 , respectively. The next ﬁgure (Fig. 5) shows us spatial grids and the distribution in space of the material temperature values at several times for the reference grid of 25 columns and with maximum partition into 4 cells. The space grid concentration is seen near the radiation wave front only, i.e. in the area of maximum solution gradient. Computations before the wave front and after it are carried out on the reference space grid, because there are no strong gradients in solution there. The diﬀerence between the levels of partitioning the neighboring cells doesn’t exceed 1. The next ﬁgure (Fig. 6) shows the results of computations with adaptivity in which the maximum partition of the reference space grid in all its cells results in 200 intervals in Z variable. These are computations 25(8) and 50(4). For comparison, the ﬁgure also shows the results of computations without adaptivity, with 25, 50, and 200 intervals in Z variable. It is seen that the results of computations with adaptivity appear to be very close to the one obtained using a ﬁne grid of 200 space cells.

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t=0.03

t=0.07

t=0.1 Fig. 5. Computation results for the reference grid NZ=25 at various times

Fig. 6. Results of comparative computations using various grids

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Fig. 7. Results of comparative computations for adaptive grids of a higher adaptability level

Figure 7 below gives the similar comparison for 400 intervals in Z variable. The material temperature curves, in this case, are given for adaptive computations 25(16) and 50(8) with maximum partition of the reference cells into 16 and 8 adaptive cells, respectively. Though the spatial grid reﬁnement, in this case, takes place in the wave front passage area alone, computations using the adaptive code actually coincide with that one using the ﬁnest grid of 400 columns and with the analytical solution, as well. The result conﬁrms the adaptive algorithm operation validity. Comparison between the running times shows that the computation with adaptivity and with maximum partition of the reference grid into 8 cells requires less time (by a factor of 6.2) than the computation using the standard technique on the space grid of 200 columns, with the results of these computations being actually the same, and the computation with adaptivity and with maximum partition of the reference grid into 4 cells is 3.1 times faster. The similar comparison between the standard computations without adaptivity using the grid of 400 columns and computations with adaptivity and with partition of reference cells into 16 and 8 adaptive cells shows that the achieved saving of time is by a factor of 9 and 5.6, respectively, with the given precision preserved.

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5.2 Results of Computations for the Dense-Layer Problem There is a cylindrical layer (1.0 cm ≤ Z ≤ 1.3 cm; 100.0 cm ≤ R ≤ 102.0 cm) which consists of one physical region and is R = 100.0 cm distant from Z-axis of revolution and Z = 1.0 cm distant from R-axis (Fig. 8).

R 102.0

a

A T3

100.0

T=1

1.0

1.3

Z

Fig. 8. The system geometry

The benchmark is based on the modiﬁcation to the well-known threeregion Fleck benchmark [8]. Namely, the benchmark oﬀered below considers one of the Fleck’s regions (optically dense) with slightly changed dimensions. Besides, the benchmark oﬀered is solved in one-group “gray matter” approximation. The following parameters are speciﬁed for the given region: • coordinate 1.0 cm ≤ Z ≤ 1.3 cm, • coordinate 100.0 cm ≤ R ≤ 102.0 cm, • absorption cross-section χa = TA3 , where A = 50.890585, T is the material temperature, • no dissipation, • internal energy versus temperature is described by equation E = 0.81 ∗ T . The incoming unilateral radiation energy ﬂux isotropically distributed over the solid body and correspondent to the radiation temperature T = 1 KeV is speciﬁed on the lower surface (1.0 cm ≤ Z ≤ 1.3 cm, R = 100.0 cm). For the upper surface (1.0 cm ≤ Z ≤ 1.3 cm, R = 102.0 cm), the boundary condition “free surface” is speciﬁed (zero incoming ﬂux). The boundary condition “mirror reﬂection” is speciﬁed for lateral surfaces.

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The initial energy corresponds to temperature T = 0.0001 KeV. The initial radiation intensity is ε(t, z, r, µ, ϕ) |t=t0 = 0. Radiation propagation over the system is calculated. In view of onedimensional behavior of solution, the radiation temperature proﬁle along R-axis at ﬁnal time is considered to be the result of computations. • To study the eﬃciency of the space adaptive algorithm for solving the equation of radiation transport and interaction with medium, the results of computations using the conventional method (4, 8, and 9) and the results of two computations with adaptivity (6 and 7) were compared: • Computation 4. The space grid consists of 22 columns and 160 rows, time step is ∆t = 0.00002. The adaptive algorithm is not used. • Computation 6. The space grid consists of 22 columns and 40 rows. The adaptive algorithm of adaptability level 2 is used (maximum partition into 4 adaptive cells). • Computation 7. The space grid consists of 22 columns and 20 rows. The algorithm of adaptability level 3 is used (maximum partition into 8 adaptive cells). • Computation 8. The space grid consists of 10 columns and 20 rows. No adaptive algorithm is used. • Computation 9. The space grid consists of 10 columns and 40 rows. No adaptive algorithm is used. Figure 9 shows the results of computation 4 and computations 6–9. 0.8 0.7 0.6

, keV

0.5 0.4 0.3 0.2 0.1 0 101

101.05

101.1

101.15

101.2

101.25

101.3

101.35

R, sm Computation 4

Computation 6

Computation 7

Computation 8

Computation 9

Fig. 9. The radiation temperature proﬁle along R-axis at time t = 0.05

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Analysis of the results above shows that results of computations 6 and 7 with the use of the adaptive technique are almost the same, with a very insigniﬁcant deviation from the result obtained using the reﬁned grid (computation 4). Figure 10 shows the space grid and the radiation temperature near the radiation wave front for computations 6 and 7.

Fig. 10. The space grid along R-axis

Further, the eﬃciency of using the adaptive algorithm is discussed. The time of computation using the reﬁned grid with no adaptation (160 rows, computation 4) exceeds the time of computation on the grid of adaptability level 2 (40 rows of the reference grid, computation 6) by a factor of 3.97 and is 4.3 times more prolong as compared to the time of computation using the grid of adaptability level 3 (20 rows of the reference grid, computation 7), with the given precision of computations preserved. More detailed consideration of the computational process shows that this saving of computation time can be attributed to the fact that concentration of the space grid (owing to the use of the adaptive technique) takes place near the radiation wave front only. Computations for the remained sub-regions of

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the system are carried out using a more coarse reference grid. It means that in computation 6 with regard to adaptive cells the numerical solution process is actually carried out using the space grid of 45 rows, and in computation 7 the space grid of 35 rows is actually used. 5.3 Results of Computations for a 2D Problem A rectangular region of length 5 cm and height 1.2 cm is considered in 2D axial symmetry geometry (Fig. 11). The computational domain consists of two physical regions: the 1st physical region is a dense casing {0 ≤ Z ≤ 5, 1 ≤ R ≤ 1.2}, the 2nd region is transparent {0 ≤ Z ≤ 5, 0 ≤ R ≤ 1}. R 1.2 1.0 G3 1

0.0

G1 Physical region 1

G4

Physical region 2

G2

5.0

Z

Fig. 11. The system geometry

The material density in the system is ρ = 1. The material energy dependence on temperature is described by equation E = Cν T , where Cν = 0.81. Computations are carried out with consideration of photon absorption, with the photon absorption cross-section being speciﬁed by the formula: χa = TA3 . The optically dense (the 1st physical region) and the optically transparent (the 2nd physical region) regions are speciﬁed using various values of factor A. In the 1st region (optically dense) A = 50.89. In the 2nd region (optically transparent) A = 0.1374. The scattering cross-section was taken zero during computations (χs = 0). The initial temperature T at all points of the system was assumed to be 0.0001. The following boundary conditions for radiation are speciﬁed at the left end of the rectangular region (boundary G3): the boundary condition “mirror reﬂection” for the boundary section belonging to the dense casing and the incoming isotropic radiation ﬂux corresponding to temperature T = 1 for the boundary section belonging to the transparent region. Zero incoming radiation ﬂux is speciﬁed at the upper boundary and the right end.

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T

The result is that solution to the equation of photon transport and interaction with medium, for the given problem, has essentially 2D behavior. A grid of 10 rows (5 rows per each physical region) and 50 columns (NR = 10, NZ = 50) is taken as the reference grid in space variables. The adaptive partition level is taken so as to provide the maximum partition corresponding to the grids in space variables with partition NR = 40, NZ = 200 and NR = 80, NZ = 400. Computation results – the material temperature values along cross sections Z = 2 cm and Z = 3 cm at time t = 0.01 – are shown in Figs. 12, 13. 0.9 0.8 0.7 0.6 0.5

NR=10, NZ=50 0.4

NR=20, NZ=100

0.3

NR=40, NZ=200 NR=60, NZ=300

0.2

Adaptive, NR=10(4), NZ=50(4) 0.1

Adaptive, NR=10(8), NZ=50(8)

0 0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2 R

Fig. 12. The material temperature values along line Z = 2 cm,t = 0.01

In Fig. 12, the solution obtained on the grid NR = 10(8), NZ = 50(8) using the adaptive technique appears to be close enough to the result of computations on the grid NR = 40, NZ = 200 using the standard technique, the adaptive computation time is 2.8 times less. In Fig. 13, the solution obtained on the adaptive grid NR = 10(8), NZ = 50(8) is close to the result of computations using the ﬁnest space grid (NR = 60, NZ = 300), while the adaptive computation time is 4.8 times less. The results obtained for a lower adaptability level also demonstrate that computation on the grid NR = 10(4), NZ = 50(4) gives the solution close to solution obtained on the grid NR = 40, NZ = 200, while the saving of running time is 8.9 times. The same adaptive computation (NR=10(4), NZ=50(4)) provides a signiﬁcantly more accurate solution than the computation on the grid NR = 20,

T

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0.8 0.7 0.6 0.5 0.4

NR=10, NZ=50 NR=20, NZ=100

0.3

NR=40, NZ=200 NR=60, NZ=300

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Adaptive, NR=10(4), NZ=50(4) 0.1

Adaptive, NR=10(8), NZ=50(8)

0 0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2 R

Fig. 13. The material temperature values along line Z = 3 cm, t = 0.01

NZ = 100 (see, especially, Fig. 13), however, even in this case the adaptive computation is 1.4 times faster. As an illustration demonstrating the results of operation of the adaptive grid generation codes, Fig. 14 shows the distribution of the radiation temperature ﬁeld over the system at three times. Black lines correspond to the reference space grid, and white lines correspond to the adaptive cells. The maximum adaptability level in this computation is 2 (four adaptive cells comprise a reference one) in the both directions. It is clearly seen that the reference space grid is adaptively partitioned at the wave front only and the partitions are diﬀerent in various spatial directions. The results presented allow us to conclude, by the example of the 2D test problem, that the adaptive algorithm of reﬁning space grids demonstrates a noticeable increase in the eﬃciency of computations. 5.4 Statement of a Demo Problem and Results of Computations This section gives the description of the problem solved to demonstrate operation of algorithms using an adaptive grid. A cylindrical layer (0.0 cm ≤ Z ≤ 4.0 cm; 100.0 cm ≤ R ≤ 103.0 cm) with the speciﬁed medium features is considered. The problem is solved in one-group “gray matter” approximation. The following parameters are speciﬁed for the computational domain: • Coordinate Z: 0.0 cm ≤ Z ≤ 4.0 cm, • Coordinate R: 100.0 cm ≤ R ≤ 103.0 cm;

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t=0.004

t=0.006

t=0.008 Fig. 14. The adaptive reﬁnement of the space grid NR = 10(4), NZ = 50(4)

• Density of material: ρ = 1.0 g/cm3; • Absorption cross-section: χa = TA3 , where A = 50.890585, T is temperature of material; • No dissipation; • The dependence of internal energy on temperature is described as E = 0.81 ∗ T . The system geometry is shown in Fig. 15. The incoming unilateral radiation energy ﬂux isotropically distributed over the solid body and correspondent to radiation temperature T1 = 2 KeV is speciﬁed for the section of the left boundary (Z = 0; 101 ≤ R ≤ 102) (solid line in Fig. 15). The incoming unilateral radiation energy ﬂux isotropically distributed over the solid body and correspondent to radiation temperature T1 = 1.8 KeV is speciﬁed for the section of the upper boundary (R = 103; 2 ≤ Z ≤ 4.0). The boundary condition “free surface” (the incoming ﬂux equals 0) is speciﬁed for the remained boundaries. Therefore, the problem is essentially 2D and allows the adaptive technique capabilities to be clearly demonstrated.

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R T2 103

102 T1 101

100 0

2

4

Z

Fig. 15. The system geometry

The initial energy corresponds to temperature T = 0.0001 KeV. The initial radiation intensity is ε(t, z, r, µ, ϕ) |t=t0 = 0. The reference space grid consisted of 20 columns and 15 rows. Construction of adaptive grids of no more than the third order (maximum partition into 8 cells in both space directions) was allowed during the computation. Figures 16–18 show some fragments of the computation results at three times. The radiation temperature distribution and lines of the space grid cells are represented, with the reference space grid’s lines being shown in black color and the adaptive space grids lines – in white color. One can clearly see from the ﬁgures above that a more ﬁne space grid resultant from operation of the adaptive algorithms is generated on the radiation wave front only, i.e. in places of maximum solution gradients. The level of reﬁnement is determined by the gradient value. After the wave has passed and the appropriate parts of the system have been heated, the computation is carried out the reference grid again, because the solution gradient disappears.

6 Conclusion The comparative computation results for all the test problems described in the paper conﬁrm that the method of adaptively reﬁned grids is quite serviceable. The results of computations using the adaptive scheme with dynamically reﬁning a grid in areas of solution changes appear to be close enough to those obtained using the standard scheme on signiﬁcantly ﬁner space grids, or to the exact analytical solution.

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Fig. 16. The initial spatial distribution of the radiation temperature

Fig. 17. The spatial distribution of the radiation temperature at intermediate time

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Fig. 18. The spatial distribution of the radiation temperature at ﬁnal time

The use of the adaptive scheme allows signiﬁcant reduction of the time required for computation while providing the speciﬁed accuracy of numerical solution. Thus, in various test problems saving of running time in computations using the adaptive scheme was by a factor of 3 to 9 (depending on the mode of computation), as compared to computations using the standard scheme. The maximum saving of running time has been achieved for the 2D problem described with the adaptive algorithms used to reﬁne cells of the reference grid in two space directions simultaneously. This provides significant reduction of the total number of cells along with preservation of the reﬁned diﬀerence grid in the system’s sub-regions having the primary eﬀect, at a given time, on the grid solution accuracy. The results obtained allow us to conclude that the method developed for solving the 2D transport equation by adaptively choosing a space grid is eﬃcient enough. The use of the adaptive algorithms makes it possible to solve problems with essentially less running times, as compared to the standard scheme, though a comparable precision of solution results is achieved.

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References 1. Shagaliev R. M., Shumilin V. A., Alekseyev A. V., Belyakov I. M., Evdokimov V. V., Zvenigorodskaya O. A., Moskvin A. N., Pleteneva N. P., Fedotova L. P. Numerical simulation of and the methods for solving multidimensional particle and energy transport problems implemented in SATURN code package / VANT, Ser.: Math. Model. Phys. Process., 1999, Is. 4, p. 20–26 2. Shagaliev R. M., Alekseyev A. V., Belyakov I. M., Gichuk A. V., Evdokimov V. V., Moskvin A .N., Pleteneva N. P., Shumilin V. A. Numerical simulation of the 2D equation of radiation transport and interaction with medium using the space adaptive algorithm: Results of computational investigations. / Report for Phase 3 eﬀorts under Contract No.37713-000-02-35, Task 014 between LANL and RFNC-VNIIEF, 2003. 3. Alekseyev A. V., Mzhachikh S. V., Pleteneva N. P., Shagaliev R. M. Some method of approximating a 2D transport equation in angular variables / VANT, Ser.: Math. Model. Phys. Process., 1999, Is. 4, pp. 20–26. 4. Carlson B. G. The Numerical Theory of Neutron Transport // Methods in Computational Physics. -1963.- Vol. 1.- P. 9. 5. Bass A. P., Voloshchenko A. M., Germogenova T. A. Methods of Discrete Ordinate in Radiation Transport Problems.- Moscow, 1986 (Preprint/ M. V. Keldysh IAM of the USSR Academy of Sciences) 6. Pleteneva N. P., Shagaliev R. M. Approximation of the 2D transport equation on quadrangular and polygonal space grids using the extended-template diﬀerence scheme / VANT, Ser.: Math. Model. Phys. Process., Is. 3, 1989. 7. Shagaliev R. M., Alekseyev A. V., Belyakov I. M., Vlasova O. E., Gichuk A. V., Evdokimov V. V., Moskvin A. N., Pleteneva N. P., Shumilin V. A. Numerical simulation of the 2D equation of radiation transport and interaction with medium using the space adaptive algorithm: First computation results. / Report for Phase 2 eﬀorts under Contract No.37713-000-02-35, Task 014 between LANL and RFNC-VNIIEF, 2003. 8. Fleck J. A., Cummings J. D. An Implicit Monte Carlo Scheme for Calculating Time and Frequency Dependent Nonlinear Radiation Transport // J. of Comput. Phys.-1971.-V.8.-P. 313–342.

Accurate and Eﬃcient Radiation Transport in Optically Thick Media – by Means of the Symbolic Implicit Monte Carlo Method in the Diﬀerence Formulation∗ Abraham Sz˝ oke, Eugene D. Brooks III, Michael Scott McKinley, and Frank C. Daﬃn University of California, Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94550, USA [email protected], [email protected], [email protected], [email protected]

The equations of radiation transport for thermal photons are notoriously diﬃcult to solve in thick media without resorting to asymptotic approximations such as the diﬀusion limit. One source of this diﬃculty is that in thick, absorbing media thermal emission is almost completely balanced by strong absorption. In a previous publication [SB03], the photon transport equation was written in terms of the deviation of the speciﬁc intensity from the local equilibrium ﬁeld. We called the new form of the equations the diﬀerence formulation. The diﬀerence formulation is rigorously equivalent to the original transport equation. It is particularly advantageous in thick media, where the radiation ﬁeld approaches local equilibrium and the deviations from the Planck distribution are small. The diﬀerence formulation for photon transport also clariﬁes the diﬀusion limit. In this paper, the transport equation is solved by the Symbolic Implicit Monte Carlo (SIMC) method and a comparison is made between the standard formulation and the diﬀerence formulation. The SIMC method is easily adapted to the derivative source terms of the difference formulation, and a remarkable reduction in noise is obtained when the diﬀerence formulation is applied to problems involving thick media.

1 Introduction The transport of thermal photons in thick media is of suﬃcient importance that substantial eﬀort has been expended in developing both deterministic [Mih78] and Monte Carlo [FC71] methods for its solution. The diﬃculties associated with thick media have been severe enough to necessitate solving asymptotic approximations, such as the Eddington and diﬀusion approximations [PB83], instead of solving the full transport equation. ∗

This work was performed under the auspices of the U.S. Department of Energy by University of California, Lawrence Livermore National Laboratory under Contract W-7405-Eng-48. UCRL-PROC-210979

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Asymptotic methods do give the right solution to the transport equation in uniformly thick media, like stellar interiors. Nevertheless, in many problems of interest the medium is a mixture of thick and thin regions; moreover, some regions of interest may be thin for some radiation frequencies and thick for others. Although a lot of progress has been made in numerical calculations for such complicated systems using asymptotic methods, they suﬀer from several defects. One of them is an unphysical energy propagation rate when the method is applied outside its proper domain, e.g. to optically thin regions. This led to the development of ad hoc corrections such as ﬂux limiters [Pom82]. Another defect is that asymptotic methods are unable to satisfy correct boundary conditions. Time honored “ﬁxes” are the Marshak [Mrs47] and Mark [Mrk47] boundary conditions, but these incorrect boundary conditions distort ubiquitous boundary layers. More importantly, it is diﬃcult to estimate or measure the errors incurred by the approximations. Only an accurate solution of the transport equation is able to eliminate the above defects. Several hurdles have stood in the way of producing accurate Monte Carlo solutions of the transport equation in thick media. The ﬁrst one was overcome by the development of a Monte Carlo technique that is numerically stable and provides correct treatment of the stiﬀ coupling between the radiation and the material in the thick limit. Several authors have shown that the radiation matter coupling is properly treated in the Symbolic Implicit Monte Carlo method [Bro89,Nka91], producing a correct implicit solution of the radiation ﬁeld and the material temperature at the end of a time step [DL04], while eﬀective scattering techniques [FC71, CF73] possess a signiﬁcant deﬁciency in this regard. A second hurdle has been the very signiﬁcant noise problem, or the equivalent problem of computational eﬃciency, when Monte Carlo methods are pressed into service for thick systems. The energy is emitted in a zone uniformly, but only particles born within a few mean free paths of a zone boundary have any chance of contributing to the ﬂux across the boundary. Most of the emitted particles are absorbed within the same zone and serve only to compute the equilibrium values of the radiation intensity and temperature in that zone. This situation for the Monte Carlo method, as applied to thermal photon transport, has been a source of frustration for a long time. The local equilibrium value of the radiation intensity in the thick limit is, of course, the black body ﬁeld for the given local temperature. One would prefer not to waste a lot of processing power computing it. In an earlier work [SB03], a new formulation was proposed for the transport of thermal photons, referred to as the diﬀerence formulation. The main considerations of this paper will now be repeated. The natural way of deriving the transport equation is to follow the propagation of narrow beams of photons as they are emitted, propagate in vacuo, are scattered and, ﬁnally,

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are absorbed by matter [Cas00,MM84]. In transparent media the absorption, emission and scattering of photons is weak and the transport equation describes the overall propagation very well. Mathematically, the equations of propagation are hyperbolic partial diﬀerential equations and their numerical solution is relatively easy and stable in optically thin media. In optically thick media, however, the probability that a photon propagates in a straight line, unhindered, is very small; radiation transport is dominated by a large number of scattering, absorption and re-emission events. As a result, the solution of the transport equation in thick media is not straightforward. An important example is hot, dense matter with a high absorption coeﬃcient. It results in conditions of local thermodynamic equilibrium (LTE) and very strong emission of photons. The emitted photons, in turn, are quickly re-absorbed, maintaining the temperature of the medium. The net emission (or absorption) is then a small diﬀerence between two large terms. The process leads to stiﬀness of the transport equation: the material and radiation temperature come into equilibrium much faster than any excess energy is transported away. In any numerical method that uses explicit diﬀerencing to balance thermal emission with absorption, the stiﬀness can cause instability, as well as a signiﬁcant increase in noise for Monte Carlo methods. If the scattering coeﬃcient is high a photon does not propagate in a straight path. This poses a diﬃculty for methods that are highly dependent upon eﬃcient streaming of photons. The transformation to the diﬀerence formulation, proposed in [SB03], is achieved by considering the diﬀerence between the radiation ﬁeld and the local equilibrium ﬁeld at each point in the problem domain. The local equilibrium ﬁeld is a function of the matter temperature, and therefore a function of both space and time. It results in a transport equation that contains only quantities that are small when the system is thick. In particular, the large emission term and its (almost) compensating absorption term are replaced by a pure absorption term for the “diﬀerence ﬁeld”. The only sources for the diﬀerence ﬁeld come from the variation of the material temperature in space and time. Why is the diﬀerence formulation interesting? To summarize, the equations are written in terms of quantities that are “natural” in thick media. (The traditional formulation is written in terms of variables that are natural in thin media.) In hot, dense matter the terms describing the nearly equal emission and absorption of photons are eliminated and only the small, net transport terms appear in the equation. We expect that this change of variables will aid in its numerical solution: it will make it less stiﬀ, more numerically stable, and it will reduce the noise in Monte Carlo methods. In fact, preliminary results shown here conﬁrm our expectations. Derivation of the diﬀusive behavior of the transport equation in thick media is simpliﬁed and clariﬁed by the diﬀerence formulation. As the diﬀerence equation is able to satisfy the correct physical boundary conditions, we hope to ﬁnd a fast

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and accurate alternative to the radiation diﬀusion equation. Finally, the new formalism might lead to the development of new numerical methods.

2 Radiation Transport in LTE The radiation transport equations will be written down in both the traditional and the diﬀerence formulation – in order to introduce the notation and for completeness. 2.1 Traditional Formulation Radiation transport and its coupling to matter is described by the equations of radiation hydrodynamics. In their general form, they consist of the equations of hydrodynamics coupled to those of radiation transport and to the interaction of radiation with matter. Excellent treatises have been written by Pomraning [Pom73], Mihalas [MM84] and Castor [Cas00]. In this paper we deal only with a subset of those equations. They are the radiation transport equation, the material energy balance equation and the conservation equation for the sum of the radiation and material energy. Furthermore we assume local thermodynamic equilibrium (LTE) – i.e. that the material has a well deﬁned temperature – it emits radiation thermally. We also assume that the material is at rest or that it moves with constant velocity. In real hydrodynamic cases, where diﬀerent parts of the material move at diﬀerent velocities, the “co-moving frame transformation” has to be used and proper account has to be given to kinetic energy and hydrodynamic work [Cas00]. When the local acceleration of the material is signiﬁcant, general relativity has to be invoked [MA00]. Our equations are written in the rest frame of the material, assumed to be an inertial frame. Otherwise, the scattering terms would have a more complicated angle and frequency dependence. The transport equation describes the propagation of the radiation ﬁeld in terms of the speciﬁc intensity, I(x, t; ν, Ω), where x, t are the space and time variables, ν is the radiation frequency and Ω is a unit vector in the direction of propagation. 1 ∂I(x, t; ν, Ω) + Ω·∇I(x, t; ν, Ω) c ∂t = σa (ν, T (x, t))[B(ν, T (x, t)) − I(x, t; ν, Ω)] + Q(I) (1) B(ν, T ) is the thermal (Planck) distribution at the material temperature, T (x, t), and c is the speed of light. The absorption coeﬃcient, σa , and the scattering term, Q(I), will be deﬁned below. The speciﬁc intensity is related to the photon distribution function f (x, t; ν, Ω) by

Accurate and Eﬃcient Radiation Transport in Optically Thick Media

I(x, t; ν, Ω) = chνf (x, t; ν, Ω) ,

259

(2)

where hν is the photon energy. In (1), all the variables, I, σa , B are functions of the independent variables, x, t; ν, Ω and/or T (x, t). In the following, the independent variables will mostly be suppressed. The emission function and the absorption cross sections, corrected for stimulated emission, are −1 2hν 3 hν/kT e − 1 , c2 σa (ν, T ) = σa (ν, T ) 1 − e−hν/kT , B(ν, T ) =

with σa being the “ordinary” absorption coeﬃcient, per unit distance. The scattering terms are denoted by Q(I) ∞ ν c2 I(ν, Ω) dν dΩ σs (ν → ν, Ω·Ω )I(ν , Ω ) 1 + Q(I) = ν 2hν 3 4π 0 ∞ c2 I(ν , Ω ) dν dΩ σs (ν → ν , Ω·Ω )I(ν, Ω) 1 + − , 2hν 3 4π 0

(3) (4)

(5)

where the x, t; T dependence of σs has been suppressed. In LTE there are thermodynamic relations among the partial scattering cross sections in (5). These follow from the observation that, in complete thermal equilibrium, the radiation ﬁeld reduces to the black body spectrum no matter what the scattering cross sections are. See (30) below. The zeroth moment of the intensity gives the radiation energy density 1 ∞ dν dΩ I (6) Erad = c 0 4π and its ﬁrst moment is the radiation ﬂux vector ∞ Frad = dν dΩ Ω I . 0

(7)

4π

Interaction of radiation with matter is expressed by the conservation law ∞ ∞ ∂Emat = dν dΩ σa [I − B(ν, T )] − dν dΩ Q(I) + G , (8) ∂t 0 0 4π 4π where Emat is the energy per unit volume of the material and G is a volume source of energy. In the absence of hydrodynamic work terms or thermal conductivity, the total energy of the radiation ﬁeld and the material are conserved ∂(Emat + Erad ) + ∇·Frad = G . ∂t

(9)

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2.2 Thick Media We all have a common-sense concept of a thick medium; we attempt to clarify it here. One property of radiation in thick media is that its distribution is almost isotropic. Another property of thick media is that the transport of energy by radiation is severely hindered. In the spirit of the ﬁrst property we deﬁne a streaming parameter, stream ; it is the ratio of the magnitude of the actual radiation ﬂux, |Frad |, to the maximum possible one |Frad | stream := . (10) c Erad It is clear that 0 ≤ stream ≤ 1 and that stream = 1 only if the radiation streams in one well-deﬁned direction. In thick media, stream is a small parameter: stream 1. In the spirit of the second property, we look at the ratio of the photon mean free path, lrad and some scale length, L. The scale length deﬁnes the distance of signiﬁcant variation in the properties of the material. We deﬁne space :=

4 lrad . 3 L

(11)

In thick media space 1. In thick media that strongly absorbs and emits radiation, far from any boundary layer, the diﬀusion approximation is valid. In the diﬀusion limit, the photon mean free path is determined by the Rosseland mean opacity, lrad = 1/σR and the scale length is set by the rate of change in the temperature: 1/L = (1/4T 4 )|∇(T 4 )|. In the diﬀusive regime the radiation energy density is that of a black body, Erad = a T 4 and the diﬀusion ﬂux is Frad = −(ac/3σR )∇(T 4 ). (Both of the preceding formulas are valid to ﬁrst order in the small parameter space .) Simple algebra shows that in the interior of a thick, strongly absorbing and emitting region, without scattering, the two approaches give the same result space ≈ stream .

(12)

The time rate of change of conditions in thick media can be estimated in a similar manner. We deﬁne a small parameter that is the ratio of the free ﬂight time of a photon to the time rate of change of the temperature time :=

1 1 ∂T 4 . c σR T 4 ∂t

(13)

Heating of the material results from radiation transport. Using the smallness of the energy ﬂux, stream 1, from the energy balance in a small volume we get the estimate time ≈ 2space

3Erad . Erad + ∂Emat /∂(T 4 )

(14)

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The parameters stream , space and time are not small in some boundary layers and in the leading edge of thermal waves.

3 The Diﬀerence Formulation In the introduction we discussed the diﬃculties of solving the transport equation, (1), in thick media. In the previous section we identiﬁed the streaming parameter, stream , that is small in thick media. We now show a very simple exact transformation of the transport equation so that it is written in terms of variables that are small in thick media. We call the result of the transformation the “diﬀerence formulation.” In the following we show the transformation in a simple case and discuss its remarkable properties. 3.1 The Diﬀerence Formulation without Scattering We start by repeating the transport equation, (1), without scattering 1 ∂I(x, t; ν, Ω) + Ω·∇I(x, t; ν, Ω) c ∂t = −σa (ν, T (x, t))[I(x, t; ν, Ω) − B(ν, T (x, t))] .

(15)

The equation is written in terms of the speciﬁc intensity carried by photons, I(x, t; ν, Ω). The left-hand side of the equation describes their unhindered propagation, the ﬁrst term on the right-hand side describes their attenuation. The two terms on the left-hand side and the ﬁrst term on the right-hand side constitute a homogeneous equation. The last term on the right-hand side, σa B, is a “source term” that makes the full equation inhomogeneous. It describes the emission of radiation by matter. From our considerations in the previous section, we expect that the diﬀerence between the two terms on the right-hand side is of the order of stream in thick media, even though each term by itself is of the relative order unity. We introduce now a “diﬀerence intensity” D(x, t; ν, Ω) := I(x, t; ν, Ω) − B(ν, T (x, t))

(16)

and subtract (1/c)(∂B/∂t) + Ω·∇B form both sides of (15). 1 ∂D(x, t; ν, Ω) + Ω·∇D(x, t; ν, Ω) = −σa (ν, T (x, t))D(x, t; ν, Ω) c ∂t 1 ∂B(ν, T (x, t)) − Ω·∇B(ν, T (x, t)) (17) − c ∂t Let us rewrite it with the independent variables suppressed for clarity. 1 ∂D 1 ∂B + Ω·∇D = −σa D − − Ω·∇B c ∂t c ∂t

(18)

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It should be emphasized that (18) and (15) are completely equivalent. In particular, they are able to satisfy equivalent initial and boundary conditions. The positivity constraint, I ≥ 0, translates into D ≥ −B. For completeness, we write the radiation energy density and its ﬁrst moment, the radiation ﬂux vector, in terms of D 1 ∞ 1 ∞ dν dΩI = dν dΩ(D + B) , (19) Erad = c 0 c 0 4π 4π ∞ ∞ Frad = dν dΩ Ω I = dν dΩ Ω D , (20) 0

4π

0

4π

and the coupling of the radiation to the material, from (8) ∞ ∂Emat = dν dΩσa D + G . ∂t 4π 0

(21)

The energy conservation equation, (9), is unchanged. We will now discuss the properties of the new transport equation, (18), and compare it to its traditional counterpart, (15). The left hand side of the propagation equation in the standard formulation, (15), is a propagation operator acting on the intensity, I. It can be written concisely as dI/ds, where ds is the path length along the light ray. If there is no material present, the straight line propagation of light can be expressed as dI/ds = 0 or dI = 0. In the presence of matter, light is attenuated as expressed by the −σa I term on the right hand side. The source term, σa B, also comes from the interaction of the light beam with the material; the material is assumed to be in LTE, a state with a well deﬁned temperature, T. The terms on the left-hand side of the propagation equation in the diﬀerence formulation, (18), are completely analogous. The left hand side propagates D in a straight line. The presence of material gives rise to an attenuation term, −σa D on the right hand side. We conclude that the intensity I and the diﬀerence intensity D propagate the same way and, in particular, their Green’s functions (propagators) are the same. Here is where the analogy stops. While in the standard formulation interaction with the material both attenuates and emits the I ﬁeld, in the diﬀerence formulation, interaction with the material only attenuates the D ﬁeld by the term −σa D. It expresses that, in the absence of external drive, the radiation ﬁeld relaxes to the local material temperature. In order to get better understanding, let us step back and consider the D ﬁeld. The radiation intensity, I = B + D, was separated into a local black body distribution B and a deviation, D. So D is the part of the radiation ﬁeld that is not in equilibrium with the material. It can be positive or negative, although it cannot be too negative, D ≥ −B. Compared to the traditional transport equation, (15), the inhomogeneous source terms have been changed drastically. One of their properties

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is that they are independent of the material properties. The source terms −(1/c)∂B/∂t − Ω·∇B are a propagation operator acting on −B. They can be written concisely as −dB/ds, where s is the path length along the path of the photon. In the absence of matter, dD/ds = −dB/ds, therefore, dI/ds = dB/ds+dD/ds = 0, expressing the straight line propagation of light. It follows then from (19) that the source terms in the propagation equation do not change the energy of the radiation ﬁeld. They come solely from the changes of the local reference radiation ﬁeld, B, along the path of the D photon. An interesting consequence is that the source terms that generate the D ﬁeld do not deposit energy or momentum in the material. That is a property of the absorption of the D ﬁeld only. Although B is tied to the temperature of the material through which the radiation is propagating, the photons represented by B stream. This property is captured by the source terms in the diﬀerence formulation. The source terms in the diﬀerence formulation are a simple conversion of the representation of photons from B to D, providing no direct energy or momentum exchange with the material. Let us now explore the source terms in some more detail. The source term in the traditional formulation for photon transport, σa B, accounts for spontaneous emission and is balanced by absorption in a thick system. In the diﬀerence formulation, the reference value for the radiation ﬁeld is B, not zero. This reference value is a function of the local temperature, T (x, t), and is therefore a function of both space and time. The new source terms in the diﬀerence formulation have a straightforward, intuitive interpretation. The term involving the time derivative of B can be understood from energy conservation. If the local temperature changes, the resultant change in B, all else remaining constant, must be accounted for by a change in the diﬀerence ﬁeld, D, in order to maintain (locally) the energy in the radiation ﬁeld. The term involving the space derivative of B is more interesting. To understand this term, consider transport in one-dimensional slab geometry where this term is now written µ dB/dx; the direction cosine of the propagation direction is µ = Ω·ˆ x, where x ˆ is a unit vector perpendicular to the slab. If the temperature is uniform, dB/dx is zero and there are no sources. Consider, however, the case where there is a positive step in the value of B, of magnitude b, at the origin. The source term, −µ dB/dx, is now −µ b δ(x). The diﬀerence ﬁeld has a source term only at the origin, with a negative source for positive µ and a positive source for negative µ. The right-moving negative source is interpreted as the missing photons that would have been streaming across the origin if the step in B did not exist. The negative sources are “photon holes”, borrowing a term from solid state physics. The left-moving positive source is simply the photons being emitted from the hotter region into the cooler region. More succinctly, the µ dB/dx term generates the transport between the hotter and cooler regions that would otherwise not occur. The total “photon” energy emitted at the origin integrates to zero.

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Another signiﬁcant diﬀerence between (15) and (18) is in the angular dependence of the source terms. The source term in the traditional formulation is σa B; it is spherically symmetric, i.e. of P0 symmetry. The source term in the diﬀerence formulation, that is dominant in thick media is Ω·∇B; it is antisymmetric in angle; more accurately it is of P1 symmetry. In the diﬀerence formulation there is also a source term, (1/c)(∂B/∂t) of P0 symmetry. While, in the standard formulation the σa B term adds energy to the radiation ﬁeld, in the diﬀerence formulation neither of the source terms add (or subtract) to the total energy of the radiation ﬁeld. The diﬀerence formulation has been developed with thick media in mind. Let us now compare the magnitudes of the terms in both formulations in some detail. The source term in (15) is σa B ≈ B/lrad , while the last source term in (18) is Ω·∇B ≈ B/L. In thin media, where lrad L, the ﬁrst version of the source term is small, while in thick media, where lrad L, it is the other way around. In addition to the question of asymptotic behavior, the source terms in the diﬀerence formulation are smooth in the frequency domain as they do not involve a factor of σa . The formulas in the previous section can be used to estimate the orders of the terms in (18) in optically thick regions. Let us divide the equation by σa B and consider the D/B term as the unknown. In thick media, the dominant source term is |Ω·∇B|/σa B ≈ space ; therefore we conclude that D/B ≈ space . We can then estimate that the other terms (1/c)(∂B/∂t)/σa B ≈ time ≈ 2space and Ω·∇D/σa B ≈ 2space . Finally, the (1/c)(∂D/∂t)/σa B term is of order 3space . 3.2 The Diﬀusion Limit, without Scattering In thick media, in LTE, far from boundaries, after suﬃcient time, radiation tends to the diﬀusion limit. This is a well established result of asymptotic analysis; nevertheless even very recently a reanalysis was published by Morel [Mor00]. We show now how the diﬀerence formulation leads to the diffusion limit. In fact we will show it in two diﬀerent ways. First, we formally integrate the transport equation; second, we show that the traditional asymptotic expansion yields the same result to ﬁrst order. It has to be emphasized that we show the diﬀusion limit of the exact transport equation; therefore it includes all terms, it is able to satisfy boundary conditions correctly and it includes the treatment of boundary layers. Formal Solution Equation (18) has a formal solution. We deﬁne a path variable, s, by x = x0 + Ω s

;

t = t0 + s/c .

It is easy to see that (18) can be written as

(22)

Accurate and Eﬃcient Radiation Transport in Optically Thick Media

dD dB = −σa D − , ds ds

265

(23)

giving the formal solution

s D(s) = D(0)exp − σa (s )ds 0 s s dB(s ) ds exp − σa (s )ds . − ds 0 s

(24)

The formal solution shows that the boundary condition, D(0), decays in a few absorption lengths. Deep in the material σa varies slowly. In fact both σa and dB/ds are constant to ﬁrst order in stream . Equation (24) can then be integrated. The result is 1 dB 1 dB . (25) exp[−σa s] − D(s) = D(0) + σa ds σa ds It shows that D(s) = −

1 σa (s)

dB(s) ds

(26)

is the steady-state solution of (24) and that any boundary value of D(0) decays to it in a few absorption lengths. A result that is correct to second order in stream was given in [SB03]. Asymptotic Expansion The relative orders of various terms in the transport equation, (18), in thick media were estimated in Sect. 2.2. The estimation is valid far from boundary layers and time transients. To ﬁrst order in stream , there are only two terms 0 = −σa D − Ω·∇B ,

(27)

giving the solution D=−

1 1 ∂B Ω·∇B = − Ω·∇(T 4 ) . σa σa ∂T 4

The radiation ﬂux, from (20), is ∞ 1 ∂B 1 ac Frad = − ∇(T 4 ) = − dν ∇(T 4 ) . 4 3 σ ∂T 3σ R 0 a

(28)

(29)

This is the correct diﬀusion limit of the transport equation. We also recovered the correct deﬁnition of the Rosseland mean opacity, σR ; see [Cas00, MM84]. To ﬁrst order (29) is identical to (25). It conﬁrms the ﬁrst order accuracy of the diﬀusion ﬂux [Mor00]. Note the utter simplicity of the derivation. An expansion in higher orders of stream can also be carried out. The results are similar to those of Morel [Mor00], but they are slightly diﬀerent and more consistent. A short discussion was given in [SB03].

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3.3 The Diﬀerence Formulation, in LTE, with Scattering The Scattering Term The scattering term was displayed in (5). In LTE, the Planck distribution at the material temperature is stationary and it also satisﬁes detailed balance. This imposes thermodynamic conditions on the scattering cross sections ν c2 B(ν) σs (ν → ν, Ω·Ω )B(ν ) 1 + ν 2hν 3 c2 B(ν ) = σs (ν → ν , Ω·Ω ) B(ν) 1 + , (30) 2hν 3 where the x, t; T dependence of σs and B has been suppressed. The scattering term expressed in terms of the diﬀerence ﬁeld, D, and the material temperature, T, is obtained by substituting I = B + D in (5) and then using (30) to rewrite it in the following form c2 D(ν, Ω) ν 1 − e−hν /kT Q(D) = dν dΩ σs (ν → ν, Ω·Ω )D(ν , Ω ) + ν 2hν 3 1 − e−hν/kT 0 4π ∞ c2 D(ν , Ω ) 1 − e−hν/kT dν dΩ σs (ν → ν , Ω·Ω )D(ν, Ω) + − . 2hν 3 1 − e−hν /kT 0 4π (31)

∞

Regrettably, there was an error in (31) in our original paper, [SB03]. If scattering does not change the radiation energy, e.g. in Thomson scattering, the stimulated emission terms in (5) cancel identically and an isotropic distribution is stationary under those conditions. In fact, we deﬁne, as usual 1 dΩ I(x, t; ν, Ω) . (32) J(x, t; ν) := 4π 4π Then I = J is stationary and (5) can be written as dΩ σs (ν, Ω·Ω ) [I(ν, Ω ) − J(ν)] Qmono (I − J) = 4π dΩ σs (ν, Ω·Ω ) [I(ν, Ω) − J(ν)] . −

(33)

4π

In the scattering terms, (5), (31), (33), both σs (ν → ν, Ω·Ω ) and I(ν, Ω) or D(ν, Ω) can be expanded in spherical harmonics [Brn95]. The integrals are then reduced to relaxation equations for the spherical harmonic components of D, or I − J, respectively. In particular, if the scattering is isotropic, (33) reduces to (34) Qmono (I − J) = σs (ν)(I − J) . Finally, we note that the scattering terms are always proportional to dΩ σs (ν, Ω·Ω ) , (35) σs (ν) = 4π

and to the analogous expressions in (5), (31).

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The Full Equations The full equations in the diﬀerence formulation, in LTE, are obtained by adding the right-hand side of (31) to the right-hand side of (17). As the change of the radiation energy caused by scattering comes from the matter, the integral of (31) over frequency has to be subtracted from (21), in analogy to (8). They have to be solved together with the conservation equations, (9). In the more general case they have to be solved together with the full set of equations of radiation hydrodynamics [Cas00], [MM84, Pom73]. Rather than presenting the formal development of the equations, we will sketch their simpliﬁed version that gives some insight into the relaxation behavior of the radiation intensity. We start by rewriting (1) in terms of the path variable, s, as deﬁned in (22). We also take liberties with the scattering term; we tacitly assume it to be monochromatic and isotropic, as in (34). dI = −σa (I − B) − σs (I − J) . ds

(36)

We deﬁne the total extinction coeﬃcient, σt = σa + σs . Obviously, σa σs dI = −σa (I − B) − σs (I − J) . + (37) σt σt ds In analogy to the diﬀerence formulation, we subtract appropriate terms from the equation to give σa d(I − B) σs d(I − J) + σt ds σt ds σ dB dJ σ s − . = −σa (I − B) − σs (I − J) − a σt ds σt ds

(38)

Depending on the relative magnitudes of σa and σs , it is easy to see that asymptotically, in thick media, the equation approaches either the diﬀusion approximation or the Eddington approximation. An alternative rearrangement gives dI = −σa (J − B) − (σa + σs )(I − J) . ds

(39)

We now subtract dB/ds from both sides of the equation, to give d(I − J) d(J − B) dB + = −σa (J − B) − (σa + σs )(I − J) − . ds ds ds

(40)

With the further assumption of dB/ds = 0 and the constancy of σa and σs along the radiation path, it is easy to verify that (40) has the solution J − B = [J(0) − B(0)]exp[−σa s] ,

(41)

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I − J = [I(0) − J(0)]exp[−(σa + σs )s] .

(42)

We note at this point that the material relaxation equation, (8), can always be written as ∞ ∂Emat = 4π dνσa (J − B) + G , (43) ∂t 0 so J − B is a natural variable to consider. We emphasize again that our derivation of the solution of (40) was highly simplistic. In particular it did not take into account material relaxation and the possibility of scattering with change of photon frequency, e.g. Compton scattering [Cas00]. We hope to report later on further developments along these lines.

4 Test Problems A variety of test problems have been investigated in [Bro05] in order to evaluate the computational eﬃciency and accuracy of the diﬀerence formulation, employing the Symbolic Implicit Monte Carlo (SIMC) method. Our goal was to analyze some simple situations that indicate the potential impact of the diﬀerence formulation in more complex physics environments. The key issues are accuracy and eﬃciency for both thick and thin media, and speciﬁcally the frequent occurrence of media which are thick at one frequency while being thin at others. In addition to comparing the Monte Carlo solution of the two formulations of transport, we compared them to analytic, or semi-analytic solutions where they are available. The main results of that study will be summarized below. 4.1 Behavior of a Finite Slab Heated from the Outside A very basic test of a time-dependent thermal transport algorithm is whether it correctly approaches the steady state solution for a ﬁnite slab immersed in a heat bath with diﬀerent temperatures on either side. The fourth power of the temperature should become a linear function of the optical depth in a gray, thick medium, in steady state. Deviations from such a straight line are indicative of boundary layers when they occur within a mean free path or so of a surface, but otherwise indicate serious problems in the numerical solution. Teleportation errors result in a wrong slope and cause curvature in the interior solution [MBS03]; for this reason zone thickness were limited to one mean free path in our piecewise constant treatment of the material temperature. In addition to the issues noted above, the time dependent approach to steady state oﬀers the opportunity to check the correctness of the implementation of interior source terms as well as initial and boundary conditions. That

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the steady state temperature is independent of time step, reﬂecting implicit behavior of the time integration, can also be checked. When the spectrum of the radiation ﬁeld is examined, even for a grey opacity, the correctness of the frequency sampling algorithm can be checked. Agreement between the standard and diﬀerence formulation is non-trivial due to the diﬀerent nature of the source terms. In the presentation of our computational results below, we provide a clear demonstration of rigorous agreement between the two formulations for transport, in terms of their approach to steady state, along with a measure of increase in computational eﬃciency for the diﬀerence formulation. The magnitude of this increase in computational eﬃciency, in the form of greatly reduced Monte Carlo noise as the optical thickness of the problem is increased, is somewhat surprising even to the authors who were prospecting for it. In the ﬁrst set of simulations we considered a ﬁnite slab heated from the left side, with an open boundary on the right, allowing the radiation ﬂowing through the slab to enter free space. The slab was composed of a uniform, static material having a frequency-independent (gray) opacity. We calculated the time dependence of the temperature and of the radiation ﬁeld after a 1 keV black-body source on the left side of the slab is turned on at time t = 0. ( 1 keV ≈ 1.2 107 ◦ K.) During the time dependent execution of the problem, a thermal wave, also known as Marshak wave, sweeps the problem domain and the solution then approaches steady state. We compared the solutions provided by the two formulations and their relative noise, for identical problem run times, in order to obtain a measure of the accuracy and relative computational eﬃciency of the two formulations for transport, under conditions that the Monte Carlo portion of the code dominates execution time. Four instances of this problem are presented below. The slab is composed of a uniform material having a frequency independent (gray) opacity of 0.1, 1, 10, and 100 mean free paths per cm respectively. The slab is 10 cm thick so the four opacities correspond to total optical depths of 1, 10, 100 and 1000 mean free paths. The speciﬁc heat of the material is a constant 0.1 jerk/(keV cm3 ), where jerk is an energy unit (1 jerk = 1016 ergs), and temperature is measured in energy units of kT = 1 keV. The slab is initially at a temperature of 0.01 keV, with the radiation ﬁeld being a Planckian in equilibrium with this temperature. All four problems used a time step of 0.2 sh, where 1 sh = 10−8 sec. The problem 1 mean free path thick was run to 20 sh in order to get close to steady state, as was the problem 10 mean free paths thick. The problem 100 mean free paths thick was run to 40 sh in order to approach steady state. The problem 1000 mean free paths heated up very slowly due to the diﬀusive nature of the solution, requiring 320 sh in order to suitably approach steady state. The problems 1 and 10 mean free paths thick employed 20 zones, while the thicker problems employed zones one mean free path thick in order to prevent teleportation error from inﬂuencing the results, and to

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prevent anomalous performance results for the diﬀerence formulation. Equal thickness zones were used everywhere. Geometric zoning has a role only if piecewise linear treatment of the material temperature is available to remove teleportation error. The speed of light is 300 cm/sh. This provides 6 traversals of the slab for a time step of 0.2 sh, reﬂecting on the stability of the method. Relaxing the need for implicit treatment of the source terms was one of the hopes of the authors, given that implicit treatment requires the solution of a non-linear system of equations for each time step. Our experiences in this regard, documented in [Daf05], were made even more diﬃcult by the T 4 term for thermal emission. Explicit treatment of the ∂B/∂x source terms was abandoned as a result, but may be worth revisiting in mixed physics applications where the time step size is limited for reasons outside of transport physics. We would like to note that the Monte Carlo solution for the two transport formulations have entirely diﬀerent requirements for spatial importance sampling if uniform statistical noise, as a function of position, is to be obtained. In the standard formulation most of the computational eﬀort is spent computing the balance between emission and absorption that produces the local equilibrium black body ﬁeld. As a result, a scheme that samples particles with a uniform density in space produces a relatively ﬂat statistical noise across the slab. The number of source particles born in each zone, during each time step, is proportional to the thickness of the zone. The statistical properties of the Monte Carlo solution for the diﬀerence formulation are in sharp contrast to this. For thicker problems, the statistical noise for the difference formulation is miniscule near the hot side and increases very rapidly towards the cold boundary. Suitable importance sampling could ﬂatten out this growth in noise. An exposition of this topic is beyond the scope of this paper. In the results shown below, the unit of source sampling was one ∂B/∂t particle, and one ∂B/∂x particle pair, for each zone, or zone interface, respectively, for the diﬀerence formulation. We have not attempted to tune the ratio between ∂B/∂t and ∂B/∂x source particles, nor have we attempted to tune the relative importance of source sampling across the volume of the problem. In the standard formulation, the unit of source sampling is one thermally emitted particle per zone. In Fig. 1 (a) we show the steady state solution for the material temperature, as a function of position in the slab, for the problem instance that is one mean free path thick. The average of 100 randomly seeded runs using the standard formulation, and the average of the same number of instances using the diﬀerence formulation, is shown by the boxes and the × symbols, respectively. The two formulations are in perfect agreement. The standard deviation of the results for the standard formulation, as well as the same statistic for the diﬀerence formulation, both multiplied by 400, are plotted using diamonds and triangles, respectively. The Monte Carlo source parti-

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Fig. 1. Temperature distribution in a slab in steady state. The slab is heated from the left by a 1 keV black body, and it radiates freely on the right. The total optical depth (OD) of the slab is 1 in (a), 10 in (b), 100 in (c) and 1000 in (d). The standard deviation of the standard formulation is denoted by diamonds and that of the diﬀerence formulation by triangles. Note the change in their relative scale with optical depth. The noise in the standard formulation increases dramatically with optical depth, while in the diﬀerence formulation it does not

cle counts were selected for equal run times for the two formulations, and were high enough that the cost of Monte Carlo transport dominated execution time. The relative performance of the methods, then, is the ratio of the squares of the standard deviations. There is a small advantage in favor of the diﬀerence formulation, except at the very right hand side of the slab. Note that the optical thickness of each zone is only 0.05 optical depths. In Fig. 1 (b) we show the results for a slab that is 10 mean free paths thick. Again, we obtain perfect agreement for the two formulations when the average of the equilibrium material temperature is examined. In order to see the standard deviation for the two formulations on the same plot, we now

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have to apply a scale factor of 100 for the standard formulation and 1000 for the diﬀerence formulation. The trend of growth for the standard deviation for the diﬀerence formulation, as the temperature gradient increases in the slab from left to right, is becoming apparent. In Fig. 1 (c) we show the results for a slab that is 100 mean free paths thick. There are 100 zones in this problem in order to avoid teleportation error, with every ﬁfth point plotted using a symbol. The other points are included in the solid lines drawn. The growth in noise for the standard formulation is now becoming visible in the plot. We apply a scale factor of only 10 to see the standard deviation for the standard formulation. The growing trend of the standard deviation for the diﬀerence formulation, as one traverses the slab from left to right is now quite clear, but the computational advantage for the diﬀerence formulation is large. In Fig. 1 (d) we show the results for a slab that is 1000 mean free paths thick. There are 1000 zones in this problem, a requirement to avoid signiﬁcant teleportation error, with every 50th zone plotted using a symbol. The disagreement between the two formulations appearing towards the left side of the slab are statistical ﬂuctuations. The computational advantage of the diﬀerence formulation is extreme. At this point, the reader may note that the results for the diﬀerence formulation have the appearance√of smooth curves, regardless of the optical thickness of the problem. The 1/ N noise behavior typical of a Monte Carlo solution is still present, it is just that the amplitude of the noise is small and remains small as the optical thickness of the problem increases. Evidence of the scaling behavior of the computational advantage is shown in the next ﬁgure. As noted above, the number of Monte Carlo particles for each optical thickness is set so that the diﬀerence formulation and standard formulation exhibit the same run time, and that the computer time is dominated by the Monte Carlo, so that the standard deviation is scaling like one over the square root of the computational work. Under these conditions, the relative computational eﬃciency of the two methods is the ratio of the squares of their standard deviations. In Fig. 2 we show the computational advantage exhibited in the standard deviation of the material temperature, as a function of position, for the four optical thicknesses. The computational advantage scales, roughly, as the square of the optical depth of the problem. Finally, in Fig. 3 we show the penetration of the thermal wave into the material, 1000 optical depths thick, at an intermediate time. As in Fig. 1 (d), the material temperature is plotted vs. distance. Reﬁning the time step and zone size demonstrates that the solution shown is fully converged. This ﬁgure illustrates the gains that the diﬀerence formulation provides, clearly visible to the reader.

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Fig. 2. The relative computational advantage of the diﬀerence formulation compared to that of the standard formulation, plotted as a function of position for various optical depths of the slab. In the 1 OD case, each zone is only 1/20 OD thick, nevertheless, the diﬀerence formulation is better than the standard one except near the surface of the slab on the right hand side. There is a sharp decrease in computational advantage where the temperature gradient is large. This could easily have been remedied by spatial importance sampling of the source particles

4.2 Comparison to Analytic Diﬀusion Solution for a Linearized Problem In order to check the correctness of our numerical implementation, we have compared results with the diﬀusion solution appearing in [SO96]. This solution requires that the material energy take the form Emat =

αT 4 4

(44)

or, equivalently, a speciﬁc heat of the form ∂Emat = αT 3 . ∂T

(45)

The purpose of this form for the material energy is to remove the nonlinearity, T 4 , that otherwise prevents an analytic solution. The resulting analytic solution can then be used to check for correct convergent behavior in the

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Fig. 3. Thermal wave (Marshak wave) penetrating a uniform, gray slab of 1000 OD at an early time, 40 sh. The standard formulation gives a noisy temperature proﬁle whereas that of the diﬀerence formulation is many orders of magnitude smoother. The slight diﬀerence in the position of the leading edge is a statistical ﬂuctuation for the standard formulation

numerical simulation. The behavior of the speciﬁc heat at T = 0, however, makes things quite fragile unless the numerical code itself is transformed to handle things in linearized form. To some extent, this defeats the purpose of checking the original code, much of which had to be modiﬁed to resolve the issue. For this problem, α and the cross section were chosen so that the values in the tables of analytical results appearing in [SO96] could be used directly. In Fig. 4 we plot the material temperature produced by our Monte Carlo solution of the standard and diﬀerence formulation, along with data from Su and Olson’s analytical solution corresponding to a late time, τ = 10. We obtain good agreement where analytical data are available. This is expected because for τ = 10 the diﬀusion approximation assumed by Su and Olson is valid. In Fig. 5 we plot the material temperature for the same problem at a much earlier time, along with data from Su and Olson’s analytical solution for a τ = .01. In this case, not surprisingly, there is sharp disagreement

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Fig. 4. Temperature distribution in the Su & Olson problem [SO96] at a late time τ = 10. There is excellent agreement of the calculations with analytic results

between the diﬀusion solution and the fully overlapped and converged Monte Carlo transport solutions for the standard and the diﬀerence formulation. Here the fundamental limitation is the speed of light. It is fully respected by the transport solution but ignored by the diﬀusion solution. 4.3 Time Dependent Marshak Wave Problem, with a Non-trivial Opacity Our Monte Carlo solutions of the standard and diﬀerence formulations of transport fully implement the details of the thermal frequency spectrum; the spectral properties of the derivative sources make the agreement between the two formulations non-trivial even in the case of a gray opacity. Once the spectral sampling of the source terms in the diﬀerence formulation is done correctly, as described in Appendix A of [Bro05], there is no more to be done for the correct treatment of a frequency dependent opacity in the diﬀerence formulation other than to use the correct absorption cross section for the given Monte Carlo particle. The accuracy of the treatment of the frequency dependent cross section is as good as the cross section itself. The emission term for the standard formulation, on the other hand, appears as σB and in the most general case it must be numerically integrated

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Fig. 5. Temperature distribution in the Su & Olson problem [SO96] at an early time τ = 0.01. The transport solution is limited by the speed of light, that is 1 in the units of τ . Note that there is no signiﬁcant spreading of the radiation front at the leading edge of the thermal wave

across a frequency group structure in each zone, for the temperature at the start of each time step, in order to produce the probability distribution for the emission spectrum that, then, must be sampled. This requirement provides another place where numerical errors must be controlled in the implementation of the standard formulation. We now turn to a relatively simple frequency dependent opacity. The opacity is constant (gray and 1000 mean free paths for the slab) for frequencies below 1 keV and again constant (gray and 10 mean free paths for the slab) for frequencies above 1 keV. This corresponds, roughly, to the precipitous drop in opacity that can occur in real materials, as a function of frequency. The portion of the emitted spectrum below 1 keV is strongly re-absorbed, while the portion of the emitted spectrum above 1 keV encounters a lower opacity and transports more freely. This is a diﬃcult non-linear problem because, as the trapped radiation heats the material, the Planckian emission spectrum moves towards higher frequencies where radiation ﬂows more freely. In Fig. 6 we show the material temperature for one instance of this problem, at an intermediate time where the thermal wave is still propagating through the slab. Again, as before, the standard formulation is the one

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Fig. 6. Thermal wave penetrating a uniform slab, shown at a time t = 1 sh. The optical thickness of the slab below 1 keV is 1000 mean free paths, but there is a precipitous drop in opacity by a factor of 100 above 1 keV. The long “foot” of the thermal wave stems from high frequency photons that penetrate deeply

exhibiting a high level of statistical noise. Note, however, that this noise disappears in the “foot” of the advancing thermal wave. The reduction in noise for the standard formulation is due to the fact that this feature is caused by photons at high frequencies where the material has a reduced opacity. Those photons are absorbed, but not reemitted by the cold matter.

5 Summary and Directions for Further Work 5.1 Theory In earlier work [SB03], a new analytical formulation was introduced for the transport equation. The new formulation is for the transport of the diﬀerence between the speciﬁc intensity and the local black body equilibrium radiation at the matter temperature, at any point in space, time and direction. Accordingly, we called the new transport equation the diﬀerence formulation to distinguish it from the traditional formalism. We have shown that the difference formulation is expressed in terms of quantities that become small in

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optically thick media. The transformation is a simple one and results in a completely equivalent system of equations, without approximation. The most important distinction between the two formulations is in the source terms. In the traditional formulation, the source term is the spontaneous emission of the medium. It is small in optically thin regions, resulting in straight line propagation of photons. The traditional formulation is well suited for this regime. In the diﬀerence formulation, the source term is the space-time gradient of the Planck function at the material temperature. The latter gets small in optically thick regions. In addition to this important diﬀerence in asymptotic behavior, the two formulations diﬀer in that the spontaneous emission depends upon the absorption cross sections for the emitting medium, while the source term in the diﬀerence formulation depends only upon the temperature of the medium, as a function of space and time. The two formulations are able to satisfy equivalent boundary conditions and initial conditions. Even the largest terms in the diﬀerence formulation are of the order of space , i.e. the ratio of the photon mean free path to the gradient length. In optically thick regions this ratio is a small quantity. We have shown that the equations reduce to the diﬀusion limit in the proper circumstances. We have also discussed brieﬂy the extensions needed when scattering is important. 5.2 Computations The Symbolic Implicit Monte Carlo method, SIMC, [Bro89] is an attractive framework for the calculation of radiation transport in complex media and geometries; it provides a basis for accurate and stable numerical schemes [Nka91] [DL04]. In our previous paper, [Bro05], we have demonstrated that the diﬀerence formulation [SB03] is eminently suitable for numerical simulations of radiation transport, employing the SIMC technique. Theoretical expectations were that the traditional formulation would be good for thin regions, while the diﬀerence formulation would be advantageous in thick media. We have demonstrated that the diﬀerence formulation, when employing the SIMC technique, oﬀers signiﬁcant noise reduction for thick systems. The expected cross-over vis-a-vis the standard formulation occurs for very thin systems, thin enough that the diﬀerence formulation might become a panacea for Monte Carlo treatment of thermal radiation transport in practical problems. Its advantage for thick systems scales like the square of the optical depth of the system. The character of the source terms is very diﬀerent in the traditional and the diﬀerence formulations: thermal emission in the former is replaced by derivative source terms in the latter. Therefore a key issue for the accuracy and stability of the diﬀerence formulation is the successful treatment of those derivative source terms. We have developed eﬃcient, accurate analytic techniques for sampling the frequency spectrum of the source terms for the

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diﬀerence formulation. Frequency sampling in the diﬀerence formulation depends only on the space and time derivative of the material temperature, not on detailed material optical properties. This oﬀers a signiﬁcant advantage for problems with complicated material optical properties. The gain in terms of code complexity became clear when we implemented the test problem with the step in opacity. Although the test problems presented were very simple, in order to be able to clearly identify the advantages of the method, extensions to more complex situations do not present conceptual diﬃculties. We would like to note that the computational gain for the diﬀerence formulation demonstrated in this paper is for the Monte Carlo portion of the problem. In order to obtain implicit treatment of the source terms, the SIMC technique requires the solution of a non-linear system of equations in order to perform the temperature update at the end of the time step. The cost of this can become signiﬁcant when the number of zones in the problem is large. We used band-limited Gaussian elimination, at the core of a Newton-Raphson solver, in our numerical work on problems with as many as 1000 zones. The bandwidth was limited both by the time step size and by the death of Monte Carlo particles when they became too small, relative to their birth weight. Multi-dimensional problems with signiﬁcantly larger numbers of zones will pose a challenge, requiring the use of suitable iterative solution techniques. We believe that the demonstrated noise reduction in thick systems will be worth the eﬀort involved. We would like to note that our test problems have been relative simple, suitable as the ﬁrst numerical tests for the diﬀerence formulation for thermal radiation transport. In our presented results, it is clear that the importance sampling requirements for the diﬀerence formulation are quite diﬀerent than those for the standard one. It is also the case that improvements such as weight vectors in frequency space, and deterministic handling of the spectral output from the interface adjoining free space, have a signiﬁcant impact on noise when spectral information associated with the photon ﬁeld is desired. The details of these enhancements, and their relative value, are dependent upon the exact nature of the problems being run and the computational results that are desired. 5.3 Work in Progress We believe that the diﬀerence formulation will help in numerical solutions of the equations of radiation hydrodynamics in optically thick regions. We expect that it will be useful regardless of the numerical method employed, be it a deterministic method, for example Sn and Pn , or a Monte Carlo method, for example the Symbolic Implicit Monte Carlo (SIMC) method of Brooks [Bro89], employed here. The dominant source of instability for Monte Carlo methods, the spontaneous emission term, is removed in the diﬀerence formulation and replaced by terms that are small in thick systems. Because of this, the well known stability problem for Monte Carlo methods in thick

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systems may, in fact, be removed. We will report on this possibility in future work. Preliminary results show that the eﬃciency for Monte Carlo methods in thick systems will be improved, due to the removal of the balance between emission and absorption in a zone that produces a relatively noisy estimate for the diﬀerence. A similar treatment may be useful in other transport problems. Neutron transport near criticality has many of the same properties as photon transport in optically thick regions. Similarly, the success of radiation therapy depends on accurate modeling of particle transport in the presence of strong absorption and scattering. We hope to be able to extend our treatment to some of those applications in the future. In our numerical work on the diﬀerence formulation, we have employed a piecewise constant discretization for the material temperature. Due to teleportation eﬀects, this discretization does not provide the correct diﬀusion limit for zone sizes that are large compared to the mean free path of a photon. It was clearly established by Clouet and Samba [CS04] that a piecewise linear treatment of the material temperature is required to obtain the correct diﬀusion limit, thereby eliminating the teleportation problem [MBS03]. We will report on the application of the method of Clouet and Samba, and related techniques, to the diﬀerence formulation in a future work. Further studies in progress include a study of the numerical stability of various treatments of the source terms of the diﬀerence formulation. Although iterative techniques for the solution of the fully implicit treatment provide a useful solution for one and two dimensional problems, the possibility of a suitably stable explicit treatment of the source terms is not yet fully explored. The time step control required for an explicit treatment of the source terms might be covered by the time step control requirements for other portions of a mixed physics application, opening up the possibility of executing the time dependent transport solution without the cost of the non-linear solver. We also intend to generalize the diﬀerence formulation to include nonmonochromatic scattering and investigate its applicability to some well known approximate treatments, e.g. the Eddington approximation and the Kompaneets equation.

Acknowledgement The authors would like to thank John Castor for pointing out the error in our expression for the scattering of the diﬀerence ﬁeld.

References [Mih78]

Mihalas, D.: Stellar Atmospheres, Freeman, San Francisco, pp 64–71 (1978).

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Fleck, J.A. and Cummings, J.D.: An implicit Monte Carlo scheme for calculating time and frequency dependent radiation transport, J. Comput. Phys. 8, 313 (1971). [PB83] Larsen, E. W., Pomraning, G. C. and Badham, V. C.: Asymptotic analysis of radiative-transfer problems, J. Quant. Spectr. Rad. Transfer 29, 285 (1983). [Pom82] Pomeraning, G.C.: Flux Limiters and Eddington Factors, J. Quant. Spectr. Rad. Transfer 27, 517–530. [Mrs47] Marshak R.E.: Note on the Spherical Harmonic Method as Applied to the Milne Problem for a Sphere, Phys. Rev. 71, 443–446 (1947). [Mrk47] Mark, J.C.: The spherical harmonic method, Technical Report, CRT-340, Atomic Energy of Canada, Ltd., Ontario, 1947. [Bro89] Brooks, E.D.: Symbolic Implicit Monte Carlo, J. Comput. Phys. 83, 433–446 (1989). [Nka91] N’kaoua, T.: Solution of the nonlinear radiative transfer equations by a fully implicit matrix Monte Carlo method coupled with the Rosseland diﬀusion equation via domain decomposition, SIAM J. Sci. Stat. Comput. 12, 505 (1991). [DL04] Densmore J.D., Larsen E. W.: Asymptotic Equilibrium Diﬀusion Analysis of Time-Dependent Monte Carlo Methods for Grey Radiative Transfer, J. Comput. Phys. 199, 175–204 (2004). [CF73] Carter, L.L. and Forest, C.A.: Nonlinear radiation transport simulation with an implicit Monte Carlo method, LA-5038, Low Alamos National Laboratory, 1973 (unpublished). [SB03] Sz˝ oke, A. and Brooks, E.D.: The transport equation in optically thick media, J. Quant. Spectr. Rad. Transfer 91, 95–110 (2005) [Bro05] Brooks, E.D, et al.: Symbolic implicit Monte Carlo radiation transport in the diﬀerent formulation: a piecewise constant discretization, J. Comput. Phys. (in press, available online) (2005) [Cas00] Castor, J.I.: Lectures on Radiation Hydrodynamics, UCRL-JC-134209, 2000. [MM84] Mihalas, D. and Mihalas, B. W.: Foundations of Radiation Hydrodynamics, (New York: Oxford University Press), 1984. [Pom73] Pomraning, G. C.: The Equations of Radiation Hydrodynamics, (Oxford: Pergamon), 1973. [MA00] Mihalas, D. and Auer, L.H.: On laboratory-frame radiation hydrodynamics, J. Quant. Spectr. Rad. Transfer, 7, 61–97 (2000). [Mor00] Morel, J.E.: Diﬀusion-limit asymptotics of the transport equation, the P1/3 equations, and two ﬂux-limited diﬀusion theories, J. Quant. Spectr. Rad. Transfer 65, 769–778 (2000). [Brn95] Brown, P. N.: A linear algebraic development of diﬀusion synthetic acceleration for 3-dimensional transport-equations, SIAM Journal of Numerical Analysis, 32, 179–214 (1995). [MBS03] McKinley, M. S., Brooks, E. D, and Szoke, A.: Comparison of Implicit and Symbolic Implicit Monte Carlo Line Transport with Frequency Weight Vector Extension, J. Comput. Phys. 205, 330–349 (2003). [Daf05] Daﬃn, F., et al.: An Evaluation of the Diﬀerence Formulation for the Transport of Atomic Lines, J. Comput. Phys. 204, 27–45 (2005).

282 [SO96]

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A. Sz˝ oke, et al. Su, B. and Olson, G.L.: Benchmark results for the non-equilibrium marshak diﬀusion problem, J. Quant. Spectr. Rad. Transfer 56, 337–351 (1996). Clouet, J.-F. and Samba, G.: Asymptotic diﬀusion limit of the symbolic Monte-Carlo method for the transport equation J. Comput. Phys. 195, 293–319 (2004).

An Evaluation of the Diﬀerence Formulation for Photon Transport in a Two Level System∗ Frank Daﬃn, Michael Scott McKinley, Eugene D. Brooks III, and Abraham Sz˝ oke University of California, Lawrence Livermore National Laboratory, Livermore, California 94550 [email protected] [email protected] [email protected] [email protected]

In this paper we extend the diﬀerence formulation for radiation transport to the case of a single atomic line. We examine the accuracy, performance and stability of the diﬀerence formulation within the framework of the Symbolic Implicit Monte Carlo method. The diﬀerence formulation, introduced for thermal radiation by some of the authors, has the unique property that the transport equation is written in terms that become small for thick systems. We ﬁnd that the diﬀerence formulation has a signiﬁcant advantage over the standard formulation for a thick system. The correct treatment of the line proﬁle, however, requires that the diﬀerence formulation in the core of the line be mixed with the standard formulation in the wings, and this may limit the advantage of the method. We bypass this problem by using the gray approximation. We develop three Monte Carlo solution methods based on diﬀerent degrees of implicitness for the treatment of the source terms, and we ﬁnd only conditional stability unless the source terms are treated fully implicitly.

1 Introduction Time-dependent transport of radiation from resonance lines is an important component of the physics of stellar atmospheres and of laser-produced plasmas. In optically thick systems, the radiation transport equation for photons is dominated by many spontaneous emission and absorption events and is tightly coupled to the level population equation. This system of equations can be diﬃcult to solve numerically in any discretized scheme in time and space due to its stiﬀness and the wide range of opacities inherent in an atomic line proﬁle. It has been known for many years that the explicit Monte Carlo solution of the radiation transport equation, coupled to the material response equation, for a strongly absorbing and emitting material, is numerically unstable. ∗

This work was performed under the auspices of the U.S. Department of Energy by University of California, Lawrence Livermore National Laboratory under Contract W-7405-Eng-48. UCRL-PROC-210977

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One reason for this is that in optically thick regions both the emission and absorption terms are large and the net emission (or absorption) of radiation is a small diﬀerence of these two quantities. Any small imbalance or inconsistency in space and time between absorption and emission terms can lead to instability. This diﬃculty requires that the source terms in the transport equation be implicitly diﬀerenced when using Monte Carlo methods for its solution [NS93]. The ﬁrst successful – and now widely used – method for addressing this diﬃculty came from Fleck and Cummings [FC71], [BF86]. Their method, called Implicit Monte Carlo (IMC), converts part of the absorption-emission cycle into instantaneous eﬀective scattering. The net eﬀect of IMC is to reduce the strength of the coupling between the photon transport equation and the material energy equation by peeling oﬀ part of the coupling and treating it as eﬀective scattering. Stability is achieved by weakening the radiationmatter coupling. This can lead to unphysical results [DL04] in addition to a signiﬁcantly increased execution time to handle the scattered photons. A second approach to the problem of numerical stability was published in [Bro89] and [NKa91]. In this scheme Monte Carlo particles are emitted and tracked with weights that remain unknown to within a multiplicative factor until the end of the integration cycle. This method, called Symbolic Implicit Monte Carlo (SIMC), removes the costly eﬀective scattering of IMC and does not artiﬁcially weaken the radiation-matter coupling. However, in thick systems the strong emission and absorption terms lead to increased Monte Carlo noise. The diﬀerence formulation for photon transport [SB05] directly addresses the stiﬀness problem by employing a transformation that replaces the spontaneous emission term with source terms that are small when the local coupling between spontaneous emission and absorption is strong. Our goal in this paper is to explore whether or not the diﬀerence formulation is cleanly applicable to the case of line transport. We implement and study a numerical application of the diﬀerence formulation for the case of the transport of a single atomic line, examining the issues of accuracy, stability and eﬃciency. In Sect. 2 we introduce the equations for line transport ﬁrst in the standard formulation, and then in the diﬀerence formulation. There, a diﬃculty for the wings of the line appears that would force us to mix the standard formulation with the diﬀerence formulation in order to treat a real line proﬁle. We conduct our numerical investigation with a gray (square) line shape function in order to sidestep the issue. The section concludes with a discussion of our treatment of boundary conditions within the new formulation. Section 3 addresses some details of the numerical treatment of the difference formulation, including the new source terms. The Symbolic Implicit Monte Carlo (SIMC) solution method [Bro89], applied to the standard formulation, requires the solution of a linear system in order to update the atomic populations at the end of each integration cycle. The correspond-

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ing population update for the diﬀerence formulation is non-linear, requiring a Newton-Raphson solver. Whether or not implicit treatment of the source terms is required in the diﬀerence formulation is an open question that we investigate. To this end, we develop three treatments of the source terms, each with diﬀering levels of implicitness. Our explicit treatment is free of a non-linear matrix solve, but is only conditionally stable. Our fully implicit treatment requires a non-linear matrix solve, but numerical evidence suggests that it is unconditionally stable. A semi-implicit method is examined and gives some insight into the numerical instabilities arising in the explicit treatment of the source terms. Next we compare the accuracy, eﬃciency and numerical stability of the SIMC method in the standard formulation to our implementations of the difference formulation in Sect. 4. We demonstrate that the diﬀerence formulation delivers a startling decrease in noise, or an equivalent increase in execution speed for a given noise ﬁgure, when compared to the Monte Carlo solution of the standard formulation for transport. Finally, we present a summary of this work in the last section.

2 The Equations for Line Transport We present the transport equations for photons for a two-level atomic system in slab geometry, where the photons are emitted and absorbed according to the same line proﬁle, φ(ν), in the regime of complete redistribution. The transport equation for photons is coupled to the population equations for the atomic levels. Motion of the medium and physical scattering of photons are not considered, but we include collisional pumping between atomic levels. 2.1 The Standard Formulation In what we refer to as the “standard formulation,” we write the photon transport equation as ∂f n2 ∂f + cµ = A21 φ − c (K12 n1 −K21 n2 ) φf , ∂t ∂x 2

(1)

where c is the speed of light, x is the position coordinate perpendicular to the slab, µ is the direction cosine of the radiation with respect to x axis, f (µ, ν, x, t) is the photon number density distribution per unit atom density, n2 (x, t) is the upper level population fraction, n1 (x, t) is the lower level population fraction, A21 is the spontaneous emission rate, φ(ν) is the line proﬁle normalized to unit integral [Mih78], and K12 = κN where κ is the lower state absorption cross section and N is the atom number density. The coeﬃcient K21 satisﬁes the Einstein relation g1 (2) K21 = K12 , g2

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where g1 and g2 are the statistical weights for levels 1 and 2, respectively. For the purposes of this paper, we consider all material parameters, C12 , C21 , A21 , K21 and K21 to be independent of x, constant in time, and assume complete redistribution within the line shape. The equations governing the atomic population fractions n1 and n2 are 1 ∞ ∂n2 = C12 n1 −C21 n2 −A21 n2 +c (K12 n1 −K21 n2 ) dµ dν φ(ν)f (µ, ν) (3) ∂t −1 0 and n1 + n2 = 1 ,

(4)

where C12 and C21 are rate constants for the collisional transitions 1 → 2 and 2 → 1, respectively. Using (4), (1) and (3) are rewritten as ∂f n ∂f + cµ = A21 φ − c [K12 − (K21 + K12 ) n] φf , ∂t ∂x 2

(5)

and ∂n = C12 − (C12 + C21 + A21 ) n ∂t + c [K12 − (K21 + K12 ) n]

1

dµ −1

∞

dν φ(ν)f (µ, ν) ,

(6)

0

respectively, where n is the upper level population fraction. We refer to these equations as the standard formulation for line transport in the context of this paper. 2.2 The Diﬀerence Formulation The diﬀerence formulation, introduced in [SB05], removes the spontaneous emission term and the trouble it causes for thick systems through a simple transformation of the transport equation. The transformation produces a transport equation with new source terms that are small for thick systems, at least in the core of the line, and leads to an eﬃcient numerical solution in optically thick media. For the case of line transport, the diﬀerence formulation is derived by considering the radiation ﬁeld that is in equilibrium with a given upper level atomic population fraction B(n(x, t)) =

n(x, t)A21 . 2c[K12 − n(x, t)(K21 + K12 )]

(7)

The equilibrium ﬁeld, B, deﬁned in (7) is independent of photon frequency.

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We begin the transformation to the diﬀerence formulation by rewriting the spontaneous emission term from (5), as well as from (6), using the equilibrium ﬁeld, (7). ∂f (x, t; ν, µ) ∂f (x, t; ν, µ) + cµ ∂t ∂x = −c [K12 − (K21 + K12 ) n(x, t)] φ(ν) [f (x, t; ν, µ) − B(n(x, t))] ,

(8)

∂n(x, t) = C12 − (C12 + C21 ) n(x, t) + c [K12 −(K21 +K12 ) n(x, t)] ∂t 1 ∞ × dµ dν φ(ν) [f (x, t; ν, µ)−B(n(x, t))] .

(9)

−1

0

Next, we deﬁne the “diﬀerence” intensity, d(x, t; ν, µ) = f (x, t; ν, µ) − B(n(x, t)) .

(10)

We note that this is our ﬁrst sign of trouble for the diﬀerence formulation when applied to the case of line transport. The fact that B does not depend upon ν means that in the wings of the line where f is small – even for a system that is thick in the core of the line – the diﬀerence ﬁeld d must be large in order to compensate. The result will be an increase in noise in the wings of the line. We will return to this issue in what follows. Substituting (10) into the transport equation gives ∂f (x, t; ν, µ) ∂f (x, t; ν, µ) + cµ ∂t ∂x = −c [K12 − (K21 + K12 ) n(x, t)] φ(ν)d(x, t; ν, µ) . (11) We now subtract the derivatives of B from both sides, giving ∂d(x, t; ν, µ) ∂d(x, t; ν, µ) + cµ ∂t ∂x = − c [K12 −(K21 +K12 ) n(x, t)] φ(ν)d(x, t; ν, µ) −

∂B(n(x, t)) ∂B(n(x, t)) − cµ . ∂t ∂x

(12)

The population equation becomes ∂n(x, t) = C12 − (C12 + C21 ) n(x, t) ∂t + c [K12 − (K21 + K12 ) n(x, t)]

1

dµ −1

∞

dν φ(ν)d(x, t; ν, µ) . (13) 0

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We refer to these equations as the diﬀerence formulation of line transport. Our formal manipulations give us two equivalent forms for the transport and atomic population equations: Equations (5), (6) and (12), (13). The two sets of equations satisfy equivalent boundary and initial conditions and were obtained without approximation. 2.3 Boundary Conditions for the Diﬀerence Formulation In order to relate the boundary conditions for the standard formulation to those for the diﬀerence formulation, we use the fact that the upper level atomic population fraction n is the same for both and use the relation d = f − B(n) to construct the d ﬁeld from f . The strict non-negativity of f translates into a lower bound for the diﬀerence ﬁeld, d ≥ −B. When an initial condition is speciﬁed for f , the corresponding condition for d can be obtained by the above relation. In this work the physical medium has ﬁnite extent with vacuum boundary conditions. We specify that n be zero in the vacuum and thus d = f there, accordingly. The emission from the surface into the vacuum is given by the −cµ ∂B/∂x term at the boundaries, in addition to the particles that escape from within. It consists of emission of positive d = f particles into the vacuum, and negative d particles into the material, cooling it, and gives a natural prescription for treating boundary conditions in the diﬀerence formulation. 2.4 The Gray Approximation The line emission proﬁle φ(ν) occurs in both the spontaneous emission and the absorption terms for line transport. This leads to the frequency independence of the equilibrium ﬁeld, B(n(x, t)), and the result that the diﬀerence ﬁeld does not become small in the wings of the line as the optical thickness of the problem is increased. In practical terms, this means that even the simplest line transport problem must employ a mixing of the diﬀerence formulation in the core of the line with the standard formulation in the wings. Wanting to evaluate the eﬀectiveness of the diﬀerence formulation in the core of the line, we apply the gray approximation to (5) and (6) giving ∂f n ∂f + cµ = A21 − c [K12 − (K21 + K12 ) n] f , ∂t ∂x 2

(14)

and ∂n = C12 − (C12 + C21 + A21 ) n + c [K12 − (K21 + K12 ) n] ∂t

1

dµ f (µ) , (15) −1

respectively. The gray approximation is φ(ν) = 1/w for |ν − ν0 | ≤ w/2 and φ(ν) = 0 for |ν − ν0 | > w/2, where ν0 is the line center frequency and w is the line width. Both f and d depend only upon the angle and position, not on

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frequency, within the line. The line width, w, is factored out of the equations by suitably redeﬁning the ﬁelds. Making the transformation to the diﬀerence ﬁeld, the counterparts to (14) and (15) are ∂d(x, t; µ) ∂d(x, t; µ) + cµ = − c [K12 −(K21 +K12 ) n(x, t)] d(x, t; µ) ∂t ∂x ∂B(n(x, t)) ∂B(n(x, t)) − cµ , − ∂t ∂x

(16)

and ∂n(x, t) = C12 − (C12 + C21 ) n(x, t) ∂t + c [K12 − (K21 + K12 ) n(x, t)]

1

dµ d(x, t; µ) ,

(17)

−1

respectively. From these equations we develop three Monte Carlo methods based upon diﬀerent treatments of the source terms −∂B/∂t and −cµ ∂B/∂x.

3 Numerical Development Let us divide the slab into N zones. The zones are labeled from 1 through N from left to right with the position of the left edge of the ith zone labeled xi . We specify an extra point, xN +1 , to mark the position of the right-hand boundary of the slab. We consider n to be piece-wise constant in space within a zone, but allow it to vary continuously in time. Since B(n) behaves likewise, let us write Bi (t) as the value of B in the ith zone at time t. Further, for the purposes of this discussion, let us deﬁne B0 and BN +1 for the two boundary regions, representing the boundary conditions to the left and right of the slab respectively, in accordance with our treatment of boundary conditions in the diﬀerence formulation introduced in the previous section. Then we may write B(x, t) = B0 +

N +1

(Bi (t) − Bi−1 (t)) u(x − xi ) ,

(18)

i=1

where u(x) is the unit-step function we deﬁne as u(x) = 1 for x > 0, u(x) = 0 for x ≤ 0. Generally, the total Monte Carlo weight to be emitted from a source S is given by the integral (19) W = S dR , R

where R is the ﬁnite volume element of the relevant phase space used in the numerical model and dR is its inﬁnitesimal. For this model R is 2∆x∆t, and so we may write

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W =− t

µ=+1

dµ

dx

W x = −c

µ=+1

dµ µ=−1

dt

∆x

µ=−1

and

∆t

∂B , ∂t

dx

∆x

dt µ ∆t

∂B , ∂x

(20)

(21)

where the superscripts t and x indicate the weight emitted by the −∂B/∂t and the −cµ ∂B/∂x source, respectively. The probability distribution function of the physical variables to be sampled is given by g=

S , W

(22)

for a source S emitting weight W . We use these relations to develop the foundation for three Monte Carlo methods for solving the diﬀerence formulation for atomic line transport, (16) and (17). 3.1 Source Terms The spontaneous emission term, nA21 /2, in the standard formulation, (14), is replaced by two new source terms, namely −∂B/∂t and −cµ ∂B/∂x, in the diﬀerence formulation, (16). The new source terms play diﬀerent roles than the spontaneous emission term of the standard formulation. The −cµ ∂B/∂x term is responsible for driving the transport of the d ﬁeld through the slab, and the −∂B/∂t term acts to compensate for changes in the reference ﬁeld B(n) by changing the d ﬁeld in order to hold f ﬁxed. Source Term −∂B/∂t We evaluate (20) for a given zone i giving the weight to be accorded to the −∂B/∂t source term: Wit = −

µ=+1 dµ

t0 +∆t

dx

µ=−1 ∆xi

dt t0

∂B ∂t

= −2∆xi [Bi (t0 + ∆t) − Bi (t0 )] , (23) i

where we have used the piece-wise constant property of B in the integral over ∆xi . Now we may write the distribution function for the source in zone i using (22) (∂B/∂t)i . (24) git = 2∆xi [Bi (t0 + ∆t) − Bi (t0 )] Further development of the distribution function depends upon assumptions about the nature of the diﬀerencing employed and varies with our construction of the Monte Carlo methods we use for the diﬀerence formulation. We will address the details of our construction later in this work.

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Source Term −cµ ∂B/∂x Now let us consider the space-derivative term. Due to the piece-wise constant treatment of n, this source term is non-zero only at a discontinuity in the value of n between two adjoining zones or at a discontinuity between the surfaces of the slab and its surroundings. The derivative ∂B/∂x gives N +1 ∂B = (Bi (t) − Bi−1 (t)) δ(x − xi ) , ∂x i=1

(25)

where δ is the Dirac delta function. Since −cµ ∂B/∂x is an odd function of µ in slab geometry, the sum of the weight emitted from this source over all angles {θ : µ = cos θ} is zero. Nevertheless, it is not correct to ignore the source; −cµ ∂B/∂x is responsible for driving the transport of d particles through the slab. Our solution is to emit d-particle pairs of equal and opposite weight in +µ and −µ directions, thereby assuring that zero net weight is emitted without statistical noise. To ﬁnd the weight to be emitted, say in the +x direction in the ith zone, we integrate the −cµ ∂B/∂x source from µ = 0 to µ = 1 Wi+x = −c =−

c 2

µ=1

t0 +∆t xi +∆xi

µ dµ

dx δ(x − xi )[Bi (t) − Bi−1 (t)]

dt

µ=0 t0 t0 +∆t

xi

dt [Bi (t) − Bi−1 (t)] .

(26)

t0

Weight emitted in the −x direction is identical, except for a change in sign. Now we may write the distribution function for the source in the +x direction in zone i using (22) 2µδ(x − xi ) [Bi (t) − Bi−1 (t)] . gi+x = t0 +∆t dt [Bi (t) − Bi−1 (t)] t0

(27)

The presence of the δ-function tells us that the particles are to be emitted at √ zone boundaries. Emission angles are sampled according to µ = ρ, where ρ is a random variate uniformly distributed between 0 and 1. 3.2 Solution Methods We construct three diﬀerent Monte Carlo solution methods employing the diﬀerence formulation for the transport of an atomic line. In the construction we address the details of the treatment of the source terms. The solution methods utilize diﬀerent degrees of implicit treatment of the source terms, with each succeeding method being more implicit than the one before it.

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We begin by integrating (17) over a time step, approximating n(t) by n(t0 + ∆t) in the collision (pumping) terms and by n(t0 ) in the absorption term, giving n(x, t0 + ∆t) = n(x, t0 ) + [C12 − (C12 + C21 ) n(x, t0 + ∆t)] ∆t t0 +∆t 1 + c [K12 − (K21 + K12 ) n(x, t0 )] dt dµ d (x, t; µ) , (28) t0

−1

where t0 is the census time of the previous Monte Carlo integration cycle and t0 + ∆t is the census time of the current cycle. This intermediate step is the common point of departure for the three Monte Carlo solution methods. We would like to note that for the diﬀerence formulation the source terms of the transport equation, (16), do not appear in the material response equation, (17), and are likewise absent in (28). This is in contrast to (14) and (15), where the spontaneous emission term, A21 n, appears in both the transport and the material response equation, causing stiﬀ coupling between them. The self-consistent diﬀerencing of the spontaneous emission term in (5) and (6), for the purpose of stability, leads to eﬀective scattering in the IMC method discussed in [BF86], and the linear system solve in the SIMC method discussed in [Bro89]. The Explicit Solution Method In this method there is no implicit treatment of the sources and, unlike SIMC, it does not require the inversion of a matrix at the end of each Monte Carlo integration cycle in order to calculate ni (t0 + ∆t). In this scheme −cµ ∂B/∂x is explicitly diﬀerenced at time t0 , and the action of −∂B/∂t is delayed until the end of the integration loop, at which point n(t0 + ∆t) is available. Starting with the −∂B/∂t source for zone i, we approximate ∂B Bi (t0 + ∆t) − Bi (t0 ) . (29) ≈ ∂t i ∆t Substituting this result into (24) gives git =

1 , 2 ∆xi ∆t

(30)

which directs us to distribute the weight given in (23) evenly within the zone. Considering the −cµ ∂B/∂x source, we take Bi (t) → Bi (t0 ), the value of Bi at the beginning of the time interval. Substituting this into (26) gives Wi+x = −

c ∆t [Bi (t0 ) − Bi−1 (t0 )] , 2

for emission in the +x-direction. And (27) becomes

(31)

An Evaluation of the Diﬀerence Formulation

gi+x =

2µδ(x − xi ) , ∆t

293

(32)

where the weight of the source is to be distributed evenly throughout the time interval ∆t, but the emission is to take place on the zone boundary xi . At the beginning of each iteration of the Monte Carlo integration loop, difference particles from the −cµ ∂B/∂x source are emitted at the zone boundaries using B(t0 ), and distributed uniformly in time across the time step interval ∆t. The particles are then propagated to census time, t0 + ∆t, according to (16) before n(t0 + ∆t) is calculated. To obtain ni (t0 + ∆t), write (28) as ni (t0 + ∆t) = γni (t0 ) + γC12 ∆t +

γc [K12 − (K21 + K12 )ni (t0 )]Di , (33) ∆xi

where i is the zone index, ∆xi is the width of the zone, γ= and where

1 , [1 + (C12 + C21 )∆t]

Di =

dt

∆t

1

dx ∆xi

(34)

dµ d (x, t; µ)

(35)

−1

is the time integral of the d-ﬁeld calculated from Monte Carlo particles traveling through zone i during the time step. Now that we have an estimate of ni (t0 +∆t) from (33), diﬀerence particles sample the −∂B/∂t source with the total weight given by (23) and are evenly distributed in space within a zone. This emission is not evenly distributed across the time step, it has a time coordinate of t0 + ∆t, the census time of the current integration interval. This sequence is then repeated for the next time step. The Semi-Implicit Solution Method We call this method “semi-implicit” because we implicitly diﬀerence the −∂B/∂t source term, but explicitly diﬀerence the −cµ ∂B/∂x source term. The implicit diﬀerencing of the −∂B/∂t source term leads to a matrix solve at the end of each iteration of the Monte Carlo integration loop. Our purpose in examining this method is to provide insight into the sources of numerical instability of the fully explicit method described previously. Once one must pay the cost of the non-linear matrix solve, one might as well extract the beneﬁts of a fully implicit solution method. In this semi-implicit treatment of the source terms for the diﬀerence formulation, the emission from −∂B/∂t is calculated at the start of the integration loop, not postponed until the end as in the explicit scheme just discussed. The weight emitted is given by (23). However, Bi (t0 + ∆t) is unknown at this point, and a portion of the weight, −2∆xi , is “symbolic” in

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the same vein as in [Bro89] and requires a non-linear matrix solve at the end of each Monte Carlo integration cycle. The remaining portion, 2∆x Bi (t0 ), contains no unknown factors and provides known (numeric) contributions to the d-ﬁeld. The particles are created with time coordinates uniformly distributed over the interval ∆t, as dictated by (30), in contrast to the explicit case where their time coordinates are set to census time. Next, the −cµ ∂B/∂x source is sampled in the same way as in the explicit treatment above, and d-particles with weight given by (31) are created. These particles, fully numeric contributions to the d-ﬁeld, are distributed in space, time and direction according to (32). This treatment of the source terms leads to the following representation of (28): ni (t0 + ∆t) = γni (t0 ) + γC12 ∆t +

⎡

N

⎤

γc [K12 − (K21 + K12 )ni (t0 )] × ⎣Di + Dij Bj (t0 + ∆t)⎦ . (36) ∆xi j=1

Here Dij is the symbolic contribution to the ith zone from particles born in the j th zone, and Di is the contribution to zone i coming from particles with numeric weights in much the same way as in (33), including the numeric contributions from −∂B/∂t and −cµ ∂B/∂x sources. The sole contributor to symbolic energy depositions is the forward-diﬀerenced portion of −∂B/∂t. Since B is non-linear in n, (36) represents a non-linear system that must be solved for nj (t0 +∆t) at the end of each cycle through the integration loop, whereas the equivalent expression in [Bro89] represented a system linear in n. We iterate the Newton-Raphson algorithm to solve the non-linear system for nj (t0 +∆t), and then use B(nj (t0 +∆t)) to convert the Monte Carlo particles with symbolic weights to numeric weights. In this way the d-ﬁeld, at census, is fully numeric and free of unknown factors before the next iteration of the Monte Carlo integration loop. The Implicit Solution Method We call this method “implicit” because we treat both the −cµ ∂B/∂x and the −∂B/∂t source terms implicitly in time. By taking Bi (t) → Bi (t0 + ∆t), instead of Bi (t) → Bi (t0 ) as in the last two methods introduced above, (26) becomes c ∆t [Bi (t0 + ∆t) − Bi−1 (t0 + ∆t)] . (37) Wi+x = − 2 Equation (32) remains unchanged. The sequence of calculations in the integration loop is similar to that used in the semi-implicit method above. First, particles sampling the −∂B/∂t source are emitted with a portion of their weight numeric, −2∆xi Bi (t0 ), and

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the remainder symbolic, −2∆xi , according to (23) and exactly like the semiimplicit method. Next, −cµ ∂B/∂x is sampled according to (32), but in this case the weight is purely symbolic. We write (28) as γc [K12 − (K21 + K12 )ni (t0 )] ni (t0 + ∆t) = γni (t0 ) + γC12 ∆t + ∆xi ⎫ ⎧ N N +1 ⎬ ⎨ t x Dij Bj (t0 + ∆t) + Dik [Bk (t0 + ∆t) − Bk−1 (t0 + ∆t)] × Di + ⎭ ⎩ j=1

k=1

(38) x , of the −cµ ∂B/∂x where we introduce the new symbolic contribution, Dik source emitted from zone k and propagated to zone i, where [Bk (t0 + ∆t) −Bk−1 (t0 + ∆t)] is the factor necessary to convert that symbolic weight into numeric weight. One may consider the last summation as one over zone interfaces while remembering that the index k in Bk and Bk−1 refers to zone t represents the symbolic contribution of the −∂B/∂t indices. The term Dij source, the Di term includes the numeric contributions from −∂B/∂t sources, and Bj (t0 + ∆t) plays the same role in this equation as it does in (36). B0 and BN +1 are prescribed boundary conditions.

4 Numerical Results in the Gray Approximation We select the SIMC solution method in the standard formulation as a point of comparison for the diﬀerence formulation [SB05]. We discuss the numerical accuracy and eﬃciency, and report on the numerical stability of each of the three Monte Carlo solution methods we developed in the previous section, with emphasis on exploring the stability characteristics of the fully explicit version, itself free of a matrix solve at the end of each integration cycle, relative to the SIMC treatment of the standard formulation for a range of optical thicknesses. We do not address the issue of teleportation error [MBS03] in this work. For the sake of brevity, we refer to each of the Monte Carlo solution methods we developed above for the diﬀerence formulation for atomic line transport as one of a trio of “diﬀerence methods” and to SIMC for the standard formulation as the “standard method.” The problems were run until equilibrium was reached. 4.1 Relative Accuracy and Eﬃciency Table 1 lists the parameters describing the initial and boundary conditions and the material parameters we use in comparing the SIMC method (standard method) for the standard formulation to each of the three Monte Carlo

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Table 1. Input parameters used for all Monte Carlo solution methods describing the initial conditions, boundary conditions, and material properties of the unit slab Parameter

Nominal Optical Depth =1

n(0 ≤ x ≤ 1, t = 0) f (x, µ, t = 0) n(x < 0, t) n(x > 1, t) A21 K12 K21 C12 C21

.25 0 0 0 10 1.125 1.125 0.245423 0.667128

Value Nominal Nominal Optical Depth Optical Depth = 10 = 100 .25 0 0 0 10 18 15.3422 0.245423 0.667128

.25 0 0 0 10 207.5 207.5 0.245423 0.667128

Nominal Optical Depth = 1000 .25 0 0 0 10 2155 2155 0.245423 0.667128

methods (diﬀerence methods) developed in the previous section for the difference formulation. In all calculations the slab is initialized with n = 0.25 for all zones and the photon ﬁelds f (t = 0) = 0. This initial state for the photon ﬁeld, f , corresponds to a non-zero initial diﬀerence ﬁeld, d, which must be sampled to properly initialize the system. While this provides a net zero photon ﬁeld in each zone, the statistical nature of sampling leads to small, local ﬂuctuations. As we will show, this in turn can lead to diﬀerences among the methods in their transient behavior, even in the limit of short time steps. The optical depth for this model depends on the value of n(x, t). We ﬁrst tune the input parameters using the standard method to obtain the desired nominal optical depth, then use the same input values for the diﬀerence methods. In order to more faithfully reproduce the boundary layer near the edges of the slab for thicker problems, we ﬁnd it necessary to modify the zoning, depending upon optical thickness, and this can inﬂuence execution time. Since the gradient of n in space varies slowly and more uniformly over the length of the slab in the thin problems – optical thicknesses 1 and 10 – we model the slab using 21 zones of uniform size in thin systems. However, for thick problems – optical thicknesses of 100 and 1000 – gradients in n are concentrated in the boundary layers. For these we use small zones in the boundary layers and increase their size in a geometric progression towards the center of the slab. Thus, we can compare accuracy and eﬃciency among methods for a given optical thickness only. Relative Accuracy Table 2 demonstrates the accuracy of the three Monte Carlo solution methods relative to the SIMC method for a simple, two-level, system in slab geometry. The data consist of the means and standard deviations of 120 statistically

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297

Table 2. The means and standard deviations of the optical thickness of a unit slab calculated using each of the three diﬀerence methods and the standard method at equilibrium. All calculations are matched in execution time Optical Thickness

Diﬀerence Method: Explicit

1 10 100 1000

1.0087 ± 2 × 10−4 10.067 ± 1 × 10−3 98.655 ± 1 × 10−3 998.726 ± 1 × 10−3

Diﬀerence Method: Semi-Implicit

Diﬀerence Method: Implicit

Standard Method

1.0088 ± 1 × 10−4 1.0087 ± 1 × 10−4 1.00873 ± 8 × 10−5 10.067 ± 1 × 10−3 10.0671 ± 9 × 10−4 10.067 ± 4 × 10−3 98.654 ± 1 × 10−3 98.653 ± 1 × 10−3 98.65 ± 8 × 10−2 998.726 ± 1 × 10−3 998.7263 ± 9 × 10−4 998.7 ± 9 × 10−1

independent calculations of the optical thickness of the slab for each method, in equilibrium, for the ﬁxed input parameters shown in Table 1. The results show that the means of the calculated optical thicknesses are within one standard deviation of each other. Therefore, the results are statistically consistent with the assertion that all three Monte Carlo solution methods in the diﬀerence formulation converge to the same result as SIMC in the standard formulation in equilibrium. It is interesting to note that the standard deviations of the diﬀerence methods are approximately independent of optical depth, whereas those of SIMC increase several orders of magnitude as optical depth increases. Relative Eﬃciency The variance in a Monte Carlo calculation scales inversely with the number of particles used, in the limit of large particle count. We use this fact as a means to evaluate the relative eﬃciency of the methods for a given discretization of the problem. We match the run-times among the methods by adjusting the number of Monte Carlo particles used in each, taking care to ensure that the Monte Carlo eﬀort dominates the calculation and that the variance scales appropriately with the number of Monte Carlo particles. Then the variances of the calculations are inversely proportional to the relative eﬃciencies of the methods. This is how we estimate the run-time advantage of the diﬀerence methods over the standard method. Table 3 consists of the variances of the optical depths presented in Table 2, and Table 4 shows the calculated speed-up factors, based upon the measurements in Table 3. All three diﬀerence methods show a clear run-time advantage over the standard formulation for thick systems. The advantage is striking at an optical depth of 1000 mean free paths. However, as Table 4 shows, for thin problems the advantage diminishes and is lost completely somewhere between optical depths of 10 and 1, corresponding to a per-zone optical depth of 0.5 and 0.05, respectively. For thick systems, the desired statistical accuracy is achieved with a much lower particle count.

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Table 3. The variances of the optical thickness of a unit slab calculated using the three diﬀerence methods and the standard method at equilibrium Nominal Optical Thickness 1 10 100 1000

Diﬀerence Method: Explicit 3.0 × 10−8 1.1 × 10−6 1.1 × 10−6 9.6 × 10−7

Diﬀerence Method: Semi-Implicit 1.5 × 10−8 1.0 × 10−6 1.2 × 10−6 1.0 × 10−6

Diﬀerence Method: Implicit 1.4 × 10−8 8.2 × 10−7 9.3 × 10−7 6.0 × 10−7

Standard Method 6.0 × 10−9 1.8 × 10−5 6.9 × 10−3 8.7 × 10−1

Table 4. Speed-up factors of the three diﬀerence methods over the standard method for various nominal optical thicknesses Nominal Optical Thickness 1 10 100 1000

Diﬀerence Method: Explicit 2.0 × 10−1 1.6 × 101 6.3 × 103 9.1 × 105

Diﬀerence Method: Semi-Implicit 4.0 × 10−1 1.8 × 101 5.8 × 103 8.7 × 105

Diﬀerence Method: Implicit 4.3 × 10−1 2.2 × 101 7.4 × 103 1.5 × 106

4.2 Transient Behavior of the Diﬀerence and Standard Formulations While we do not expect two diﬀerent solution methods, such as the standard method and any one of the three diﬀerence methods, to behave identically during the ﬁrst few time steps of the integration, we do expect their behaviors to converge for suﬃciently small time step sizes. This is indeed the case. Figures 1 through 6 show the transient behavior of n(t), the fraction of atoms in the excited state, in the central zone of a slab with an optical thickness of 10 mean free paths at equilibrium. Figures 1 through 4 show the transient behavior of each of the three diﬀerence methods, with Figs. 5 and 6 showing the transient behavior of the standard method. Each ﬁgure shows graphs of n(t) calculated with diﬀerent time step sizes in units of (slab length)/c, where c is the speed of light in the material (set to 1 in this work). Initially the slabs have uniform excitation energies corresponding to n(x, t = 0) = 0.25, but there are no photon ﬁelds. At the start, the radiation ﬁeld and the material energy are out of equilibrium, with n falling initially in order to bring about radiative equilibrium. The motion of n is then driven by the net collisional excitation and absorption, recovering on a longer time scale. Each of the diﬀerence methods and the standard method show this behavior and agree qualitatively. Note the overshoot in the standard formulation and explicit implementation of the diﬀerence formulation for long time steps in Figs. 2 and 6. One can see that while similar, the

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overshoot for the standard implementation is more pronounced. The quantitative agreement among the methods improves with decreasing time step sizes, and Fig. 7 shows good overlap for a time step size of 0.00625. Aside from the noise apparent in the standard method as the magnitude of the photon ﬁeld grows, there is a small but discernible diﬀerence in the minimum of n(t) among the four methods. We believe this is due to sampling noise. Recall that initializing the photon ﬁeld to zero in the diﬀerence formulation requires sampling d(t = 0) so that f = d + B(t = 0) = 0. Statistical ﬂuctuations in the Monte Carlo sampling of the physical coordinates of the

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particles composing this initial d-ﬁeld leads to small, localized ﬂuctuations that can aﬀect n shortly after t = 0. Table 5 shows the average and one standard deviation of the minimum n reaches for 200 statistically independent calculations using each of the three diﬀerence methods and the standard method, all matched in execution time. The time step size used in each calculation is 0.00625, the same as in Fig. 7. Also shown are the average and standard deviation of the times at which n reached its nadir in the calculations. Table 5 shows that the three diﬀerence methods and the standard method produce minima of the same magnitude

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and at the same time, within the estimated uncertainties. Thus we show that not only do the diﬀerence methods agree with the standard method in equilibrium (see Table 2) they also agree in the transient behavior of n for suﬃciently small time step sizes. 4.3 Numerical Stability of the Diﬀerence Formulation We explore the stability characteristics of the three diﬀerent treatments of the source terms in the diﬀerence formulation for line transport. Of particular interest is the numerical stability of the explicit treatment, since it is free

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Table 5. The mean and standard deviation of the minimum n and the time of its nadir. Quantities were calculated using the three versions of the diﬀerence method and the standard method Monte Carlo Solution Methods

Diﬀerence: Explicit Diﬀerence: Semi-Implicit Diﬀerence: Implicit Standard Method

Minimum n 0.204 ± 0.001 0.204 ± 0.001 0.204 ± 0.001 0.2041 ± 0.0001

Time of Nadir of n 0.197 ± 0.008 0.196 ± 0.008 0.196 ± 0.008 0.193 ± 0.004

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of a matrix solve in the Monte Carlo integration cycle and will thus remain economical as the number of zones in the problem increases. We ﬁnd the implicit treatment, (38), for the diﬀerence formulation to be numerically stable for optical thicknesses ranging from 1 to 1000, even for time step sizes on the order of 10 light travel times across the slab, and we expect the treatment to remain stable for thicker systems. This provides numerical evidence that this treatment of the source terms is unconditionally stable. We ﬁnd that both the explicit and the semi-implicit treatments, (33) and (36), respectively, are only conditionally stable. For these treatments of the source terms, stability depends upon the optical depth of the slab, the size of the zones, and the size of the time step. Figures 8 and 9 show the approximate neighborhood of the onset of instability for both treatments. The methods are numerically unstable in the regions above their graphs. Beyond a certain optical thickness, the systems become stable for practically any time step size, so the graphs terminate. The calculations were run until the systems were well equilibrated, with unstable calculations identiﬁed when we observed the characteristic, geometrically growing oscillation about equilibrium with a period of 2∆t in the graphs of n(t).

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Whereas the explicit diﬀerencing of the standard formulation is known to be stable for thin and unstable for thick systems [FC71], we ﬁnd the contrary for the semi-implicit and fully explicit diﬀerence methods. Thus, in

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the explicit treatment of the diﬀerence formulation it appears that we trade numerical stability in thin systems for numerical stability in thick systems. Figures 8 and 9 show that the regions of stability for both the explicit and the semi-implicit methods are similar in shape. It is apparent in both ﬁgures that the explicit treatment requires shorter time steps in order to obtain stability. For thin systems the stability of both treatments is insensitive to the zone size, as shown in Fig. 9. For thick systems the constraint on the time step size in order to obtain stability is relaxed as the zone size is increased. Both ﬁgures demonstrate that the optical thickness of the zones is an important factor in the stability of the calculations. It is interesting to note the weakness of the dependence of both the semiimplicit and explicit treatments of the source terms on the zone size, ∆x, for thin systems. Terms in the ﬁnite diﬀerence equations, (33) and (36), that depend upon zone size have little apparent inﬂuence upon the stability of those solution methods. Additionally, since both treatments of the source terms have similar regions of stability, a formal stability analysis of the simpler explicit formulation may give insight into the stability criterion of the more complicated, semi-implicit method. For the slab geometry, collisional-pumped, line-trapping problems studied here, the explicit treatment of the source terms, unencumbered by a nonlinear system solve at each time step, appears no more economical than the semi-implicit method, which is more stable. One should consider, however, that the cost of the non-linear system solve grows rapidly as one scales the number of zones in the problem. Further, while the implicit scheme demonstrates superior stability characteristics, it too relies upon a non-linear system solve at each time step. The primary diﬀerence between the conditionally stable semi-implicit method and the unconditionally stable implicit method is in the treatment of the −cµ ∂B/∂x-term. In addition, since the −∂B/∂tterm is explicitly treated in the explicit method and implicitly treated in the semi-implicit method without a great diﬀerence in the stability regions for the two, we believe that the explicit diﬀerencing of the −cµ ∂B/∂x-term is responsible for driving the numerical instability.

5 Concluding Remarks In this paper we examined the accuracy and performance of the diﬀerence formulation [SB05] relative to the Symbolic Implicit Monte Carlo (SIMC) [Bro89] solution method applied to the standard formulation of photon transport in a strongly absorbing/emitting two level system using the gray approximation. We developed three diﬀerent numerical treatments of the diﬀerence formulation and presented evidence of their superior computational eﬃciency for thick systems. We found that to an equivalent noise ﬁgure, the diﬀerence methods were 106 times faster than the standard method for slabs 1000 mean

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free paths thick, or equivalently, provide a 103 reduction in Monte Carlo noise for a given execution time. We demonstrated that the three implementations of the diﬀerence formulation we developed were in excellent agreement with the SIMC implementation of standard formulation. Additionally, we showed through a detailed comparison that while their transient behavior diﬀers for large time steps there is good numerical evidence that all the treatments of the source converge for suﬃciently small time steps. We found that the fully implicit version of the diﬀerence formulation is stable, and we believe it to be unconditionally so. The fully explicit version, although free of any matrix solve, is only conditionally stable. Moreover, it possess a stability region similar to the semi-implicit diﬀerence method which may provide insight into a formal stability analysis. For both conditionally stable versions of the diﬀerence formulation, stability appears to depend strongly upon the optical thickness of the zones dividing the material. Finally, we believe that it is the explicit treatment of the −cµ ∂B/∂x term that drives the instability in the explicit diﬀerence method. As a ﬁnal note, the explicit treatment of the source terms in the standard formulation is stable in the limit of optically thin systems, while the explicit source term treatment of the diﬀerence formulation is stable in the limit of optically thick systems. This leaves open the possibility that the non-linear matrix solve might be avoided when applying the diﬀerence formulation to practical problems involving thick media.

References [BF86]

Brooks, E.D., III, Fleck, J.A., Jr.: An Implicit Monte Carlo Scheme for Calculating Time-Dependent Line Transport. Journal of Computational Physics, 67, 1, 59–72 (1986) [Bro89] Brooks, E.D., III: Symbolic Implicit Monte Carlo. Journal of Computational Physics, 83, 2, 433–446 (1989) [DL04] Densmore, J.D., Larsen, E.W.: Asymptotic Equilibrium Diﬀusion Analysis of Time-Dependent Monte Carlo Methods for Grey Radiative Transfer. Journal of Computational Physics, 199, 175–204 (2004) [FC71] Fleck J.A., Jr., Cummings, J.D.: An implicit Monte Carlo scheme for calculating time and frequency dependent non linear radiation transport. Journal of Computational Physics, 8, 313–342 (1971) [MBS03] M.S. McKinley, M.S., Brooks, E.D., III, Sz˝oke, A.: Comparison of Implicit and Symbolic Implicit Monte Carlo Line Transport with Frequency Weight Extension, Journal of Computational Physics, 189, 330– 349 (2003) [Mih78] Mihalas, D.: Stellar Atmospheres. W.H. Freeman and Company, San Francisco (1978) [NKa91] N’Kaoua, T.:Solution of the Nonlinear Radiative Transfer Equations by a Fully Implicit Matrix Monte Carlo Method Coupled with the Rosseland

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F. Daﬃn et al. Diﬀusion Equation Via Domain Decomposition. SIAM Journal Scientiﬁc and Statistical Computing, 12, 3, 505–520 (1991) N’Kaoua, T., Sentis, R.: A New Time Discretization for the Radiative Transfer Equations: Analysis and Comparison with the Classical Discretization. SIAM Journal on Numerical Analysis, 30, 733–748 (1993) Sz˝ oke, A., Brooks, E.D., III: The Transport Equation in Optically Thick Media. Journal of Quantitative Spectroscopy and Radiative Transfer, 91, 95–110 (2005)

Non-LTE Radiation Transport in High Radiation Plasmas Howard A. Scott Lawrence Livermore National Laboratory, P.O. Box 808, L-18, Livermore, CA 94551, USA [email protected] Abstract. A primary goal of numerical radiation transport is obtaining a selfconsistent solution for both the radiation ﬁeld and plasma properties, which requires consideration of the coupling between the radiation and the plasma. The diﬀerent characteristics of this coupling for continuum and line radiation have resulted in two separate sub-disciplines of radiation transport with distinct emphases and computational techniques. LTE radiation transfer focuses on energy transport and exchange through broadband radiation, primarily aﬀecting temperature and ionization balance. Non-LTE line transfer focuses on narrowband radiation and the response of individual level populations, primarily aﬀecting spectral properties. Many high energy density applications, particularly those with high-Z materials, incorporate characteristics of both these regimes. Applications where the radiation ﬁelds play an important role in the energy balance and include strong line components require a non-LTE broadband treatment of energy transport and exchange. We discuss these issues and present a radiation transport treatment which combines features of both approaches by explicitly incorporating the dependence of material properties on both temperature and radiation ﬁelds. The additional terms generated by the radiation dependence do not change the character of the system of equations and can easily be added to a numerical transport implementation. A numerical example from a Z-pinch application demonstrates that this method improves both the stability and convergence of the calculations. The information needed to characterize the material response to radiation is closely related to that used by the Linear Response Matrix (LRM) approach to near-LTE simulation, and we investigate the use of the LRM for these calculations.

1 Introduction Radiation transport methods have been applied to a wide variety of physical systems over the last several decades. Solution techniques are now suﬃciently well developed to allow routine usage of multidimensional simulations in the ﬁelds of astrophysics and high energy density physics. In particular, applications in two regimes have been well studied in these ﬁelds. Energy transport by radiation in high temperature plasmas in local thermodynamic equilibrium (LTE) is an important, if not dominant, phenomenon in applications such as stellar interiors and inertial conﬁnement fusion (ICF), in which the radiation interacts with matter and transports energy over a wide range of frequencies. The importance of this broadband radiation transport has led to

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a great deal of work over the last few decades, resulting in the development of very eﬀective numerical schemes [22]. The situation is similar for transport of line radiation in non-LTE applications such as stellar atmospheres and laboratory plasma spectroscopy. In the regime of interest for these applications, a very narrow frequency range of radiation interacts with the matter, altering the populations of atomic energy levels, ultimately aﬀecting the spectrum emitted by the matter. Eﬀective numerical algorithms now allow the solution of extremely large and complex systems [Kal84,MO98]. Despite the successes achieved in these two ﬁelds, much work remains in the development of eﬃcient, robust and general solution methods. Non-LTE materials in high radiation ﬁelds, common in high energy density physics applications, respond strongly to radiation in both their energy content and spectral characteristics. The motivation behind this paper is the extension of a class of commonly used numerical radiation transport methods to handle such applications. In Sect. 2, we consider the non-LTE energetics of radiation interacting with matter, determining the relationship between energy density and temperature. The generalization of the LTE relationship separates the dependence of the energy density on the temperature from the direct eﬀect of radiative interactions. A numerical example then identiﬁes the regime in which the radiation spectrum can signiﬁcantly alter the material response. The equations and solution methods considered in this paper are presented in Sect. 3. The set of equations used for broadband radiation transport and material energetics is given in Sect. 3a, along with a straightforward solution scheme. Similarly, Sect. 3b presents the equations used for line radiation transport, along with a very common solution technique. For each of these cases, we brieﬂy discuss those aspects of the physical situation and numerical techniques that contribute to the eﬀectiveness of the solution methods. The emphasis here is on the coupling of the radiation to the material, and on the self-consistent solution for the radiation ﬁeld and material properties. The discussion leads naturally to an extension of the broadband algorithm which explicitly incorporates the dependence of material properties on radiation. The extended algorithm is presented and discussed in Sect. 3c, along with a low-density approximation used to calculate the additional material response terms. Both the original solution scheme for the broadband equations and the extended algorithm use linearization to incorporate material response information. A majority of the discussion and conclusions contained in this paper should apply to any solution scheme that shares this characteristic. Solution techniques that do not explicitly incorporate response information, e.g. those depending on a robust non-linear solver, will not beneﬁt directly, but the insights aﬀorded by this work should still be valuable.

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Since our interest here is in handling the coupling between radiation and matter, we do not explicitly consider the numerical solution of the radiative transport equation itself. We will actually use a diﬀusion operator rather than a true transport operator for numerical work, with the assumption that this does not alter the character of the radiation-matter coupling. Section 4 presents results from a test application based upon a dynamic hohlraum, for both the standard broadband algorithm and for the extended algorithm. The additional terms required by the extended algorithm are very expensive to compute, and add to the cost of the already-expensive non-LTE calculations. In Sect. 5, we address the possibility of using tabular information for the material properties and responses, which would greatly decrease the cost of the non-LTE simulations. The linear response matrix (LRM) approach to near-LTE simulations [MK01] tabulates response information similar to that required by the extended algorithm, and we consider a generalization of this approach that would be suitable for our purposes. All numerical results presented in this paper use material properties calculated self-consistently with the radiation ﬁeld by a collisional-radiative model [Sco01]. The electronic structure and transition rates are calculated using a modiﬁed screened-hydrogenic atomic model very similar to that described in [CC05].

2 Non-LTE Energetics For matter that is not in LTE, describing the response to radiation is more complicated than for the corresponding LTE case. We consider the relationship between the material energy density Em , the material properties and the radiation ﬁeld: Em =

3 (ne + ni )kT + Eint (T, Jν , t) 2

(1)

Here, ne (ni ) is the number density of the free electrons (ions), which are assumed to have a thermal distribution corresponding to the material temperature T . Eint is the material internal energy, which depends not only on the temperature and density, but also on the radiation ﬁeld, denoted by Jν , and on the time t. For the remainder of this paper, we ignore the density dependence as unimportant to the discussion and focus on the temperature and radiation. We also adopt a single-temperature description of the material for simplicity of exposition. For material in LTE, the internal energy depends only on temperature, then the rate of change in material energy density and temperature are related through the speciﬁc heat (at constant density) cV : ∂Em dEm E dT LT E = cLT , c = (2) V V dt dt ∂T Jν =Bν

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Implicit in this formulation is the assumption that either radiative interactions are completely unimportant or that the extant radiation also has a thermal distribution, i.e. Jν = Bν , where Bν is the Planck distribution. In the more general non-LTE formulation, the rate of change of material energy density is comprised of three diﬀerent types of terms: ∂Em ∂Jν ∂Em ∂Em dEm ∂T = + + (3) dt ∂T Jν ∂t ∂Jν T ∂t ∂t Jν ,T ν The ﬁrst term on the RHS of (3) describes the response of the material energy density to a change in temperature, but with ﬁxed radiation densities, while the second term describes the material response to a change in radiation at ﬁxed temperature. The coeﬃcient of the ﬁrst term plays the part of the non-LTE speciﬁc heat, which is related to the LTE speciﬁc heat by ∂Em ∂Bν ∂Em LT E N LT E N LT E , cV = cV + = (4) cV ∂Jν T ∂T ∂T Jν ν The last term on the RHS of (3) arises from evolution of the material at ﬁxed temperature and radiation, and acts as a source or sink of energy. This term can be quite important in following the thermal evolution of matter at very low densities and temperatures, but for the remainder of this discussion we assume this term is negligible and do not consider it further. Non-LTE eﬀects will become signiﬁcant at densities low enough for important radiative transition rates to become comparable to the corresponding collisional rates. A numerical example illustrates the relative importance of the temperature and radiative responses to the speciﬁc heat. For this example, we calculate the speciﬁc heat of a Lu plasma at three diﬀerent densities. Figures 1a—1c show the speciﬁc heat as a function of temperature for ion number densities of 1018 , 1020 and 1022 cm−3 , respectively. In each ﬁgure, E , while the thin solid line the thick solid line gives the LTE speciﬁc heat, cLT V N LT E evaluated assuming a Planckian ragives the non-LTE speciﬁc heat cV LT E diation ﬁeld at the given temperature, and the thin dashed line gives cN V evaluated assuming no radiation ﬁeld. The dotted line gives the speciﬁc heat obtained by evaluating (4) using approximate values for the derivatives with respect to Jν . This low-density approximation (the “diagonal” approximation) will be explained in Sect. 3.3. At the highest of the three densities, LTE is a good approximation and the speciﬁc heat varies little with the radiation. As the density decreases, E LT E and cN increases, and it becomes apparent the diﬀerence between cLT V V that the material radiative response dominates the temperature response. Regardless of the other considerations in this paper, use of the LTE speciﬁc heat at low densities in the presence of non-Planckian radiation ﬁelds will not describe the material energetics correctly.

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3 Radiation Transport In the ﬁrst two parts of this section, we consider salient features of numerical radiation transport algorithms as employed in common applications for both broadband (continuum) and line radiation. For both these cases, we list the basic set of equations to be solved, i.e. the radiative transfer equation together with the appropriate material equation(s), and brieﬂy discuss certain

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aspects of their solution. The emphasis here is on those aspects critical to constructing a self-consistent solution method with reasonable convergence properties. Although these subsections repeat information broadly available in the literature, they provide the components for the last subsection, which proposes an extension of the broadband algorithm that incorporates features from both cases. 3.1 Broadband Radiation Transport The system of equations describing energy transport by broadband radiation is comprised of the radiation transport equation 1 ∂Iν → + Ω •∇Iν = −(αν Iν − ην ) = −αν (Iν − Sν ) c ∂t and the material energy equation ∂E = 4π αν (Jν − Sν ) dν + Q ∂t

(5)

(6)

where Iν is the speciﬁc intensity at frequency ν, αν and η ν are the absorption coeﬃcient and emissivity, Q represents other energy sources, Jν is the angleaveraged intensity 1 Iν dΩ (7) Jν = 4π and Sν = ην /αν is the source function. In LTE, the source function is the Planck function, Bν , and is a function of temperature only. A common method of solving this non-linear set of equations is to discretize in time and linearize about the current temperature T 0 . Applying these operations to (6) gives αν (Jν − Sν0 ) dν + Q∆t (8) cT (T − T 0 ) = ∆t

where cT = cV + ∆t

αν

∂Sν dν ∂T

(9)

acts as a speciﬁc heat for the total system of matter and radiation. Equation (8) can be analytically combined with the linearized and discretized version of (5), resulting in 1 Iν − Iν0 → ∂Sν + Ω •∇Iν = −αν (Iν − Sν0 ) + αν (T − T 0 ) c ∆t ∂T ∆t ∂Sν αν (Jν − Sν0 )dν (10) αν = −αν (Iν − Sν0 ) + cT ∂T ∆t ∂Sν + Q αν cT ∂T

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We have assumed a fully-implicit time discretization, since applications usually require at least a partially-implicit treatment for stability. The superscript “0” denotes values at the beginning of the time interval. This treatment can be, and in LTE often is, generalized to an iterative procedure to converge the nonlinear dependence of Sν on the temperature. The integral terms in (10) couple intensities for all angles and all frequencies. Our primary interest here is the coupling of the diﬀerent radiation frequencies to the material, so we simplify the equations further by eliminating the angular dimensions. Replacing the transport operator with a diﬀusion operator through the substitutions →

Iν ← Jν , Ω •∇Iν ← −∇ • Dν ∇Jν

(11)

in (10), where Dν = c/3αν is the diﬀusion coeﬃcient, simpliﬁes both the analysis while leaving the essential aspects of coupling between radiation and material unchanged. The resulting multigroup radiation diﬀusion equations converge very slowly under a straightforward iterative procedure, but can be made to converge quickly upon application of multifrequency-grey acceleration [MLM85]. As discussed in this paper, a stability analysis reveals that without acceleration, short wavelength modes can be marginally convergent. The grey acceleration operator provides much faster convergence, eliminating the error in these modes by transporting a correction term with a spectral shape determined by the absorption coeﬃcient spectrum. The correction accounts for redistribution of radiation into optically thin high frequencies, where it can propagate freely. The eﬃciency of grey acceleration does depend upon the spectrum of the absorption coeﬃcient. If the absorption coeﬃcient falls oﬀ too quickly with frequency¯, the acceleration degrades or fails. However, physically realistic continuum opacities tend to fall of as ∼ν −3 at high frequencies, slowly enough for grey acceleration to be eﬀective. 3.2 Line Radiation Transport In this case, the radiation transport equation (5) is combined with the atomic kinetics rate equation dy = Ay (12) dt where the vector y represents the population densities of the atomic levels and A is the rate matrix. The total rate Aij connecting two atomic levels i and j includes collisional and radiative transitions, both discrete (boundbound) and continuous (free-bound). The populations respond to the radiation through the eﬀects on the transition rates. The prototypical example of this type of system is the steady-state twolevel atom [Mih78], consisting of two levels connected by a single discrete

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radiative transition and a collisional transition. The source function for such a system, obtained under the approximation that the width of the spectral line is very narrow (and assuming complete redistribution), has the form: Sν =

3 3 2hνij 2hνij yj ¯ = Sij (J) c2 [yi − (gj /gi ) yj ] c2

(13)

where νij is the transition frequency and gi is the statistical weight of level i. The frequency-independent source function Sij depends on the radiation ﬁeld only through the angle-integrated, frequency-averaged quantity ∞ ¯ J= Jν φ (v) dv (14) 0

which enters into the transition rate, where φ is the line proﬁle function. For this simple system, Sij has the form ¯ = (1 − ε)J¯ + εBij Sij (J)

(15)

where Bij is the non-dimensional Planck function at the transition energy. ε is determined by the ratio of the collisional and spontaneous radiative transition rates and can be very small for a strong radiative transition. The more general source function accounting for all overlapping radiative transitions (including free-free transitions) is somewhat more complicated, but in the low-density regime it remains true that for a frequency corresponding to a strong transition i → j, the dominant radiative dependence will be on the ¯ corresponding J. As with broadband radiation transport, there are a variety of techniques designed to produce self-consistent converged solutions with reasonable convergence rates. Over the last couple decades, the most versatile and economical methods have been based on the concept of approximate operators [Kal87]. Rather than invert the full transport operator, these methods invert an approximation to the full operator, using the inverse in an iterative procedure to obtain the full solution. The closer the approximate operator is to the full operator, the faster the procedure converges. There are many ways to choose an approximate operator, and several dimensions to work in, but the near-universal choice for multidimensional problems is to use the (spatial) diagonal of the full operator [OAB86]. This is trivial to invert, being a scalar, and relatively simple to form. These tradeoﬀs compensate for a somewhat slower convergence rate. The diagonal approximate operator contains contributions from all frequencies contributing to the line radiation, but in contrast to the broadband acceleration operator, it contains only local information about the material response. This does not contradict the broadband analysis, as the absorption spectrum for a spectral line violates the conditions for grey acceleration to be eﬀective. The troublesome high frequency modes do not exist in line radiation and it suﬃces to handle the frequency-frequency coupling locally.

Non-LTE Radiation Transport in High Radiation Plasmas

315

3.3 Combined Broadband + Line Radiation The solution techniques described in the last two subsections work quite well in their respective regimes, but this success does not carry over between regimes. For instance, an approximate operator technique applied to broadband radiation with a standard absorption spectrum converges very slowly. If a high-frequency cutoﬀ is added to the absorption spectrum, mimicking a line spectrum, the approximate operator converges quickly but grey acceleration becomes ineﬀective. The considerations of the previous two subsections suggest that a simple combined approach might be eﬀective in situations where both broadband radiation and line radiation are important. We retain the structure of the broadband treatment, including a grey acceleration step to treat highfrequency modes, but add the local dependence of the material properties on the radiation spectrum by linearizing the source function in the radiation ﬁeld as well in the temperature: Sν = Sν (T, Jν ) ≈ Sν (T 0 , Jν0 ) +

∂Sν ∂Sν (T − T 0 ) + (Jν − Jν0 ) (16) ∂T ∂J ν ν

This linearization of the source term is a straightforward generalization along the lines of the discussion of energetics in Sect. 2. In the regime where the radiation spectrum signiﬁcantly inﬂuences the energetics, it is reasonable to expect a similarly signiﬁcant dependence in the source function. The considerations of the previous two subsections provide cause for optimism that this simple extension will produce a stable convergent solution to radiationdominated non-LTE problems. Section 4 presents a numerical application of this method demonstrating support for this viewpoint. The new response terms ∂Sν /∂Jν involve all frequencies. These terms do not introduce any new complications into the solution method, but computing all the additional derivatives is extremely expensive. In the low-density regime where we expect strong line radiation to dominate the radiative response, we can make the additional approximation that each bound-bound radiative transition Rij responds to radiation of a single frequency. Implicit in this approximation is the assumption that each strong line is contained within a single frequency bin, and we make no attempt to resolve any of the lines. Under these conditions, we can easily calculate the required derivatives from the atomic kinetics equations. We refer to this as the “diagonal” approximation, as it uses only the diagonal terms from the complete response matrix. Figures (1a)–(1c) demonstrate that this approximation indeed does very well at low densities, but poorly near LTE. Extending the solution method described in Sect. 3a to include the new response terms is straightforward. Linearizing the equations about the current temperature and radiation spectrum, using the diagonal approximation, produces

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∂Sν cT (T − T 0 ) = ∆t αν (Jν − Sν0 − (Jν − Jν0 )) dν ∂Jν ∂Em − (Jν − Jν0 ) + Q∆t ∂J ν T ν 1 Jν − Jν0 − ∇ • Dν ∇Jν = −αν (Jν − Sν0 ) c ∆t ∂Sν ∂Sν (T − T 0 ) + + αν (Jν − Jν0 ) ∂T ∂Jν

(17)

(18)

where cT is deﬁned as in (9) using the non-LTE speciﬁc heat. Equations (17) and (18) retain the same multigroup structure as before, and may be solved in the same manner. A numerical implementation of the broadband equations can be extended in a very simple manner. Most of the changes in the implementation are captured by the substitutions: ∂Sν 0 ∂Sν J (19) , Sν0 ← S˜ν0 = Sν0 − αν Jν ← α˜ν Jν = αν Jν 1 − ∂Jν ∂Jν ν In the absence of the diagonal approximation, these changes include the obvious sums over frequencies. Besides these substitutions, only the term involving ∂Em /∂Jν remains to be handled separately. One signiﬁcant change in solution procedure from standard LTE practice is necessitated by the non-LTE nature of the material properties. The source function is no longer a Planckian and need not have a simple form, so any iterative procedure that attempts to produce a self-consistent solution will necessarily entail iterating the atomic kinetics equations as well. This would be prohibitively expensive in most cases, although Sect. 5 discusses one approach towards a tabular solution. A related issue has to do with the grey acceleration step, which uses Planckian weights and is intended for use within the iterative procedure. However, an alternative procedure is to directly solve (10) for all frequencies simultaneously. A reduction procedure makes this quite economical in one dimension. Test cases, including the examples presented in the following section, have shown that using a single grey acceleration step produces nearly the same results as the direct solution procedure. Grey acceleration remains remarkably eﬀective for non-LTE problems as well as for LTE problems.

4 Test Case: Radiation-driven Cylinder As a test of the extended transport algorithm, we consider a case that is based upon a Z-pinch dynamic hohlraum experiment [Mat97]. The speciﬁcations are very similar to those used in [NZ03] to model tungsten liners. The chosen conﬁguration, illustrated in Fig. 2, consists of a hollow cylinder

Non-LTE Radiation Transport in High Radiation Plasmas

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vacuum

250 eV

r0=0.16cm r r r3=0.36cm 1 2

Fig. 2. Diagram of geometry for example application, consisting of an annular cylinder comprised of uniform density Lu between radii r0 = 0.16 cm and r3 = 0.36 cm, illuminated at r = r0 by a 250 eV Planckian radiation source

composed of uniform density Lu. The cylinder is illuminated from the interior by a blackbody radiation source of temperature 250 eV, with vacuum on the exterior. The inner surface of the cylinder has a radius of r0 = 0.16 cm and the outer surface has a radius of r3 = 0.36 cm. The radii r1 and r2 , equally spaced between r0 and r3 , are identiﬁed for purposes of displaying results. The goal is to calculate the self-consistent temperature and radiation distribution throughout the cylinder. Although this is a static geometry and we seek the steady-state solution, we perform time-dependent calculations with each time step corresponding to a single atomic kinetics evaluation followed by a single application of either the broadband or extended transport algorithm, using one grey acceleration correction. We consider three diﬀerent densities for the cylinder. At the lowest of these, with number density Ni = 1018 cm−3 , the cylinder is optically thick only at frequencies corresponding to strong line transitions. The highest density, Ni = 1020 cm−3 , corresponding to the middle density used for the speciﬁc heat evaluations in Sect. 2, is not yet in LTE, but has moderate to high optical depths over most of the frequency range. Figure 3 shows the optical depth along a radial line between the inner and outer radii for these two cases, evaluated from fully converged NLTE solutions, as well as the optical depth for the lower density case, evaluated from an LTE solution. Figures 4a and 4b show the radiation intensity, Jν , at the inner and outer radii for the low-density and the high-density cases, respectively. The Planck function appropriate to the material temperature is included for reference. The low-density case is very far from LTE, and the intensity spectra largely reﬂect the driving radiation, with some modiﬁcations due to strong line transitions. The high-density case is reasonably close to LTE, although a number of non-Planckian features are visible in the intensity spectra. The source function, Sν , provides a better indication of the non-LTE nature of these cases, as shown in Figs. 5a and 5b. Again, the appropriate Planck functions are included for reference.

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3

10

1

-3

Ni = 10 cm 18 -3 Ni = 10 cm 18 -3 Ni = 10 cm (LTE)

τ

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-1

10

-3

10

-5

10

2

3

10 energy (eV)

10

4

Fig. 3. Optical depth as a function of photon energy along a radial line between r0 and r3 . The upper solid curve is for a number density of 1020 cm−3 and the lower solid curve is for a density of 1018 cm−3 . The dotted curve is for a density of 1018 cm−3 , assuming LTE

intensity (cgs)

1.5x10

3

4.5x10

3

1.0x10

3

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3

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J ν , r = r0 Bν, r = r0 J ν , r = r3 Bν, r = r3

J ν , r = r0 Bν, r = r0 J ν , r = r3 Bν, r = r3

intensity (cgs)

2.0x10

0 10

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3

10 energy (eV)

Fig. 4a. Radiation intensity (thick line), Jν , and Planck function (thin line), Bν , at r = r0 (solid lines) and r = r3 (dotted lines) for a number density of 1018 cm−3

10

4

3.0x10

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3

0 10

2

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10 energy (eV)

Fig. 4b. Same as Fig. 4a for a number density of 1020 cm−3

10

4

Non-LTE Radiation Transport in High Radiation Plasmas 6x10

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2x10

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Sν, r = r0 Bν, r = r0 Sν, r = r3 Bν, r = r3

intensity (cgs)

intensity (cgs)

Sν, r = r0 Bν, r = r0 Sν, r = r3 Bν, r = r3

319

3.0x10

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10 energy (eV)

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4

10

Fig. 5a. Source function (thick line), Sν , and Planck function (thin line), Bν , at r = r0 (solid lines) and r = r3 (dotted lines) for a number density of 1018 cm−3

2

3

10 energy (eV)

10

4

Fig. 5b. Same as Fig. 5a for a number density of 1020 cm−3

A more interesting measure of the non-LTE nature of these cases is given in Figs. 6a and 6b. These ﬁgures display the diagonal of the source response function, ∂Sν /∂Jν , as a function of frequency, at both the inner and outer radii for the low-density and the high-density cases. This quantity is strictly zero in LTE, as the source function is independent of the radiation ﬁeld. For the two-level atom, this quantity reaches a maximum value of one when collisional rates are negligible. A large value in this context also indicates 1.0

1.0

r0 r3

r0 r3 0.8

0.6

0.6

ν

ν

∂S /∂J

∂S /∂J

ν

ν

0.8

0.4

0.4

0.2

0.2

0

0 10

2

10 energy (eV)

3

10

4

Fig. 6a. Diagonal source response function, ∂Sν /∂Jν , at r = r0 (solid lines) and r = r3 (dotted lines) for a number density of 1018 cm−3

10

2

10 energy (eV)

3

10

4

Fig. 6b. Same as Fig. 5a for a number density of 1020 cm−3

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highly radiation-dominated transitions. A value too close to one can cause problems in the numerical implementation, as evidenced in (19), so we restrict the maximum value of this quantity to 0.95. This limit is also reﬂected in the ﬁgures. It is evident that various narrow spectral ranges of the source function are very strongly dependent on the radiation ﬁeld, indicating that these spectral ranges are dominated by strong line transitions. We also deﬁne an average value of this quantity by integrating over the spectrum, weighted by the radiation intensity: ∂Sν ∂S Jν dν Jν dν (20) = ∂J ∂Jν For the low-density case, the average value is about 0.47 over the entire spatial domain, indicating a very strong dependence on the radiation ﬁeld. For the high-density case, the average value varies from 0.27–0.37. Since the diagonal approximation is not accurate for this case, these values are not reliable, but may still indicate a degree of sensitivity to the radiation ﬁeld. The converged material and radiation temperature proﬁles for the lowdensity case, for both non-LTE and LTE calculations, are given in Fig. 7a. The radiation temperature is insensitive to the material treatment and changes very slowly with radius. The material temperature, however, diﬀers considerably for these two treatments. The diﬀerence in behavior of the broadband and extended algorithms, as shown in Fig. 7b, is quite dramatic. The solutions were obtained through time-dependent evolution, starting from a uniform material temperature of 200 eV. The timesteps were initially very small, gradually increasing while attempting to keep temperature changes small within each timestep. Figure 7b displays temperature histories for four equally spaced points from the inner radius to the outer radius. The histories generated by the extended algorithm are smooth and well behaved. The broadband algorithm, however, evolves in the wrong direction early in time and quickly goes unstable. Stabilizing the evolution requires timesteps small enough for an explicit algorithm. Figures 8a and 8b give the corresponding results for the mid-density case, with Ni = 1019 cm−3 . The spatial temperature proﬁles still diﬀer signiﬁcantly between the non-LTE and LTE solutions, particularly near the outer boundary of the cylinder. The extended algorithm again performs well, smoothly evolving to the steady-state solution. The broadband algorithm experiences some diﬃculty, experiencing a mild instability at low temperatures, but stabilizes and smoothly evolves to late times. However, slow evolution continues at very late times and the solution does not reach steady state. The results for the high-density case are given in Figs. 9a and 9b. Here, the non-LTE and LTE spatial temperature proﬁles are indistinguishable, although the material and radiation temperatures still diﬀer noticeably near the boundaries. For this case, the broadband algorithm performs well, evolving smoothly to the steady-state solution. The extended algorithm does not

250

250

200

200

temperature (eV)

temperature (eV)

Non-LTE Radiation Transport in High Radiation Plasmas

150 Te - NLTE Tr - NLTE Te - LTE Tr - LTE

100

0.20

321

r0 r1 r2

150

r3

extended broadband

1 0.25 r (cm)

0.30

0.35

Fig. 7a. Final material temperature (solid curves) and radiation temperature (dashed curves) proﬁles for a number density of 1018 cm−3 . The heavy curves correspond to a converged non-LTE calculation and the light curves correspond to an LTE calculation

100 -13 10

10

-11

-9

10 time (s)

10

-7

10

-5

Fig. 7b. Material temperature as a function of time at positions r = r0 , r1 , r2 and r3 for a number density of 1018 cm−3 . The solid lines correspond to a non-LTE calculation using the extended algorithm as described in the text, while the dashed lines correspond to a non-LTE calculation using the broadband algorithm

250

250 extended broadband

temperature (eV)

temperature (eV)

r0 200

150

200

r1 r2

150

Te - NLTE Tr - NLTE Te - LTE Tr - LTE

100

0.20

r3

0.25 r (cm)

0.30

0.35

Fig. 8a. Same as Fig. 7a for a number density of 1019 cm−3

100 -13 10

10

-11

-9

10 time (s)

10

-7

10

-5

Fig. 8b. Same as Fig. 7b for a number density of 1019 cm−3

perform quite as well at early times and goes unstable at late times. This behavior is almost certainly due to the failure of the diagonal approximation at this high density. The extended algorithm does expand the range of conditions for which this type of radiation transport algorithm can be successfully applied. It improves

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250

250 r0

temperature (eV)

temperature (eV)

r1 200

150

200

r2

extended broadband

150

r3

Te - NLTE Tr - NLTE Te - LTE Tr - LTE

100

0.20

0.25 r (cm)

0.30

0.35

Fig. 9a. Same as Fig. 7a for a number density of 1020 cm−3 . The non-LTE results were obtained without the intensity derivatives

100 -13 10

10

-11

-9

10 time (s)

10

-7

10

-5

Fig. 9b. Same as Fig. 7a for a number density of 1020 cm−3

both the stability and the convergence of the solution, at least when the diagonal approximation to the response function is appropriate. Unfortunately, it is not necessarily clear when this approximation is valid. Similarly, in mildly non-LTE situations it may not be apparent when the broadband algorithm becomes inaccurate. In addition, calculating the diagonal response function increases the cost of expensive non-LTE simulations, although these additional calculations could undoubtedly be optimized. Developing criteria for determining when to apply the diagonal response function would be one path towards applying the extended algorithm as a general-purpose algorithm. Alternatively, using the full response function would bypass this diﬃculty, but at a high computational cost since all (or many) cross-derivatives would also be required. To avoid this expense, we are investigating the use of tabulated material properties, including full response functions, using a technique closely related to the linear response matrix method for near-LTE conditions.

5 Linear Response Matrix The linear response matrix (LRM) is a symmetric matrix Rνν describing the change in energy absorption and emission at frequency ν due to the deviation of the radiation ﬁeld from a Planckian at frequency ν [MK01]. Rνν = 4π

∂ (ην − αν Bν ) ∂Bν /∂Te ∂Jν

(21)

The symmetry of the LRM follows from the general principle of detailed balance [FM03]. An equivalent deﬁnition is given by the relationship

Non-LTE Radiation Transport in High Radiation Plasmas

4π (ην − αν Bν ) =

Rνν

Jν − Bν dν ∂Bν /∂T e

323

(22)

where the LHS of (22) is recognized as the quantity entering into the material energy equation. The computational utility of the LRM derives from the framework it provides for calculating near-LTE radiation transport problems using only tabular information. For our purposes, we require the individual derivatives from (21), instead of just the combination. The extent to which the linear description assumed by the LRM remains valid under non-LTE conditions will be the subject of a future discussion. For our present purpose, we note that the components of the LRM, are the response quantities required by the extended transport algorithm, including all cross-derivatives. In this context, we view the LRM as a tabulation of these derivatives. The questions to be addressed here are whether the derivatives evaluated at LTE can be used advantageously in the extended algorithm, and whether the tabular approach can be extended to situations far from LTE. In practice, the tabulated derivatives must be used carefully, with limits placed on both the intensity deviations (Jν − Bν ) used to calculate corrections, and on the corrections themselves. With these constraints, the method works quite well for the high-density test case of Sect. 4, reproducing the results of Fig. 9a. However, since an LTE treatment does just as well, the derivatives have no signiﬁcant eﬀect here. For the mid-density case, the resulting temperature proﬁles are shown in Fig. 10 along with the LTE and non-LTE results. Using the tabulated derivatives reproduces some of the features of the non-LTE solution, but overall the results are slightly worse than

temperature (eV)

250

200

150 Te - NLTE Te - LTE Te - LRM

100

0.20

0.25

0.30

0.35

r (cm)

Fig. 10. Final material temperature for a number density of 1019 cm−3 . The heavy curve corresponds to a converged non-LTE calculation, the light solid curve corresponds to an LTE calculation, and the light dashed curve corresponds to a calculation using tabulated LRM information

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the strictly LTE treatment. Similar results are obtained at lower densities. The cause of this behavior is under investigation. One possible failure of the LRM approach is overstepping the linear regime. Extending the reach of the tabular approach will require tabulating information not only at LTE, but for non-LTE conditions as well. We are now experimenting with evaluating the required quantities at points characterized by separate material and radiation temperatures. The additional parameter used in the tables is the ratio of the radiation temperature to the material temperature. Early results from this work are encouraging, but much work remains to be done.

6 Summary Solving the radiation transport equation also involves evaluating material properties that can depend on the radiation ﬁeld. In LTE, the material properties depend only on the material temperature, and the computational focus lies in calculating energy transport by broadband radiation and energy exchange between the material and radiation. In non-LTE, the material properties can depend directly on the radiation ﬁeld, and the coupling between the material and radiation takes on a diﬀerent character, which is reﬂected in the computational algorithms. In this paper, we have demonstrated an extended version of a broadband radiation transport algorithm that combines features of both types of transport schemes. The extended algorithm successfully handles applications with strong broadband and line radiation. Incorporating line radiation features into the algorithm improves both the stability and convergence of simulations with strong radiation ﬁelds, as demonstrated by an example based on a Z-pinch dynamic hohlraum. The numerical implementation of the extended algorithm requires additional information about the response of the material properties to radiation. The full set of response information is prohibitively expensive to calculate inline with the radiation transport, but a low-density approximation using only the diagonal of the response matrix, only modestly increases the computational cost. Preliminary investigations into the use of tabulated material information, including the full response matrix, have been only slightly encouraging. However, a successful approach of this type would not only permit routine use of the extended algorithm, but would greatly speed up broadband non-LTE calculations. This will be a topic for future research.

Acknowledgments This work was performed under the auspices of the U.S. Department of Energy, by the University of California, Lawrence Livermore National Laboratory under contract W-7405-ENG-48.

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References [AL02]

Adams ML, Larsen EW: Fast iterative methods for discrete-ordinates particle transport calculations. Prog in Nucl Energy 40, 3–159 (2002) [Kal84] Kalkofen W, editor: Methods in radiative transfer. Cambridge University Press, Cambridge (1984) [MO98] Molisch AF, Oehry, BP: Radiation trapping in atomic vapours. Oxford University Press, Oxford (1998) [MK01] More RM, Kato T, Libby SB, Faussurier G: Atomic processes in nearequilibrium plasmas. J Quant Spectrosc Radiat Transfer 71, 505-518 (2001) [Sco01] Scott HA: Cretin – a radiative transfer capability for laboratory plasmas. J Quant Spectrosc Radiat Transfer 71, 689–701 (2001) [CC05] Chung HK, Chen MH, Morgan WL, Lee RW: FLYCHK: Simple but generalized population kinetics and spectral model. Submitted to J Quant Spectrosc Radiat Transfer (2005) [MLM85] Morel JE, Larsen EW, Matzen MK: A synthetic acceleration scheme for radiative diﬀusion calculations. J Quant Spectrosc Radiat Transfer 34, 243–261 (1985) [Mih78] Mihalas D: Stellar atmospheres. W.H. Freeman and Co., San Francisco (1978) [Kal87] Kalkofen W: Survey of operator perturbation methods. In: Kalkofen W, editor. Numerical radiative transfer. Cambridge University Press, Cambridge (1987) [OAB86] Olson GA, Auer LH, Buchler JR: A rapidly convergent iterative solution of the non-LTE line radiation transfer problem. J Quant Spectrosc Radiat Transfer 35, 431–442 (1986) [Mat97] Matzen MK: Z pinches as intense x-ray sources for high-energy density physics applications. Phys Plasmas 4, 1519–1527 (1997) [NZ03] Novikov VG, Zakharov SV: Modeling of nonequilibrium radiating tungsten liners. J Quant Spectrosc Radiat Transfer 81, 339–354 (2003) [FM03] Faussurier G, More RM: Non-local thermodynamic equilibrium response matrix. J Quant Spectrosc Radiat Transfer 76, 269–288 (2003)

Finite-Diﬀerence Methods Implemented in SATURN Complex to Solve Multidimensional Time-Dependent Transport Problems R.M. Shagaliev, A.V. Alekseyev, A.V. Gichuk, A.A. Nuzhdin, N.P. Pleteneva, and L.P. Fedotova Numerical simulation of multidimensional particle transport problems falls into the category of the most complex and labor-intensive application problems. The paper brieﬂy describes the numerical methods implemented in SATURN codes to solve time-dependent problems. The codes are intended for solving 2D and 3D both linear and nonlinear spectral transport problems in the physics of high densities and high energies in multigroup approximation: The system of multigroup neutron–transport equations written in cylindrical coordinates: ∂ Ni + LNi + αi Ni = Fi ∂t νi ⎞1 ⎛ i1 Fi = ⎝ βij n0j + Qi ⎠ 4π j=1

∂ " ∂ ∂ LNi = (µNi ) + r 1 − µ2 cos ϕ · Ni + r∂r ∂z ∂Φ " 1 − µ2 ∂ sin ϕ · Ni − ∂ϕ r 1 n0i

=

1 − µ2 sin ϕ · Ni r

2π dµ

−1

"

Ni dϕ;

i = 1, 2, . . . , i1

0

The system of multigroup radiation transport equations in cylindrical in cylindrical coordinates: i1 1 ∂εi χai (0) + Lεi + χ·i εi = εip + aij χsj εj c ∂t 2π j=1

Here the transport operation is 1 ∂ " 1 ∂ " ∂εi + Lεi = µ 1 − µ2 · sin ϕεi r · 1 − µ2 · cos ϕεi − ∂z r ∂r r ∂ϕ

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R

V = V(r,z,Φ,t) – velocity; α = α(r,z,Φ,t) – collision factor; β = β(r,z,Φ,t) – particle multiplication constant; Q=Q(r,z,Φ,t) – independent source of particles; Ω = Ω(µ ,ϕ ) – a unit vector in the neutron flight direction; N=N(r,z,Φ,t,µ,ϕ) – flux of neutrons at point (r,z,Φ,t) flying in a given direction; µ = cos θ

(–1 < µ < 1,

Ω ϕ θ

0

Z

0 < ϕ < 2π).

Energy equation: i1 i1 i1 i1 1 ∂Ee (0) (0) = χni εi ∆ωi − χai εip ∆ωi − aij χsj εj j ∆ωi ρ ∂t i=1 i=1 i=1 j=1 εi = εi (r, z, µ, ϕ, ωi , t) – radiation intensity function; ωi – mean energy of photons in group i; ∆ωi – the range width in energy variable; T = T(r, Z) – temperature of medium; ρ – densiy of medium;

χai = χai (ρ, Te , ωi ) – absorption cross-section; χsi = χsi (ρ, Te , ωi ) – scattering cross-section; χ·i = χai + χsi – full cross-section; εip = εip (Te , ωi ). – Planck function; 1 p (0) ej dµdf ej = −1 0

Ee = Ee (ρ, Te ) – internal energy; aij – coeﬃcients describing energy exchange during Compton scattering.

Qi = Qi (r, z, ωi ) – independent source.

The main ideas of the numerical method implemented in SATURN to sole multidimensional time-dependent transport problems are: 1. Approximation to the transport equation in space variables is constructed using non-orthogonal space grids; in 2D cases: – Using regular non-orthogonal grids of convex quadrangles (Fig. 1);

Fig. 1. A structured quadrangular (regular) grid

Finite-Diﬀerence Methods Implemented in SATURN Complex

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Fig. 2. Arbitrary polygons are components of a grid

– Using irregular non-orthogonal grids of arbitrary-shape convex polygons (Fig. 2). 2. The following various conservative ﬁnite-diﬀerence schemes depending on the category of problems to be solved are used for the grids above to approximate transport equations: – scheme with extended template; – scheme with introduction of closing relations basing on moment equations; – scheme based on the use of adaptively reﬁned grids in phase space. The schemes above have the following common feature: with the use of non-orthogonal grids they preserve important features of DSn – schemes, such as the transport equation approximation within a single phase space cell and, consequently, a possibility to resolve systems of grid equations using sweep (point-to-point) method of computations. Nevertheless, they diﬀer from each other in the accuracy of approximation using essentially non-orthogonal grids, in monotone behavior of grid solution and some other features. 3. The method of source iterations is used to solve the system of multiplegroup grid transport equations numerically. The following methods are used to accelerate the iterative process convergence: – FCA method for the category of time-independent linear multiple-group problems of calculating the critical parameter Keﬀ ; – KM method for the category of time-dependent nonlinear multiplegroup problems of radiation transport.

330

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P˝1

P2

P12

P1

P˝0 P'2

P23

P˝3

P0

P3

P'1

P˝4

P4

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Fig. 3.

4. The eﬃcient parallelization algorithms have been developed and successfully used for numerically solving transport equations on multiprocessors: – for 2D geometry – combined algorithm with ﬁne-grain parallelization in space variables; – for 3D geometry – pipeline-type algorithm. Finite-diﬀerence schemes used to approximate 2D transport equations using non-orthogonal space grids Extended-template scheme. The scheme is built using the grid function values in a cell, on sides and at vertexes of a quadrangular cell (Figs. 3, 4).

P(1)

P(12) ×

P(2)

P0 ×

P(23) P(3)

×

P(34) Fig. 4.

P(4)

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1 Multiple-Group Transport Equation Approximation • in time: implicit scheme with weighting multipliers; • in angular variables: method of discrete ordinates (sn -quadratures); • in space: using the extended template for non-orthogonal space grids The extended template scheme features: • the scheme is conservative; • convergence to the transport equation solution with the second order of accuracy using non-orthogonal space grids; • the scheme complies with the diﬀusion limit condition in optically dense media; • it uses DSn-method quadratures to approximate the transport in angular variables. The system of grid equations includes the following: 1. Grid equations for particle balance in the grid cells: ∆VP0 ·

n − NPn0 NPn+1 n+γ n+γ 0 + RP (l,l+1) · NPn+γ − Rq−1 Nq−1 (l,l+1) + Rq Nq ∆t l=1

+ αP0 · NPn+γ · VP0 = ∆VP0 · FP0 , 0 where rP (l) + rP (l+1) 2 " × µm rP (l+1) − rP (l) − 1 − µ2m zP (l+1) − zP (l) cos ϕq , l = 1, 2, 3, 4 , " " 1 − µ2m · sin ϕq−1 1 − µ2m sin ϕq Rq = ∆SP0 · , Rq−1 = ∆ SP0 · ∆ϕq ∆ϕq

RP (l, l+1) =

2. Additional correlations in time variable and angular variable ϕ NPn+γ = γNPn+1 + (1 − γ) NPn0 , 0 0 NP0 = ηNq + (1 − η) Nq−1 ,

0.5 ≤ γ ≤ 1 0.5 ≤ η ≤ 1

3. Additional space relations between the values of the desired function in a cell, on sides and at nodes of a cell. The number of relations in the scheme considered depends on the number of illuminated (exposed to particles) quadrangular cell’s sides. Here the following three variants shown below in Fig. 5 are possible depending on the values of µm , ϕq, m . The following additional correlations correspond to the three illumination variants in the extended-template scheme:

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R.M. Shagaliev et al. Ωm,q

P1

P1

P2

P2 P3

(a)

P4

P4

P3

(b)

P2

P1

P3 P4

(c)

Fig. 5. Examples of a cell exposure to particles (illumination)

(a) NP (2,3) = δNP 3 + (1 − δ) NP , NP0 = δNP 3 + (1 − δ) NP NP (3,4) = 0.5 (NP 3 + NP 4 ) , NP0 = δNP 4 + (1 − δ) NP NP (4,1) = δNP 4 + (1 − δ) NP 1 (b) NP 6 = 0.5 (3 δ − 1) NP 3 + 0.5 (1 − δ) (NP 5 + NP 2 + NP 4 ) (c) NP0 = δNP (3,4) + (1 − δ) NP (1,2) Below given are the results for one 2D test problem demonstrating the extended-template scheme accuracy during numerical solution of problems using essentially non-orthogonal space grids. One-group time-independent transport equation speciﬁed in a cylinder with dimensions 0 = R = 1, 0 = Z = 2 is considered in (r, z) geometry. Full cross-section α and multiplication coeﬃcient ß are α = 1.34 and ß = 2.25 respectively. The boundary condition is a zero incoming ﬂow. The proper parameter λ has to be calculated. Numerical computations for the problem were carried out in 24 directions of particle ﬂights (6 ranges in µ). The following three methods were used to choose a space grid and its further reﬁnement in order to study the scheme convergence rate: Method 1. Rectangular uniform grids with steps 12 hz = hr = 1/7, 1/14, 1/21, 1/28 were chosen. Method 2. Some initial essentially non-orthogonal and non-uniform grid was constructed (Fig. 6). The subsequent grids were obtained by reﬁnement of the initial ones with preservation of a “bad” form of cells (angles of quadrangles remained the same). With reﬁnement using method 2, evenness of the numbers of rows and columns was provided, they were taken equal to 8, 14, 20, and 28. The Table 1 below shows the results of computations. It is clearly seen that the calculated values of λ satisfy the formula corresponding to the second-order convergence: λ = 0.1647 − kA2 , where A becomes equal to 1 or 1.16 using method 1, or method 2 of grid selection and reﬁnement, respectively.

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Fig. 6. A non-orthogonal grid 7 × 7 Table 1. Number of Rows and Columns (method 1)

Rectangular Grids (method 1)

Non-orthogonal Grids (method 2)

Number of Rows and Columns (method 2)

7×7 14 × 14 21 × 21 28 × 28

0.153134 0.161915 0.163499 0.164045

0.154008 0.160656 0.162438 0.163346

8×8 14 × 14 20 × 20 28 × 28

1.1 Scheme with Introduction of Closing Relations Basing on Moment Equations During such scheme construction the solution inside a grid cell is represented in the form of bilinear decomposition in space variable and in angular variable ϕ : N (τ, η, ϕ , µ) ≈ N0 + × +

ηϕ Nηϕ 4

τ η ϕ τη τ ϕ N τ + Nη + N ϕ + Nτ η + Nτ ϕ 2 2 2 4 4

ϕ where ϕ = ∆ϕq−1/2 , η, τ are the coordinates of a point of space inside a cell during bilinear transformation of a quadrangle into a unit square (Fig. 7).

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(-1,1)

P1

(1,1)

P2

P3

P4

(-1,-1)

r

(1,-1)

τ z

η Fig. 7.

The balance equations for particles in space grid cells and the corresponding moment equations are used to ﬁnd the unknown coeﬃcients of the expansion in series. To construct the diﬀerence analogs to moment equations, the (1) is replaced by some simpliﬁed expansions in series. 1. Moment diﬀerence equation in angular variable f . To construct the equations of interest, we use the following representation of solution inside the diﬀerence grid range in variable f : N (τ, η, ϕ , µ) ≈ N0 + ϕ2 Nϕ + τ ϕ ηϕ ϕ 4 Nτ ϕ + 4 Nηϕ ϕ = ∆ϕq−1/2 . This expansion in series is apparently obtained by omitting the terms without variable ϕ in the original bilinear expansion. Using the given expansion in series we construct a rather simple grid moment equation in variable f that has the following form: 1 " 1 " 1 − µ2 S0 sin ϕq Nq − 6 1 − µ2 S0 sin ϕq−1 Nq−1 ∆ϕ ∆ϕ " " 1 " +12 1 − µ2 S0 P1 N0 + 12 1 − µ2 S0 P2 Nϕ + 1 − µ2 S0 P3 Nϕ ∆ϕ 1 " +12 1 − µ2 S0 P4 N0 + αV0 Nϕ = 0 ∆ϕ " 1 − µ2 1 1 sin ϕq J0 Nη + Nηϕ 4r0 Nηϕ B1 + A1 − 12 2 ∆ϕ 2 " 2 1−µ 1 + Jη N0 + Nϕ sin ϕq Jτ Nτ η − 2 ∆ϕ

−6

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" " 1 − µ2 2 + 12 ϕ cos ϕdϕ sin ϕdϕ + 6 1 − µ ∆ϕ 1 × J0 Nη + Jη N0 + Jτ Nτ η 12 " 2 " 1−µ + 6 ϕ sin ϕdϕ + 3 1 − µ2 (ϕ )2 cos ϕdϕ ∆ϕ " 1 − µ2 × (J0 Nηϕ + Jη Nϕ ) + 2r0 α(J0 Nηϕ + Jη Nϕ ) = 12 ∆ϕ × sin ϕq−1 Jη Nq−1 " 1 − µ2 1 1 sin ϕq J0 Nτ + Nτ ϕ y 4r0 Nτ ϕ A1 + B1 − 12 2 ∆ϕ 2 " 2 1−µ 1 +Jτ N0 + Nϕ sin ϕq Jη Nτ η − 2 ∆ϕ " " 1 − µ2 sin ϕdϕ + 6 1 − µ2 ϕ cos ϕdϕ + 12 ∆ϕ 1 × J0 Nτ + Jτ N0 + Jη Nτ η 12 " " 1 − µ2 2 2 (ϕ ) cos ϕdϕ + 6 ϕ sin ϕdϕ + 3 1 − µ ∆ϕ × (J0 Nτ ϕ + Jτ Nϕ )

"

+2r0 α(J0 Nτ ϕ + Jτ Nϕ ) = 12

1 P1 =

ϕ cos ϕdϕ −1

1 − µ2 sin ϕq−1 Jτ Nq−1 ∆ϕ 1

P2 =

ϕ sin ϕdϕ

−1

sin ϕq − sin ϕq−1 cos ϕq−1 − cos ϕq P3 = ; P4 = ; ∆ϕ 1 ∆ϕ 1 q− /2 q− /2 1 1 ϕ = (1 + ϕ )ϕq + (1 − ϕ )ϕq−1 2 2 2. Moment diﬀerence equations in space variables. Similarly to the case described above, the simpliﬁed expansion in series containing only the dependences on space bilinear variables τ, η is used to construct such equations:

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∂ ∂ (B(η)N ) + (A(τ )N ) + J(τ, η)ραN = J(τ, η)ρF ∂η ∂τ ¯ p(3,4) − 6N0 (R ¯ p(1,2) Np(1,2) − 6 N0 − 1 Nη R ¯ p(1,2) − R ¯ p(3,4) ) 6R 2 ¯ p(2,3) + SαNη = SFη ¯ p(1,2) + R ¯ p(3,4) ) + 3R ¯ p(3,4) Nη,p(1,4) + Nη R −Nη (R ¯ p(2,3) − 6N0 (R ¯ p(4,1) Np(4,1) − 6 N0 − 1 Nτ R ¯ p(4,1) − R ¯ p(2,3) ) 6R 2 ¯ p(3,4) + SαNτ = SFτ ¯ p(4,1) + R ¯ p(2,3) ) + 3R ¯ p(1,2) Nτ,p(1,2) + Nτ R −Nτ (R 1 3r0 χ ¯12 Nτ,12 − r0 χ ¯34 Nτ − Nτ η − r0 (B0 Nτ + A0 Nη ) + 3r0 χ ¯41 Nη,41 2 " 1 1 1 − µ2 sin ϕdϕ (Jτ Nηϕ + Jη Nτ ϕ ) −r0 χ ¯23 Nη − Nτ η − 2 12 ∆ϕ 1 1 + r0 α(J0 Nτ η + Jη Nτ + Jτ Nη ) = r0 (J0 Fτ η + Jη Fτ + Jτ Fη ) 6 6 Some results of computational investigations for one test problem giving a preliminary image of the main features of the scheme under consideration are presented below. The results obtained using LMS scheme and the extended template scheme used at present in SATURN-3 complex are compared. Statement of the Test Problem The linear time-independent particle transport problem (1)–(3) is considered in axially symmetric region {0 ≤ R ≤ 1, 0 ≤ Z ≤ 2}. The source and the equation coeﬃcients are taken as Q = 1, α = 1, β = 1. The incoming ﬂow equal to zero is speciﬁed for the boundaries parallel to R-axis (on the bases of cylinder) and the boundary condition “mirror reﬂection” is speciﬁed for the boundary parallel to Z-axis. A series of computations was carried out using grids concentrating in angular and spatial variables. An orthogonal uniform space grid was generated for all the computations below. 1. Computation 1 (z1). The space grid has 10 rows and 10 columns. The angular grid has 24 equal-area solid angles (the number of ranges in angular variable µ is 6). 2. Computation 2 (z2). A space grid has 20 rows and 20 columns. The angular grid has 84 equal-area solid angles (the number of ranges in angular variable µ is 12). 3. Computation 3 (z3). A space grid has 40 rows and 40 columns. The angular grid has 312 equal-area solid angles (the number of ranges in angular variable µ is 12).

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4.5

4

3.5

3

2.5

2 0

0.2

0.4

cx.1 _ z1

0.6

0.8

cx.1 _ z2

1

cx.1 _ z3

1.2

1.4

lms_z1

1.6

lms_2

1.8

2

lms_z3

Fig. 8. Solution along the ﬁrst row

The distribution of particle density function n(0) obtained using the extended template scheme (scheme 1 in the following description) and LMS scheme is considered as a result of computations. Figure 8 shows the solution (density of particles n(0) ) along the ﬁrst row corresponding to R = 1 that has been obtained in 3 computations. As one can see from the ﬁgure above, the solutions in the ﬁrst row obtained using 2 various schemes coincide to a high precision (∼0.13% in computation 1) and converge to the single solution. Figure 9 shows the distribution of the particle density function along the last row lying on Z-axis. 4.5

4

3.5

3

2.5

2 0

0.2

cx.1 _ z1

0.4

0.6

cx.1 _ z2

0.8

1

cx.1 _ z3

1.2

lms _ z1

1.4

1.6

lms _ z2

Fig. 9. Solution along the last row

1.8

2

lms _ z3

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4.14 4.13 4.12 4.11 4.1 4.09 4.08 4.07 4.06 4.05 4.04 4.03

0

0.1

cx.1 _ z1

0.2

0.3

cx.1 _ z2

0.4

cx.1 _ z3

0.5

0.6

lms _ z1

0.7

0.8

0.9

lms _ z2

1

lms _ z3

Fig. 10. Solution along the central column

One can see here that solutions in the last row obtained using two various schemes diﬀer from each other. For example, in computation 1, i.e. on the coarsest grid, the discrepancy in solution is about 1.1%. Figure 10 shows the proﬁle of solution along the central column. Note that the solution along R-axis is one-dimensional (this follows from the problem statement). As one can see from the ﬁgure above, the use of LMS scheme does not lead to the solution decrease along Z-axis and gives the solution constant to a high precision. At the same time, there was some discrepancy in one-dimensional behavior of solution in computations using the extended-template scheme (∼1% in computation 1) typical of DSn–type schemes. 1.2 Adaptive Method of Reﬁned Grids in the Phase Space The idea of the method under discussion is that in the phase region of the problem solution some subregions are singled out, in which the original (reference) grid cells are reﬁned. The major requirement imposed on the reﬁnement criteria and algorithms is that of construction of grids adapted to the problem solution at a given time in the phase space. This should eventually ensure a signiﬁcant saving of the computational costs and, simultaneously, a needed accuracy of the resultant numerical solution.

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The reﬁnement can be: – in space variables; – in angular variables; – in energy variable. In time-dependent problems the reference grid cell reﬁnement region is redeﬁned at timesteps. In so doing special algorithms for selection of the subregions and reﬁnement levels are involved. The transport equation approximation on nonorthogonal spatial grids using the adaptive reﬁned grids entails the problem of preservation of the principal properties of the scheme used for the numerical solution to the transport equation on the reference grid, such as the transport equation approximation within a single computational cell, conservatism of the scheme, the possibility to solve the grid transport equations with the point-to-point algorithm, the possibility to use acceleration algorithms, and some others. An important feature of the constructed adaptive method of reﬁned grids is that it ensures the solution to the above problem. Below are two examples of the 2D benchmark problem solution with the adaptive method of reﬁned grids. Benchmark problem. A rectangular region in 2D axisymmetric geometry is considered. The region is presented in Fig. 11. The computational domain is composed of two physical regions: Physical region 1 is a dense casing {0 = Z = 5; 1 = R = 1.2}; Physical region 2 is a transparent region {0 = Z = 5; 0 = R = 1}. R 1,2 1

Physical region 2

Physical region 1 0

5

Z

Fig. 11. The system geometry in the 2D benchmark problem

The material density in the system is ρ = 1, the material energy as a function of temperature is given with equation E = 0.81 T . The computations include photon absorption, with the photon absorption cross section being given as χa = A/T 3 . Diﬀerent values of the coeﬃcient A in the system give optically dense region (Physical region 1) and optically transparent region (Physical region 2). In Physical (optically dense) region 1 A = 50.89. In Physical (optically transparent) region 2 A = 0.1374. The scattering cross section was taken zero (χs = 0) in the computations. Initial temperature T was assumed equal to 0.0001 at all the system points.

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The following boundary conditions in radiation are given on the left end of the rectangular domain {Z = 0; 1 = R = 1.2}: the “specular reﬂection” condition is given on the boundary part referring to the dense casing and the incoming isotropic radiation ﬂux corresponding to temperature T = 1 on the part referring to the transparent region. On the upper boundary and on the right end the incoming radiation ﬂux was given zero. As a result, the solution to the equation of photon transport and radiationmedium interaction is of a highly 2D nature in the problem under consideration. The computations for the above problem employed the reference spatial grid composed of 10 rows (5 in each of the physical regions) and 50 columns, the angular grid included 16 spacings in angle µ and 6 spacings in ϕ, the timestep was ∆t = 0.00002. The computation on the spatial grid containing 40 × 200 cells was taken as the base computation and the resultant solution is considered accurate. Results of the computations using the adaptive technique. The spatial grids, which the computations were performed on, are described below using notation Nr (Pr ) × Nz (Pz ), where Nr is the number of rows, Nz is the number of columns, Pr is the maximum adaptivity level in rows (MaxAdapt), Pz is the maximum adaptivity level in columns. If no adaptivity is used in one of the directions, the parameter in brackets is not speciﬁed. The results of the 2D benchmark problem computation on the adaptive grids 10(4) × 50(4) and 10(8) × 50(8) are plotted in Fig. 12. For comparison that same ﬁgure presents the numerical solutions on some spatial grids without the adaptivity. As seen from the plots, the solution on grid 10(8) × 50(8) using the adaptive technique proves close to the result of the computation on grid 40 × 200 using the standard technique, with the adaptive computation requiring time less by a factor of 2.8. As an illustration demonstrating the results from the adaptive grid formation programs, Fig. 13 presents the radiation temperature ﬁeld distribution in the system for three diﬀerent times. The black lines correspond to the reference spatial grid and the white lines to the adaptive cells. In the computation the maximum adaptivity level is 2 (four adaptive cells in the reference grid) along either direction. The adaptive partition of the spatial grid is seen to proceed only at the wave front, with this being in diﬀerent ways in diﬀerent spatial directions. 1.3 Algorithms for the Iterative Process Convergence Acceleration in Complex SATURN The numerical solution of many application problem classes requires methods for acceleration of convergence of iterations in source (in the right-hand side of the transport equation) to ensure the computation eﬃciency. For the computations of linear time-independent problems of critical parameter calculation we have developed and are successfully using a ﬂow

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Fig. 12. Material temperature proﬁle along line Z = 2 at time 0.01 in the 2D benchmark problem for diﬀerent reference grids: ——– solution on the base grid 40 × 200; – – – calculation 20 × 100; · · · · · · – calculation 10 × 50; – – – calculation 10(4) × 50(4); —— calculation 10(8) × 50(8)

consistent acceleration method (FCA method). The brief formulation of the method and some results of its numerical studies are presented below. FCA Method (Flow Consistent Acceleration) In the FCA method the following items are desired: functions of one-sided ﬂow ⎧ + ¯ n Pk = Ω, ¯ k Nk dΩ ⎪ ⎪ ⎪ ¯ nk )>0 ⎨ (Ω,¯ , ⎪ − ¯ n ⎪ Ω, ¯ N = − dΩ P ⎪ k k k ⎩ ¯ nk )<0 (Ω,¯ whose average grid values are introduced on the edges and medians of the grid cells, function of total ﬂux Wk = Pk+ − Pk− given at the same points. Below we introduce the average values on the scalar ﬂux function n(0) = Ω N dΩ cell medians (Fig. 14).

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t = 0.004

t = 0.006

t = 0.008

Fig. 13. Adaptive partition of spatial grid 10(8) × 50(8) at diﬀerent times

Pi −

i, j+1

i, j ni+1/2 + j

P

Pj−+1

nj+1/2

i+1, j+1 i+1, j

Pi ++1 Fig. 14. The spatial grid cell

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Under these assumptions the diﬀerence equation system of the FSA method is constructed using the integral moment equations, namely, the equations for zero and ﬁrst moments of function N of the solution to the initial transport equation that include the compensating sources of the simple iteration. To coordinate the solutions to the diﬀerence equations 1 ˜ S+1 + α LN ˜ N S+1 = (βn0S + Q) 4π used at the simple iteration phase and at the simple iteration convergence acceleration phase, special grid functions of the “compensating sources” are introduced to the right-hand sides of the diﬀerence equations of the FCA method. The introduction of the functions ensures convergence of the grid solution obtained by iterations with the FCA method to the solution with the simple iteration method. (SLk Wk − SLk+1 Wk+1 ) + (α − β) n0 = (Q + F ) k∈(i,j,l)

+ + − − Pk + Pk− + α · m · Wk+1/2 = F Wk+1/2 , + Pk+1 Dk+1/2 Pk+1 where k ∈ (i, j, l) ⎧ + ¯ n Pk = Ω, ¯ k Nk dΩ ⎪ ⎪ ⎪ ¯ nk )>0 ⎨ (Ω,¯ ⎪ − ¯ n ⎪ Ω, ¯ k Nk dΩ ⎪ ⎩ Pk = − ¯ nk )<0 (Ω,¯ Wk = Pk+ − Pk− , where k ∈ (i, j, l, i + 1/2, j + 1/2, l + 1/2) The grid equation set of the FCA method is closed with the relations of the grid values given at diﬀerent points of the spatial cell: % + + + F δk+ Pk+1/2 = δ · Pk+ + (1 − δ)Pk+1 , where k ∈ (i, j, l) − − Pk+1/2 = δ · Pk+1 + (1 − δ)Pk− + F δk− + − + qP k +1/2 . + P n0k+1/ = APk+1/2 Pk+ 1/ k+1/ 2

2

2

In the right-hand sides of the relations there are the grid functions of the “compensating sources”. Like in the above equations, these sources are introduced for the coordination of the grid solution obtained from the difference transport equations and the correction obtained from the diﬀerence equations of the FCA method. As the experience of using the FCA method suggests, in solution of complex 2D application problems it reduces the time costs of the computations by a factor up to 500. Below are examples of the computations for several benchmark problems with the FCA method.

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3D

α

β

N/A

FCA

N/A

FCA

10. 10. 10. 10.

9. 9.9 9.99 10.

81 331 463 482

7 7 7 7

116 300 361 370

12 13 13 13

Results of the computations for reactor SNR-300 Method

Number of Iterations

Kellog method Direct method Method of inverse iteration Method of inverse iteration + FCA

143 141 112 20

Results of the computations for the RBMK reactor channel Method

Number of Iterations

Kellog method Direct method Source iteration method Source iteration method + FCA Source iteration method + FCA + Chebyshev method

229 412 555 42 24

Homogeneous sphere R = 10 Number of Iterations α=β

Simple Iterations

DSA Method

FCA Method

1. 2. 4. 8. 12. 16.

221 597 1627 3927 6003 7658

8 8 9 8 12 24

6 6 6 6 6 9

Numerical Studies of the FCA Method for the Iterative Process Convergence Acceleration Particularly high requirements to the convergence and eﬃciency of the simple iteration convergence acceleration methods are imposed in solution to nonlinear time-dependent multi-group X radiation transport problems.

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For this problem class, a so-called KM method was constructed by us as early as the 1980s and improved in the recent years. The brief formulations of the KM method and the improved KM method (KM3 method) and the comparative results of the calculations with these methods for one timedependent benchmark problem are presented below. The KM method is a two-step iterative method. At step 1 of the KM method (predictor), the transport equation is solved with the method of simple iterations (by source iterations).

1 νi

1 s+ 2 n+1 εi

1

1

s+ 2 s+ 2 NI s −εni 1 (0)n+γ n+γ + L εi +αi εn+γ = βij εj , i ∆t 2π j=1

εn+γ = γεn+1 + (1 − γ) εni , i = 1, 2, . . . , N I i i At step 2 of the KM method (corrector), the correction equation system of the following form is solved: s+1

s+

1

NI NI 2 s+1 s+1 s+1 1 1 ∆εn+1 (0)n+γ n+γ n+γ i + L ∆εn+γ +α ∆ ε − β ∆ ε = β ∆ε , i ij ij j i i j νi ∆t 2π j=1 j=1 s+1

1 s+ 2

s+1

s+1

s+1

∆ εi = εi − εi , ∆εn+γ = γ ∆εn+1 , i i 1 s+ 2 (0) ∆εi

=

1 s+ 2 (0) εi

s (0)

− εi

, i = 1, 2, . . . , N I

Mention some features of the group correction equations of KM-method step 2: 1. The KM method is a conservative iterative method, with the two major laws of conservation characteristic of the transport equation, i.e. the law of conservation relative to the particle transport and that relative to the medium-radiation energy exchange, being simultaneously satisﬁed at each iterative process step. 2. The KM-method step 2 group equations are of the same form as the original group transport equations, which allows the same diﬀerence methods as those for the governing equations to be used for their grid approximation. 3. The cost-eﬃcient “point-to-point computation” method can be extended to the numerical solution of the KM-method step 2 correction diﬀerence group equation system. KM3 Method I step: predictor – like in the KM method

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II step: corrector – iterative:

1 νi

1 s+1,ν+ 2 n+1 ∆εi

∆t

+

1 s+1,ν+ 2 L ∆εn+γ i

s+1,ν N I (0)n+γ βij ∆εj × j=1

1 + 2π

1 s+1,ν+ 2 +αi ∆ εn+γ i

NI

−l

NI

1 s+1,ν+ 2 βij ∆ εn+γ j

j=1 1 1 s+ 2 s+1, ν+ 2 (0)n+γ βij ∆εj , ∆εi

=

= (1 − l)

1 s+1, ν+ 2 εi

1 2π

s+1, ν

− εi

j=1

s+1,ν+1

NI s+1,ν+1 s+1,ν+1 s+1,ν+1 1 ∆2 εn+1 1 i + L ∆2 εn+γ +αi ∆2 εn+γ − βij ∆2 εn+γ = (1 − l) i i j vi ∆t 2π j=1

×

N I j=1

1 s+1,ν+ 2 (0)n+γ βij ∆εj −(1

1 s+ 2 , ν NI 1 (0)n+γ − l) βij ∆εj , 2π j=1

s+1, ν+1 ∆2 εi

s+1, ν+1

=

∆εi

1 s+1, ν+ 2 − ∆εi

Numerical studies of the KM3 method Benchmark problem «Tube»: R Region 2 (10×100)

1.2 1

Region 1 (10×100) 0

10 Z

Fig. 15. Problem geometry

The number of iterations in region 2 (NIterν = 1, l = const = 0.5) ∆t = 0.1 · 10−10 sec Simple Step No. Iteration KM KM3 1 5 10 t, ???

19108 3332 1559 2880

14 24 27 61

14 20 20 47

∆t = 0.5 · 10−10 sec Simple Iteration 28696 6363 8477 7020

∆t = 0.1 · 10−9 sec

Simple KM KM3 Iteration

KM KM3

26 121 101 233

35 238 179 435

26 88 75 184

26454 7960 9146 8580

35 174 135 347

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1.4 Parallel Techniques As mentioned above, the numerical solution to many classes of multidimensional time-dependent problem classes entails high computational costs. Therefore, parallel algorithms oriented to advanced high-performance multiprocessor computer systems are required. The development of highly eﬃcient algorithms for parallelization of the problem class under discussion is an involved methodological problem. There are a number of objective reasons for this, among which the following should be primarily mentioned. 1. As known, implicit schemes are mainly used for the numerical solution to the transport problems, hence, spatial grid cell computations should be performed in some strictly determined sequence. When using nonorthogonal spatial time-varying grids, the sequence of the cell computation can be diﬀerent at diﬀerent timesteps. In other words, in parallelization of this problem class over spatial variables it is very hard, in contrast to the problem classes using explicit numerical methods, to ensure a simultaneous uniform loading of all processing elements used. 2. In numerical solution to nonlinear transport equations the costs of the transport equation coeﬃcient computation are signiﬁcantly diﬀerent at diﬀerent spatial points, which leads to an additional disbalance of the parallel computations. 3. In the numerical solution of the problem class under discussion a number of other physical processes must be calculated simultaneously with the simulations of the transport processes in separate subregions, which also signiﬁcantly inﬂuences the parallel computation balance. We have developed a ﬁne-grain parallelization method to solve the above problems on multiprocessor computers in the spatial two-dimensional approximation. The method is oriented to the case of arbitrary nonorthogonal spatial grids. It admits spatial problem decomposition to a fairly large number of processors with arrangement of balanced computations. Principal concepts of the ﬁne-grain parallelization method • The principle of spatial decomposition of the initial system to subdomains (paradomains) for the data allocation over the processors • The pipeline computation of the paradomain along the particle ﬂight directions with internal boundary conditions calculated at the current iteration, which preserves the computation eﬃciency and accuracy • Loading of temporarily idling processors with proﬁtable computations relating to calculation of additional coeﬃcients, which the solution is expressed by. • Using a combined approach with simultaneous parallelization over the subdomains and by angular variable µ. • When solving the multi-group transport problems, the spatial decomposition principle is used in combination with the parallelization by energy groups.

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Fig. 16. Acceleration factor versus the number of processors

Algorithm for pipeline parallelization of 3D transport equation Time diagram of the solution to the 3D transport equation with parallelization Clock 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

µ 1 1 1 1 1 1 2 2 2 2 2 2 -

Process 1 ϕ 1 1 2 2 3 3 1 1 2 2 3 3 -

Process 2 Φ 1 2 1 2 1 2 1 2 1 2 1 2 -

µ 1 1 1 1 1 1 2 2 2 2 2 2 -

ϕ 1 1 2 2 3 3 1 1 2 2 3 3 -

Process 3 Φ 3 4 3 4 3 4 3 4 3 4 3 4 -

µ 1 1 1 1 1 1 2 2 2 2 2 2

ϕ 1 1 2 2 3 3 1 1 2 2 3 3

Φ 5 6 5 6 5 6 5 6 5 6 5 6

The 3D particle transport equation is solved in the cylindrical coordinate system R

Z

Φ

Fig. 17. Results of the numerical study of the 3D transport equation parallelization on a model problem

The results of the numerical studies of the parallelization algorithm eﬃciency on one model spherical problem are presented below. Eﬃciency estimation with the magniﬁcation method. The problem size . 900 computational cells on each processor remains constant. SpN = TT1 ∗N N per processor. The results are presented in Fig. 16.

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Results of the numerical studies of eﬃciency Nproc

Spn

En, %

2 5 10 20 40 50

1.84 4.23 8.33 15.3 28.6 34.3

92 85 83 77 71 69

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Nproc

Fig. 18. Acceleration factor versus the number of processors

Algorithm for Pipeline Parallelization of 3D Transport Equation The pipeline parallelization algorithm has been developed by us and has been being successfully used for many years in solution to three-dimensional transport problems (Fig. 17). The eﬃciency estimation with the reﬁnement method. The total size of N the problem remains constant. SpN = TTN1 , EN = Sp N ∗ 100%. Problem parameters: 8 energy groups, 96 particle ﬂight directions (S8 ), 250000 3D spatial cells. The results of the numerical studies are summarized in the next table and in Fig. 17.

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Numerical simulation of experiments on laser facility ISKRA-5 1- holder, 2- laser beams, 3- «ILLUMINATOR», 4- cylindrical box, I – III– zones where materials to be studied are located.

1

2 I II III

3 4 Fig. 19. The “illuminator” target

(a)

(b)

Fig. 20. Distribution of eﬀective temperature of radiation (a) and electrons (b) inside the cylinder at time 5 ns. Length scale: 100 µm; temperature scale: keV

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Design of the “labyrinth” target used for X-ray field generation on the sample surface in the experiments to study the equation of state with the shock compression method on facility «ISKRA-5» film

D box

d2

laser radiation flux

d1

cone

box

disk

d3 Fig. 21. The “labyrinth” target

Fig. 22. Distribution of radiation eﬀective temperature in the outlet cross section. “Labyrinth” problem

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(a)

(b)

Fig. 23. Distribution of electron temperature (a) and eﬀective temperature of radiation (b) at the time of peak laser pulse. “Labyrinth” problem

1.5 Some Examples of Computations for 2D Application Problems The above-discussed numerical methods and algorithms are widely used presently for computations of diﬀerent application problems in multidimensional geometries. To demonstrate the capabilities of the methods developed, below are some results of the computations for 2D coupled time-dependent problems that describe experiments on laser facility ISKRA-5 (Figs. 19–23).

Implicit Solution of Non-Equilibrium Radiation Diﬀusion Including Reactive Heating Source in Material Energy Equation Dana E. Shumaker1 and Carol S. Woodward2 1

2

Lawrence Livermore National Laboratory [email protected] Lawrence Livermore National Laboratory [email protected]

1 Introduction In this paper, we investigate performance of a fully implicit formulation and solution method of a diﬀusion-reaction system modeling radiation diﬀusion with material energy transfer and a fusion fuel source. In certain parameter regimes this system can lead to a rapid conversion of potential energy into material energy. Accuracy in time integration is essential for a good solution since a major fraction of the fuel can be depleted in a very short time. Such systems arise in a number of application areas including evolution of a star [1] and inertial conﬁnement fusion [2]. Previous work has addressed implicit solution of radiation diﬀusion problems [3–8]. More recent work has looked at implicit and semi-implicit solution of reaction-diﬀusion systems. In general, a fully implicit method has been found to be the most accurate method for diﬃcult coupled nonlinear equations [9, 10]. In previous work, we have demonstrated that a method of lines approach coupled with a BDF time integrator and a Newton-Krylov nonlinear solver could eﬃciently and accurately solve a large-scale, implicit radiation diﬀusion problem [7, 8]. In this paper, we extend that work to include an additional heating term in the material energy equation and an equation to model the evolution of the reactive fuel density. This system now consists of three coupled equations for radiation energy, material energy, and fuel density. The radiation energy equation includes diﬀusion and energy exchange with material energy. The material energy equation includes reaction heating and exchange with radiation energy, and the fuel density equation includes its depletion due to the fuel consumption. In many applications, the added heat source involves a reaction rate with a strong nonlinear dependence on material temperature and thus provides a good test for an implicit solution method. We use an approximation to the reaction rate valid for temperature regimes less than 1 keV where the rate has its strongest dependence on material temperature. In particular, the rate

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depends on temperature to the ﬁfth power in this regime. While the actual reaction rate drops oﬀ at high temperature, we use the the ﬁfth power ﬁt for any temperature in order to investigate the performance of implicit solution techniques on problems with strong nonlinearities. The remainder of this paper is organized as follows. The next section describes the model and presents the equations we are solving. Section 3 describes the implicit solution method we employ, and Sect. 4 presents numerical results illustrating accuracy and eﬃciency of the solution method. Conclusions are presented in the last section.

2 Mathematical Model We consider a ﬂux-limited formulation of radiation diﬀusion including a model for heating due to a fusion source term. The evolution of the fusion fuel density is determined by an equation which models the depletion of the fuel as its fusion energy is added to the material energy. The radiation diﬀusion model is given by [11, 12] 4 c ∂ER =∇· ∇ER + cρκP (TM ) · aTM − ER , (1) ∇E R ∂t 3ρκR (TR ) + ER

where ER (x, t) is the radiation energy density (x = (x, y, z)), TM (x, t) is the material temperature, ρ(x) is the material density, c is the speed of light, and a = 4σ/c where σ is the Stephan–Boltzmann constant. The Rosseland opacity, κR , is a nonlinear function of the radiation temperature, TR , which is deﬁned by the relation ER = aTR4 . The Planck opacity, κP , is a nonlinear function of material temperature, TM , which is related to the material energy through an equation of state, EM = EOS(TM ). In this paper, when we use variable opacity and speciﬁc heat their values are taken from the LEOS equation-of-state package [13] which determines opacities and speciﬁc heats via table look-up. In some simpliﬁed test problems we use the relation, EM = ρcv TM , for the speciﬁc heat and constant values for opacities. In the ﬂux limiter, the norm · is taken to be the l2 norm of the gradient vector. In the simulations presented here we use either Dirichlet or Neumann boundary conditions on the radiation energy. This equation is solved in conjunction with two other equations. One equation expresses the conservation of material energy [11, 12] given by 4 ∂EM = −cρκP (TM ) · aTM − ER + er σv ρF 2 ∂t

TM T0

5 .

(2)

The heating term is controlled by a fusion model [14](p. 13) of fuel density given by

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5

∂ ρF TM = −σv ρF 2 , (3) ∂t T0 where ρF is the fuel density. This evolution equation has been used in controlled fusion models [15]. cm3 , with a refThe ﬁt to the reaction rate gives, σv = 1.43 × 103 sec−g erence temperature, T0 , of 1 keV. For the energy per reaction we use, erg . This value corresponds to the alpha particle energy er = 1.35 × 1018 sec−g from deuterium tritium fusion. We neglect the neutron energy from the reaction and assume the alpha particle has zero range, thus depositing its energy locally. Both the material energy and the fuel density equations contain the nonlinear temperature dependence term (TM /T0 )5 . This dependency is a good ﬁt to a tritium-deuterium reaction rate at low temperature (less than a few keV) such as in a tokamak fusion experiment [16]. The reference temperature, T0 , is a normalization constant (1 keV) included to simplify the units of the ﬁtted reaction rate σv . The actual reaction is less strongly dependent on TM above 1 keV, but, as noted above, we will use this approximation above that temperature. The fusion fuel is assumed to be a 50:50 mixture of tritium and deuterium with ρF representing the tritium or deuterium density. The binary nature of the reaction leads to the ρF 2 dependence in the reaction rate. The energy per reaction added to the material energy equation is er . Obviously there are a signiﬁcant number of physical processes omitted from this simple model. One relevant to the deposition of heat, is that we assume the energy of reaction is deposited locally. The correct physical process would distribute the heat spatially due to the ﬁnite range of the charged particles in the matter. We chose not to model this nonlocal eﬀect here.

3 Numerical Methods In this section we present both the spatial and temporal discretization methods used for solution of the system (1)–(3). 3.1 Spatial Discretization For spatial discretization, we employ a cell-centered ﬁnite diﬀerence scheme. We use a tensor product grid with Nx , Ny , and Nz cells in the x, y, and z directions, respectively. Deﬁning ER,i,j,k (t) ≈ ER (xi,j,k , t), EM,i,j,k (t) ≈ EM (xi,j,k , t), and ρF,i,j,k (t) ≈ ρF (xi,j,k , t) with xi,j,k = (xi , yj , zk ), and ⎛ ⎜ ER ≡ ⎝

ER,1,1,1 .. . ER,Nx ,Ny ,Nz

⎞

⎛

⎟ ⎜ ⎠ EM ≡ ⎝

EM,1,1,1 .. . EM,Nx ,Ny ,Nz

⎞

⎛

⎟ ⎜ ⎠ ρF ≡ ⎝

ρF,1,1,1 .. .

ρF,Nx ,Ny ,Nz

⎞ ⎟ ⎠,

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we can write our discrete equations in terms of a discrete diﬀusion operator given by T L(ER ) ≡ L1,1,1 (ER ), · · · , LNx ,Ny ,Nz (ER ) , (4) a local coupling operator given by S(ER , EM ) ≡ (S1,1,1 (ER , EM ), · · · , SNx ,Ny ,Nz (ER , EM ))T ,

(5)

and the local reaction rate operator given by R(EM , ρF ) ≡ (R1,1,1 (EM , ρF ), · · · , RNx ,Ny ,Nz (EM , ρF ))T ,

(6)

where, as in [8] Li,j,k (ER ) ≡ ER,i+1,j,k − ER,i,j,k ER,i,j,k − ER,i−1,j,k − Di−1/2,j,k Di+1/2,j,k /∆xi (7) ∆xi+1/2,j,k ∆xi−1/2,j,k ER,i,j+1,k − ER,i,j,k ER,i,j,k − ER,i,j−1,k + Di,j+1/2,k − Di,j−1/2,k /∆yj ∆yi,j+1/2,k ∆yi,j−1/2,k ER,i,j,k+1 − ER,i,j,k ER,i,j,k − ER,i,j,k−1 + Di,j,k+1/2 − Di,j,k−1/2 /∆zk ∆zi,j,k+1/2 ∆zi,j,k−1/2 with the diﬀusion coeﬃcients evaluated on the face centers, c , 3ρi+1/2,j,k κR,i+1/2,j,k + ∇ER i+1/2,j,k /ER,i+1/2,j,k c ≡ , 3ρi−1/2,j,k κR,i−1/2,j,k + ∇ER i−1/2,j,k /ER,i−1/2,j,k

Di+1/2,j,k ≡ Di−1/2,j,k

with y and z terms similarly deﬁned, 4 − ER,i,j,k , Si,j,k (ER,i,j,k , EM,i,j,k ) = cρi,j,k κP,i,j,k aTM,i,j,k

(8)

and Ri,j,k (EM,i,j,k , ρF,i,j,k ) = σv ρ2F,i,j,k

TM,i,j,k T0

5 .

(9)

Thus, our discrete scheme is to ﬁnd ER (t) and EM (t) such that, dER = L(ER ) + S(ER , EM ), dt dEM = −S(ER , EM ) + er R(EM , ρF ), dt dρF = −R(EM , ρF ). dt For more details, see [7, 8].

(10) (11) (12)

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3.2 Time Integration and Solvers We formulate (10)–(12) as an implicit system of ordinary diﬀerential equations (ODEs) and use an ODE time integrator to handle the implicit time step selection. In particular, we employ the parallel ODE solver, CVODE [17], developed at Lawrence Livermore National Laboratory and included in the SUNDIALS package [18]. CVODE employs the ﬁxed leading coeﬃcient variant of the Backward Diﬀerentiation Formula (BDF) method [19, 20] and allows for variation in the order of the time discretization as well as in the time step size. Time step sizes are chosen to minimize the local truncation error and thus give a solution that obeys a user-speciﬁed accuracy bound. This time integration technique leads to a coupled, nonlinear system of equations that must be solved at each time step. For example, solving the ODE system y˙ = f (t, y), (13) with the backward Euler method leads to the following nonlinear system yn − yn−1 = f (tn , yn ) ∆t

(14)

that must be solved at each time step. For the solution of this system, we use an inexact Newton–Krylov method with Jacobian-vector products approximated by ﬁnite diﬀerences. As the methods in CVODE are predictor-corrector in nature, an explicit predictor (e.g., forward Euler in the case above) is used for an initial guess in the nonlinear solve. In the methods discussed above, we use the scaling technique incorporated into CVODE. Thus, we include an absolute tolerance (ATOL) for each unknown and a relative tolerance (RTOL) which is applied to all unknowns. These tolerances are then used to form a weight which is applied to each solution component during the time step from tn−1 to tn . This weight is given as i | + AT OLi wi = RT OL|yn−1

and is also used to weight a root mean square norm 1/2 N −1 2 (yi /wi ) yW RM S = N

(15)

(16)

1

which is applied to all error-like vectors within the solution process. We use the GMRES Krylov iterative solver for solution of the linear Jacobian system at each Newton iteration [21]. The tolerance for the Newton iteration is taken to guarantee that iteration error introduced from the nonlinear solver is smaller than the local truncation error. For more details regarding the step size and order selection strategies in CVODE, as well as acceptance of a step and nonlinear convergence, we refer the reader to the review article [18].

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3.3 Preconditioning Preconditioning is generally essential when using Krylov linear solvers. To describe our preconditioning strategy, we begin by considering the content and structure of the Jacobian matrix. In (13), set y = (ER T , EM T , ρF T )T , and then form f using the right-hand sides of (10)–(12). The Jacobian matrices used in the Newton method are of the general form F (y) = (I − γJ), where J = ∂f /∂y is the Jacobian of the nonlinear function f , and the parameter γ ≡ ∆tβ with ∆t the current time step value and β a coeﬃcient depending on the order of the BDF method. Recalling the deﬁnitions of the discrete divergence, coupling and reaction rate operators, the block form of the Jacobian of f is ⎞ ⎛ ∂S/∂EM 0 ∂L/∂ER + ∂S/∂ER −∂S/∂EM + er ∂R/∂EM er ∂R/∂ρF ⎠ −∂S/∂ER J =⎝ 0 −∂R/∂EM −∂R/∂ρF ⎛ ⎞ A+G B 0 −B + er C er H ⎠ , = ⎝ −G 0 −C −H where A = ∂L/∂ER , G = ∂S/∂ER , B = ∂S/∂EM , C = ∂R/∂EM , and H = ∂R/∂ρF . We note that G, B, C and H are diagonal matrices. In all of our preconditioning strategies, we neglect the nonlinearity in the diﬀusion term and use the approximation [7, 8] ˆ E ˆ R ) ≡ A, ˆ R )/∂ER ≈ L( ˜ A = ∂L(E ˆ R )/∂ER is the Jacobian of L evaluated at a radiation energy, E ˆ R. where ∂L(E The size of the neglected term is related to the derivatives of the Rosseland opacity and the ﬂux-limiter. Our motivation for neglecting this term arises from the fact that −A˜ is symmetric and positive deﬁnite, whereas −A is not, and we thus can use multigrid methods for solution of diﬀusion terms within the preconditioner. ˜ as Our preconditioning strategy is to factor the matrix, M ≡ I − γ J, ⎞ ⎛ ⎞ ⎛ P Q 0 I − γ(A˜ + G) −γB 0 ⎝U T V ⎠ ≡ ⎝ γG I − γ(er C − B) −γer H ⎠ = M 0 Y Z 0 γC I + γH The preconditioner solve then consists of solving MSchur x = b for x where, ⎞⎛ ⎞ ⎛ ⎞⎛ S 0 0 I 0 0 I QT˜−1 0 MSchur = ⎝ 0 0 ⎠ ⎝ T˜−1 U I 0 ⎠ I V Z −1 ⎠ ⎝ 0 T˜ −1 0 0 I 0 0 Z 0 Z Y I (17) with S = P − QT˜−1 U and T˜ = T − V Z −1 Y. Thus,

Implicit Solution of Non-Equilibrium Radiation Diﬀusion

⎛

⎞

⎛

359

⎞

x1 S −1 (b1 − QT˜−1 b˜2 ) ⎝ x2 ⎠ = ⎝ T˜−1 (b˜2 − U x1 ) ⎠ x3 Z −1 (b3 − Y x2 )

(18)

where b˜2 = b2 − V Z −1 b3 . If the Schur complement, S, is exactly inverted, there will be no error associated with this preconditioner for the non-ﬂux-limited, constant opacity case. In addition, because G, B, C, H and hence T˜ and Z are diagonal, there is no penalty associated with inverting T˜ for every iteration of a method that inverts S. Also note that S is formed by modifying the diagonal of P and thus is composed of a symmetric diﬀusion-like matrix with a modiﬁed diagonal. Hence, we can employ multigrid methods to invert this Schur complement. Dependence of opacities on temperatures can give rise to large spatial gradients and thus a very heterogeneous problem. Hence, to invert the Schur complement matrix, S, we use a multigrid method designed to handle large changes in problem coeﬃcients. In particular, we use one V-cycle of the semicoarsening multigrid algorithm developed by Schaﬀer [23,24] as our multigrid solver. Semi-coarsening multigrid methods have been found to be quite effective on highly heterogeneous problems [25]. Details of this method can be found in the cited references, and more information about multigrid methods in general can be found in [26].

4 Results In this section we demonstrate the above solution method on the implicit formulation of (1)–(3). In the ﬁrst two subsections we present illustrations of problems modeled by this system of equations. In the next two subsections we give results which verify the accuracy and convergence of the method. 4.1 1D Solution Illustration In this section we illustrate how the time evolution of the solution is affected by the initial fuel density. This 1D problem has a domain from 0 cm to 10.0 cm discretized with 100 grid points. The fuel is initialized using a step function with fuel density, ρF 0 , on the left half of the domain and 0 on the right. The initial radiation and material temperatures are equal and constant in the domain. Neumann boundary conditions are applied on the R left boundary, ∂E ∂x = 0, while Dirichlet boundary conditions are applied on the right boundary with the temperature set to the initial value. The relative tolerance requested was RTOL = 10−7 , and the absolute tolerances were set for temperatures and fuel density as 10−6 , 10−6 , and 10−24 for the radiation, material, and fuel density, respectively. Results are presented for two diﬀerent initial conditions; one with a high fuel density, and the other with a low fuel density. The high density simulation

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has ρF 0 of 0.03 g/cm3 and TR = TM of 0.5 keV. The low density simulation uses ρF 0 = 0.01 g/cm3 and TR = TM = 0.776 keV. These initial temperatures 5 , are approximately the were selected so that the initial heating rates, ρ2F TM same for both simulations. Figure 1 illustrates the evolution of the radiation and material temperature, as well as fuel density of a point near the left boundary. As can be seen, the high density case yields a very rapid increase in material temperature and consumes most of the fuel. The low density case results in a slower evolution of the material temperature. The slowly changing temperature allows for more energy to be transfered to the radiation and loss via diﬀusion. Since the temperature is lower in this case, a smaller fraction of the fuel is consumed. ρF0 = 0.01

ρF0 = 0.03 0.015 1

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Fuel Density (g/cm3)

Temperature (keV)

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0.05

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Fig. 1. Evolution of radiation temperature, material temperature, and fuel density for two simulations. Values plotted are for a point near the left boundary. Linear scale used on the low density case and a log scale used on the high density case

Figure 2 shows proﬁles of radiation temperature, material temperature, and fuel density at various times for the ρF 0 = 0.03 g/cm3 simulation. Diﬀusion loss of energy results in a lower temperature in the outer region. Thus, the inner region begins to consume fuel and heat up sooner than the outer region resulting in a steep temperature gradient that sweeps through the fuel region (see Fig. 2). These results are similar to the reaction-diﬀusion wave results presented in the next section. Figure 3 gives the history of the time step and integration order for the higher fuel density case. The smallest step size, which occurs during the rapid heating stage, is 3.26 × 10−8 µs. The longest time, in the quiescent period at the end of the simulation, is 5.31 × 10−4 µs. This large diﬀerence in step sizes illustrates the ability of our method to select time step size and integration order resulting in the largest time step possible subject to the accuracy constraints.

Implicit Solution of Non-Equilibrium Radiation Diﬀusion Radiation Temperature

Fuel Density −2

10

(g/cm3)

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(keV)

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Material Temperature 15

t =0.0002 µs t =0.0076 µs t =0.0082 µs t =0.0598 µs

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Fig. 2. Radiation temperature, material temperature and fuel density proﬁles at various times for the ρF 0 = 0.03 g/cm3 run −3

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Fig. 3. History of time step and integration order

4.2 1D Reaction Diﬀusion Wave In the previous section we initialized the simulation with a uniform initial material temperature and a step function for the initial fuel density. The simulation in this section begins with a uniform fuel density and a step function in the initial material temperature. These initial conditions produce a reaction diﬀusion wave which is propagated by the diﬀusion of radiation energy and is driven by the energy from the fusion reaction. The 1D domain for this simulation is 0.0 to 2.0 cm, with a uniform fuel density of 0.1 g/cm3 , and uses 200 grid points. The initial background radiation and material temperatures are set to 0.1 keV except for a small region on the left from 0 to 0.1 cm where the material energy is initialized to 2.0 keV. erg cm3 and er = 1.35×1018 sec−g . The reaction parameters are, σv = 1.43×103 sec−g

D.E. Shumaker and C.S. Woodward Time = 0.00030 µs

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6 4

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0.04 0.02

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Fig. 4. Proﬁles at various times for a reaction diﬀusion wave moving from left to right

This simulation also uses the LEOS equation-of-state data base [13] for hydrogen. The high density region begins to heat much faster than the remainder of the domain. Diﬀusion of radiation energy heats the material in front of the wave leading to more heating due to the fusion reactions. Figure 4 gives proﬁles at four diﬀerent times. 4.3 0D Analytic Test In this section we demonstrate accuracy of the implicit solution for fusion heating of the material and the fuel evolution by a comparison with an analytic solution. In order to obtain the analytic results, two simplifying assumptions are made. First, we assume κP = 0. This assumption eliminates the exchange of energy between radiation and material and thus removes the radiation energy equation and diﬀusion from the system. With this assumption the material energy equation reduces to, 5 TM ∂EM 2 = er σv ρF . (19) ∂t T0

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Second, the relationship between EM and TM , normally determined by tables in LEOS, is replaced by the simple linear relation, EM = ρcv TM , where the speciﬁc heat, cv , is assumed to be a constant. From the coupled pair of (19) and (3), it can be shown that the sum of the material energy, EM , and the potential nuclear energy, er ρF , W = EM + er ρF ,

(20)

is independent of time. Thus we can use the conservation relation, EM (t) + er ρF (t) = EM 0 + er ρF 0 ,

(21)

to eliminate EM (t) from (3), which becomes, ∂ ρF 5 = −ρ2F (τ ) (b − ρF (τ )) , ∂τ

(22)

where b = EM 0 /er + ρF 0 and τ = σv ( ρcevrT0 )5 t, and where the initial fuel density and material energy are ρF 0 and EM 0 , respectively. With some help from Mathematica the solution can be expressed as a transcendental equation, L(ρF 0 ) + P (ρF 0 ) − L(ρF (τ )) − P (ρF (τ )) = τ ,

(23)

where L and P are given by, L(ρF (τ )) =

5 (log(ρF (τ )) − log(ρF (τ ) − b)) , b6

(24)

and, P (ρF (τ )) =

−4b4 +

ρ

ρ

500 3 260 2 2 F (τ ) F (τ ) − 3 b 12 b 4 5 4b (b − F (τ ))

ρ

+ 70bρ3F (τ ) − 20ρ4F (τ )

ρF (τ )

.

(25) This analytic result can be used in two diﬀerent ways. We could solve the transcendental (23) for ρF (τ ) at a number of diﬀerent times, τ , then compare these values with the computed solution from our code. Or, we could use (23) to solve for τ when ρF (τ ) is some fraction, α, of the initial value, ρF (0). We chose the latter method, since it avoids any computation error associated with a transcendental solution. The time at which ρF (τ ) = αρF 0 is obtained from (23), (26) τα = L(ρF 0 ) + P (ρF 0 ) − L(αρF 0 ) − P (αρF 0 ) The τ , and thus t, determined by the above equation, is used to verify our solution. After we have established the parameters for a test run and selected a value for α, we use this equation to determine a stopping time for the simulation. The simulation computes the ﬁnal fuel density, and this density is then compared to the expected αρF 0 .

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For this test problem we use the following parameters, ρ = 1.0 g/cm3 , erg cm3 cv = 5.755 × 1014 erg/keV, er = 1.35 × 106 sec−g , and σv = 1.43 × 103 sec−g . The initial material temperature is 50.0 keV, and the initial fuel density is 0.1 g/cm3 . We chose three values of α, (0.5, 0.1 and 0.01), to determine check point times. These times are indicated on Fig. 5 which gives the fuel density versus time for this simulation.

0.1 0.09

Fuel Density (g/cm3)

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

0

0.2

0.4

0.6

0.8

1

time (µs)

1.2

1.4

1.6

1.8

2 x 10−6

Fig. 5. Evolution of fuel density for 0D analytic comparison test problem. Markers indicate the compared test values at ρF = 0.5 ρF (0), ρF = 0.1 ρF (0) and ρF = 0.01 ρF (0)

Table 1 gives the relative error in fuel density for values of RTOL from 10−4 to 10−12 along with the number of time steps, NST. Note that absolute tolerances were set to equal the relative tolerances. We see that the error decreases linearly with the RTOL value indicating good convergence of the implicit solution method. We also see a fairly large increase in the number of time steps required to resolve the solution for RTOL values less than 10−8 . As one would expect, requiring very high accuracy comes at a price in computation time.

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365

Table 1. Statistics and Error for 0D Analytic Problem

ρF

= 0.5ρF (0)

ρF

= 0.1ρF (0)

ρF

= 0.1ρF (0)

RTOL

NST

ERROR

NST

ERROR

NST

ERROR

10−4 10−6 10−8 10−10 10−12

144 157 173 258 484

−1.01 × 10−2 −1.65 × 10−3 −2.79 × 10−5 −2.93 × 10−7 −9.84 × 10−9

159 177 217 359 686

−2.09 × 10−2 −3.08 × 10−3 −5.18 × 10−5 −5.67 × 10−7 −1.72 × 10−8

196 219 289 500 967

−1.74 × 10−3 −4.81 × 10−4 −4.51 × 10−6 4.72 × 10−9 −1.87 × 10−10

NST = time steps

4.4 2D Results In this section we present results for a 2D problem illustrating the convergence with respect to reducing tolerances and varying the maximum allowed order of integration. The simulation domain consists of a square region 1 cm by 1 cm with 50 grid points in each direction. The initial fuel density is centered at x = y = 0.0 with a smooth radial distribution given by, " 3 ρF (x) = 0.1 B x2 + y 2 , 0.75 (g/cm ) , (27) where the bicubic radial distribution, B(x, ), ⎧ +x 2 6 − 8 +x , ⎪ 2 ⎨ 2 2 −x 2 −x B(x, ) ≡ 2 6 − 8 2 , 2 ⎪ ⎩ 0,

is given by, if − < x ≤ 0 ; if 0 ≤ x < ;

.

(28)

otherwise .

This simulation uses the LEOS equation-of-state data base [13] for hydro3 gen which has a uniform density of 5.0 g/cm . Neumann boundary conditions are applied at the x = 0 and y = 0 boundaries. Dirichlet boundary conditions are applied at x = 1 and y = 1 where radiation temperature is set to 0.5 keV. Initial Radiation and material temperatures were also set to this value. For erg . The this simulation, the energy per reaction, er , is set to 1.35 × 1018 sec−g 3

cm as determined for plots reaction rate ﬁt parameter, σv , is 1.43 × 103 sec−g in [16]. The reference temperature, T0 , is 1 keV. Figure 6 gives the time histories of material and radiation temperatures and fuel density for the point x = 0, y = 0. The material temperature rises 5 dependency in the heating rate leads to slowly initially; however, the TM a rapid increase resulting in a nearly complete depletion of the fuel at this point. For points in the outer region which have a lower initial fuel density, the evolution is somewhat diﬀerent. Radiation energy from the hotter central region passes through this outer region. The parameters for this simulation,

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TR TM ρF −2

10

0

10

Fuel Density (g/cm3)

Temperature (keV)

10

−3

10

0

0.005

0.01

0.015

time (µs) Fig. 6. Time dependence of radiation and material temperatures and fuel density at x = 0, y = 0

however, limit the exchange of energy between radiation and material. The combination of a large diﬀusion loss rate and a poor coupling of radiation and material energy results in a lower material temperature. Due to the strong 5 , the lower temperature dependence of the reaction rate on temperature, TM outer region can have a signiﬁcantly lower reaction rate. This lower rate results in incomplete consumption of fuel in this region. Figure 7 give contours of material temperature, radiation temperature and fuel density at t = 0.0085 µs. At this time, the fuel has been signiﬁcantly depleted in the central region. The material temperature is high around the inner edge of the remaining fuel and the central material temperature is held down by the transfer to the cooler radiation. The radiation is heated by the cylindrical shell-like region of high material temperature. This heating, along with diﬀusion, keeps the radiation temperature in the central region ﬂat. Lastly, the shell-like region of unconsumed fuel is still present at the end of the simulation. In order to study the convergence of this problem with respect to reducing the tolerance, we made several simulations varying RTOL = ATOL from 10−7 to 10−10 for maximum orders of integration of 2 and 5. For this series of runs the numerical statistical counters and parameters deﬁned by, RTOL MO NST NNI NLI RT

= relative tolerance, = maximum order allowed, = time steps, = nonlinear iterations, = linear iterations, = run time in seconds,

Implicit Solution of Non-Equilibrium Radiation Diﬀusion Fuel Density(g/cm3)

Material Temperature (keV) 0.9

0.9

0.8

0.8

0.8

0.7

0.7

0.7

0.6

0.6

0.6

0.5 0.4

y (cm)

0.9

y (cm)

y (cm)

Radiation Temperature (keV)

0.5 0.4

0.5 0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1 0.2

0.6 0.8

0.4 0.6 x (cm) 1

0.8

0.1 0.2

1.2 1.4 1.6 1.8

367

1

0.4 0.6 x (cm) 1.5

2

0.8

2.5

3

0.2

0.005

0.4 0.6 x (cm) 0.01

0.015

0.8

0.02

Fig. 7. Radiation temperature, material temperature and fuel density at t = 0.0085 µs

are given in Table 2. Here we see a substantial decrease in the run times due to using a higher maximum integration order. The time stepping algorithm is able to use the higher integration order to meet accuracy requirements with larger time steps resulting in a faster overall runtime for a given accuracy with the higher maximum integration order. This advantage is seen in savings for all integration statistics, including the cumulative numbers of linear and nonlinear iterations required for solution. Lastly, we again see a signiﬁcant increase in run time to meet accuracy requirements for RTOL values below 10−9 . Table 2. Statistics for 2D Fusion Fuel Problem RTOL −7

10 10−7 10−8 10−8 10−9 10−9 10−10 10−10

MO

NST

NNI

NLI

2 5 2 5 2 5 2 5

17,384 14,510 45,768 35,686 113,545 81,535 266,722 172,137

20,807 17,479 56,333 45,797 140,788 110,791 323,349 242,944

62,322 52,369 136,973 116,451 292,722 224,178 621,406 433,689

RT (sec) 1,730 1,449 4,152 3,347 9,313 7,414 20,564 15,500

As a measure of accuracy for these runs, we determined the maximum relative error in material temperatures over the entire 2D grid. This quantity is shown in Fig. 8. The relative error is computed with respect to the most highly resolved run, RTOL = 10−10 , MO = 5. This plot indicates that the reduction in error is approximately linearly related to RTOL. Note that Fig. 8 also indicates that MO = 5 runs are more accurate than MO = 2 for the same RTOL. The relative error plots are all similar in that the maximum is

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10

−3

Max relative error

10

−4

10

−5

10

MO = 5 RTOL = 10−7 RTOL = 10−8 RTOL = 10−9 MO = 2 RTOL = 10−7 RTOL = 10−8 RTOL = 10−9

−6

10

−7

10

−8

10

5

6

7

8 time (µs)

9

10 −3

x 10

Fig. 8. Maximum relative error in material temperature vs time for maximum order, MO, 2 and 5

during the sharp rise in material temperature. Relative error plots for the other components of the system, radiation temperature and fuel density, are similar and are thus not shown.

5 Conclusions We have presented fully implicit solutions of a highly nonlinear three equation system including: (1) radiation energy evolution with diﬀusion and exchange with material energy; (2) material energy evolution with reaction heating and exchange with radiation energy; and (3) fuel density evolution with depletion due to consumption. Our solution method makes use of high order in time integration methods, inexact Newton-Krylov nonlinear solvers, and multigrid preconditioning using a Schur complement strategy. We have demonstrated accurate solution of the fusion heating and fuel evolution by comparison with an analytic solution for a simpliﬁed problem. Using a 2D problem which encompasses all the processes included in our model, we have demonstrated convergence to a highly resolved result. Our test problems have shown that using a higher order of time integration leads to a more accurate and eﬃcient method than with lower orders.

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Acknowledgements This work was performed under the auspices of the U. S. Dept. of Energy by University of California, Lawrence Livermore National Laboratory under contract W-7405-ENG-48.

References 1. A. J. Mewadows, Stellar Evolution, Pergamon Press, 1978, p. 97. 2. J. J. Duderstadt, G. A. Moses, Inertial Conﬁnement Fusion, John Wiley and Sons, 1982, p. 131. 3. D. A. Knoll, W. J. Rider, G. L. Olson, Nonlinear convergence, accuracy, and time step control in nonequilibrium radiation diﬀusion, J. Quant. Spec. and Rad. Trans. 70 (1) (2001) 25–36. 4. V. A. Mousseau, D. A. Knoll, W. J. Rider, Physics-based preconditioning and the Newton–Krylov method for non-equilibrium radiation diﬀusion, J. of Comput. Phys. 160 (2000) 743–765. 5. D. J. Mavriplis, Multigrid approaches to non-linear diﬀusion problems on unstructured meshes, Num. Lin. Alg. with App. 8 (8) (2001) 499–512. 6. L. Stals, Comparison of non-linear solvers for the solution of radiation transport equations, Elec. Trans. Num. Anal. 15 (2003) 78–93. 7. P. N. Brown, C. S. Woodward, Preconditioning strategies for fully implicit radiation diﬀusion with material-energy transfer, SIAM J. Sci. Comput. 23 (2) (2001) 499–516. 8. P. N. Brown, D. E. Shumaker, C. S. Woodward, Fully implicit solution of largescale non-equilibrium radiation diﬀusion with high order time integration, J. Comp. Phys. 204 (2) (2005) 760–783. 9. D. L. Ropp, J. N. Shadid, C. C. Ober, Studies of the accuracy of time integration methods for reaction-diﬀusion equations, J. Comp. Phys. 194 (2004) 544–574. 10. C. C. Ober, J. N. Shadid, Studies on the accuracy of time integration methods for the radiation-diﬀusion equations, J. Comp. Phys. 195 (2004) 743–772. 11. G. C. Pomraning, The Equations of Radiation Hydrodynamics, Pergamon, New York, 1973. 12. R. L. Bowers, J. R. Wilson, Numerical Modeling in Applied Physics and Astrophysics, Jones and Bartlett, Boston, 1991. 13. E. M. Corey, D. A. Young, A new prototype equation of state data library, Tech. Rep. UCRL-JC-127698, Lawrence Livermore National Laboratory, Livermore, CA, submitted to American Physical Society Meeting (1997). 14. J. Wesson, Tokamaks, Clarendon Press, Oxford, 2004, p. 13. 15. D. J. Rose, J. Melville Clark, Plasmas and Controlled Fusion, M.I.T. Press, 1961, p. 322. 16. T. J. Dolan, Fusion Research Vol. 1, Principles, Pergamon Press, 1980, p. 29. 17. G. D. Byrne, A. C. Hindmarsh, PVODE, an ODE solver for parallel computers, Int. J. High Perf. Comput. Appl. 13 (1999) 354–365. 18. A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker, C. S. Woodward, SUNDIALS: Suite of nonlinear and diﬀerential/algebraic equation solvers, ACM Trans. Math. Softw. 31 (3) (2005) 363– 396.

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19. P. N. Brown, G. D. Byrne, A. C. Hindmarsh, VODE: A Variable-Coeﬃcient ODE Solver, SIAM J. Sci. Stat. Comput. 10 (5) (1989) 1038–1051. 20. K. R. Jackson, R. Sacks-Davis, An alternative implementation of variable stepsize multistep formulas for stiﬀ ODEs, ACM Trans. Math. Software 6 (1980) 295–318. 21. Y. Saad, M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 7 (3) (1986) 856–869. 22. S. F. Ashby, R. D. Falgout, A parallel multigrid preconditioned conjugate gradient algorithm for groundwater ﬂow simulations, Nuclear Science and Engineering 124(1) (1996) 145–159. 23. S. Schaﬀer, A semi-coarsening multigrid method for elliptic partial diﬀerential equations with highly discontinuous and anisotropic coeﬃcients, SIAM J. Sci. Comp. 20 (1) (1998) 228–242. 24. P. N. Brown, R. D. Falgout, J. E. Jones, Semicoarsening multigrid on distributed memory machines, SIAM J. Sci. Stat. Comput. 21 (5) (2000) 1823–1834. 25. J. E. Jones, C. S. Woodward, Newton-Krylov-multigrid solvers for large-scale, highly heterogeneous, variably saturated ﬂow problems, Advances in Water Resources 24 (2001) 763–774. 26. W. F. Briggs, V. E. Henson, S. F. McCormick, A Multigrid Tutorial, 2nd. Edition, SIAM, Philadelphia, PA, 2000.

Part IV

Mathematics and Computer Science

Transport Approximations in Partially Diﬀusive Media Guillaume Bal Department of Applied Physics and Applied Mathematics, Columbia University, New York NY, 10027, USA [email protected] Abstract. This paper concerns the analysis of approximations of transport equations in diﬀusive media. Firstly, we consider a variational formulation for the ﬁrstorder transport equation that has the correct diﬀusive behavior in the limit of small mean free paths. The associated bilinear form is shown to be coercive on a classical Hilbert space in transport theory with a constant of coercivity independent of the mean free path. This allows us to obtain the diﬀusion approximation of transport as an orthogonal projection onto a subspace of functions that are independent of the angular variable. Similarly, projections onto functions that are independent of the angular variable only in subsets of the full domain can be interpreted as a transport-diﬀusion coupling method. Convergence results based on averaging lemmas and error estimates are presented. Secondly, we address the problem of extended non-scattering layers or ﬁlaments surrounded by highly scattering media and derive generalized diﬀusion equations to model transport in such geometries.

1 Introduction The solution of linear transport equations of Boltzmann type describing the phase-space density of particles, is of interest in many applications ranging from neutron transport in nuclear reactor physics [14] and photon transport in human tissues to wave propagation in highly heterogeneous media [10,18,31]. Among the many deterministic methods available [23], several numerical schemes have been obtained recently by using a variational framework. Variational methods have a long history in transport theory, both in the analytical and numerical approaches of source term and eigenvalue transport problems. A very short and incomplete list of bibliographical references includes [8,11,14,30,32]. Although variational methods that involve symmetric positive deﬁnite forms have been known for a long time in the even-parity formulation of transport (see e.g. [1, 23, 27]), their derivations for the ﬁrst-order Boltzmann equation are more recent [7, 24–26]. Among the variational formulations involving a symmetric positive definite form over a Hilbert space deﬁned in [7, 25], some enjoy the property that the constant of coercivity is independent of the transport mean free path, which measures (locally) the main distance between successive interactions of the particles with the underlying media. This property ensures that discretizations of solutions of the variational problem are accurate in

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the transport regime, characterized by mean free paths comparable to the typical length of variation of the geometrical components of the transport equation, as well as in the diﬀusive regime, characterized by much smaller mean free paths. For references on the approximation of transport by diﬀusion, see e.g. [12, 19]. We consider in this paper the SAAF (Self-adjoint angular ﬂux) method presented in [26] and characterized explicitly in [7] as a variational method equivalent to the solution of the ﬁrst-order transport equation. We show that the symmetric bilinear form associated to the variational problem has indeed a constant of coercivity independent of the mean free path over a Hilbert space that is also deﬁned independently of the mean free path. One of the main advantages of symmetric variational formulations is that approximations of the solutions can be obtained by orthogonal projection (with respect to the bilinear form and onto a subspace of the considered Hilbert space). This is analyzed in detail in [24, 25] in the framework of ﬁrstorder system least squares (FOSLS). In this paper we show that the diﬀusion approximation of transport may be obtained as such an orthogonal projection (this already appears in [26]) and show that the error between the transport and the diﬀusion equations are linear in the mean free path (and quadratic in certain circumstances) as expected. The use of the variational formulation in conjunction with averaging lemmas allows us to obtain that the above error converges to zero in the limit of small mean free paths even when the limiting solution is not very regular. The advantage of the proposed method is that there is no need to construct any scaling transformation as in [24] or adding any terms accounting for boundary conditions as in [25] as both can be deduced directly from the variational formulation. In many applications, the mean free path will be small (compared to variations of the geometrical environment) in certain areas but not necessarily in the whole domain. A possible numerical approach consists then of solving the diﬀusion approximation where it is valid and the transport equation elsewhere. The main diﬃculty is then to couple both models at their common interface. Such a coupling was considered in the even-parity formulation of transport in [5]. See [16, 22, 33, 34] for additional references on the coupling problem. We consider here the transport-diﬀusion coupling as the orthogonal projection of the exact transport solution onto the subspace of the underlying Hilbert space of functions that depend only on the spatial variable in the diﬀusive domain but depend on both the spatial and angular variables in the rest of the domain. We also show the convergence of the coupled transportdiﬀusion solution (with appropriate error estimates) to the exact transport solution when diﬀusion is valid on the domain treated by the diﬀusion equation. It is interesting to observe that the ﬁrst-order transport equation is not exactly satisﬁed on the transport area (as opposed to the case where the full domain is modeled by transport as mentioned earlier; see [7]). Rather the coupling involves a second-order transport equation in the transport area.

Transport Approximations in Partially Diﬀusive Media

375

This exempliﬁes the inﬂuence of the boundary and interface conditions in the variational formulation of ﬁrst-order transport. Many more details on the full discretization of the transport equation using variational approaches can be found in [24–26]. We do not consider the discretization problem here except for a brief remark in Sect. 2.7. For general geometries of embedded domains where the diﬀusion approximation does not hold, the coupling mentioned above may be a useful alternative to solving the full transport equations. In certain situations however, a more macroscopic model can be derived. Initiated by the analysis of clear layers in optical tomography [13], generalized diﬀusion models have been derived in [4, 6] to account for the propagation of photons along straight lines in non-scattering clear layers. In this paper we generalize the analysis to non-scattering ﬁlaments in three dimensional geometries and obtain new generalized diﬀusion models. Possible applications in which similar models may be useful include γ ray propagation in astrophysics and radiation in atmospheric clouds. The rest of the paper is structured as follows. Section 2 presents the variational formulation of the ﬁrst-order transport equation. General symmetric scattering operators are considered in the two-dimensional model only where the decomposition over spherical harmonics is particularly simple as shown in Sect. 2.1. The variational formulation in the diﬀusive regime and its main properties are presented in Sect. 2.2. The diﬀusion approximation is derived in Sect. 2.3 and analyzed in the spherical harmonics expansion in Sect. 2.4. Convergence results and error estimates are given in Sects. 2.5 and 2.6. The transport-diﬀusion coupling is addressed in Sect. 3. Seen as an orthogonal projection in Sect. 3.1 the resulting local equations are show in Sect. 3.2. Finally error estimates are given in Sect. 3.3. The derivation of generalized diﬀusion models to account for thin non-scattering inclusions is addressed in Sect. 4.

2 Variational Formulation for Transport We start with the steady-state transport equation written in the form ω · ∇u + Gu = q, u = g,

X =Ω×V Γ− = {(x, ω) ∈ ∂Ω × V, ω · ν(x) < 0} .

(1)

Here u(x, ω) is the particle density in phase space, q(x, ω) is a volume source term, g(x, ω) is a boundary source term, Ω ⊂ Rn for n = 2 or n = 3 is the spatial domain, ν(x) is its outward unit normal at x ∈ ∂Ω, and V = S n−1 is the unit sphere in the monogroup-approximation of transport. As usual Γ− denotes the set of incoming conditions. The operator G is deﬁned as k(x, ω − ω)u(x, ω )dµ(ω ) . (2) Gu(x, ω) = σ(x)u(x, ω) − S n−1

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The integration measure is the normalized Lebesgue measure so that µ(S n−1 ) = 1. The total absorption σ(x) and the scattering coeﬃcient k(x, ω − ω) are positive bounded functions. The scattering coeﬃcient satisﬁes k(x, v) = k(x, −v) and is suﬃciently small so that G is a symmetric positive deﬁnite operator on L2 (S n−1 ) with bounded inverse G−1 . We now consider the variational formulation presented in [7] and mentioned in [26]. We ﬁrst recast the equation as G−1 (ω · ∇u) + u = G−1 (q), u = g,

X Γ− .

(3)

We now multiply the ﬁrst equation above by ω ·∇v for a smooth test function v(x, ω) and integrate over X to get: G−1 (ω · ∇u)ω · ∇v dp − ω · ∇uv dp + uvω · ν dq X X ∂Ω×V = G−1 (q)ω · ∇v dp . X

Here, dp = dxdµ(ω) and dq = dσ(x)dµ(ω), where dσ(x) is the surface measure on ∂Ω. Upon using the transport equation (1) we recast the above equality as ﬁnding u ∈ W such that a(u, v) = L(v), where

∀v ∈ W ,

(4)

G−1 (ω · ∇u)ω · ∇v + Guv dp + uvω · ν dq , Γ+ X (5) G−1 (q)ω · ∇v + qv dp + gv|ω · ν| dq L(v) = Γ− X W = u ∈ L2 (X), ω · ∇u ∈ L2 (X), uΓ± ∈ L2 (Γ± ; |ω · ν|dq) .

a(u, v) =

The set Γ+ = {(x, ω) ∈ ∂Ω × V, ω · ν(x) > 0}. It is easy to verify that with our assumptions on G, a(u, v) is a continuous, coercive, and symmetric bilinear form on the Hilbert space W (equipped with its natural norm) [12] and that L is continuous in the same sense. The above equation (4) thus admits a unique solution by the Lax-Milgram theory. 2.1 Harmonic Decomposition It is interesting to analyze the above variational formulation using the spherical harmonics decomposition. We concentrate on the two dimensional case n = 2 so that the velocity space V = S 1 is the unit circle. Generalizations to n = 3 using “classical” spherical harmonics are straightforward though more tedious computationally [25,26]. In this setting, directions are parameterized by ω = (cos θ, sin θ) for 0 ≤ θ < 2π. We identify u(ω) ≡ u(θ).

Transport Approximations in Partially Diﬀusive Media

Deﬁne the harmonic decomposition 2π 1 e−inθ u(ω)dθ(ω), Fu(n) = u ˆn = 2π 0

F −1 u ˆ(θ) = u(θ) =

377

einθ u ˆn .

n∈Z

(6) In this basis, the operator G may be written as a (diagonal) Fourier multiplier: G = F −1 (σ − kn )F ,

(7)

where the coeﬃcients σ and kn are such that |kn | < k0 < σ for |n| ≥ 1. Also, since G is a real-valued operator, we verify that k−n = k¯n , where the upper bar denotes complex conjugation. Because G is symmetric, we have here k−n = kn real-valued. The above decomposition means that Geinθ = (σ − kn )einθ and implies that G−1 = F −1 (σ − kn )−1 F .

(8)

We also verify the following convenient expression for the diﬀerential operator ∂ ∂ 1 ∂ 1 ∂ ¯ ω · ∇ = eiθ ∂ + e−iθ ∂, −i +i ∂= , ∂¯ = . (9) 2 ∂x ∂y 2 ∂x ∂y ¯vn+1 . We ﬁnally identify x = (x, y) This implies that (ω · ∇v)n = ∂ˆ vn−1 + ∂ˆ with z = x + iy in the complex plane. With this notation, we verify that the transport equation is equivalent to (10) eiθ ∂ + e−iθ ∂¯ u + Gu = q with appropriate boundary conditions. In the Fourier domain this is nothing but (11) ∂u ˆn−1 + ∂¯u ˆn+1 + (σ − kn )ˆ un = qˆn , Ω×Z. Recall the Parseval relation S 1 uvdµ = n∈Z u ˆn vˆn = n∈Z u ˆn vˆ−n since u and v are real-valued functions. The variational formulation (4) still holds with a(u, v) and L(v) recast as: 1 ¯v−n+1 ) (∂ u ˆn−1 + ∂¯u ˆn+1 )(∂ˆ v−n−1 + ∂ˆ a(u, v) = a − kn Ω n∈Z + (σ − kn )ˆ un vˆ−n dµ(z) + uvω · ν dq , Γ+ (12) 1 ¯v−n+1 ) + qˆn vˆ−n dµ(z) qˆn (∂ˆ v−n−1 + ∂ˆ L(v) = Ω n∈Z a − kn + gv|ω · ν| dq . Γ−

The above formulae prove very useful in the analysis of the transport solution in the diﬀusive regime.

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2.2 Variational Formulation in the Diﬀusive Regime We now consider the regime of high collisions and small absorption. This regime is characterized by replacing G, q and g by [12] 1 Gε u(x, ω) = k(x, ω − ω) u(x, ω) − u(x, ω ) dµ(ω ) + εσa u , ε S n−1 gε = εg . qε = εq, (13) Here gε is of order ε to avoid the presence of boundary layers at the leading order [12] (we will see below that a term of order ε1/2 rather than ε would have been suﬃcient). We recast Gε = ε−1 Q + εσa , where σa is uniformly bounded 2 from below by a positive constant so that G−1 ε is bounded on L (X) though not uniformly in ε. The local ﬁrst-order transport equation (1) reads in this regime: (14) ω · ∇uε + Gε uε = εq . We recast (4) in this regime as ﬁnding u ∈ W such that ∀v ∈ W ,

aε (u, v) = Lε (v), where

1 aε (u, v) = ((εGε ) (ω · ∇u)ω · ∇v + ε Gε (u)v)dp + ε X G−1 gv|ω · ν| dq . Lε (v) = ε (q)ω · ∇v + qv dp + X

−1

−1

(15) uvω · ν dq , Γ+

Γ−

(16) We now derive some properties of the above variational formulation and the above bilinear form that make them attractive theoretically and computationally. We ﬁrst assume that Gε is invertible (in L2 (V )) with inverse given by 1 G−1 u ¯ + εHε u, Hε u = 0 , (17) ε u= εσa operator in L2 (X) with norm where Hε u is a symmetric and bounded ¯ = S n−1 u(ω)dµ(ω) is the angular average bounded by α−1 < ∞, and u of u. This property holds when Gε is decomposed over spherical harmonics; see for instance (37) and (41) below. This allows us to recast (16) as 1 Gε uv dp ω · ∇u ω · ∇v + H (ω · ∇u)ω · ∇v + aε (u, v) = ε ε2 σ a ε X 1 + uvω · ν dq (18) ε Γ+ 1 Lε (v) = ω · ∇v + εHε (ω · ∇v) + v q dp + gvω · ν dq . εσa X Γ−

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Let us assume that Q and Hε are coercive on L2 (X) with coercivity constants α and σ0−1 , respectively, in the sense that: (Qu, u) ≥ αu2 ,

(Hε u, u) ≥

1 u2 . σ0

(19)

Here f is the L2 (X) norm of f . These are reasonable assumptions when Gε can be diagonalized in the basis of spherical harmonics; see (37) and (41) below. Let us also assume that the absorption 0 < σa1 ≤ σa (x) ≤ σa0 on Ω. We then verify that 1 1 ω · ∇uε 2 + 2 ω · ∇uε 2 σ0 ε σa0 α 1 + 2 uε − uε 2 + σa1 uε 2 + uε|Γ+ 2b ≤ aε (uε , uε ) . ε ε

(20)

Here f b denotes the L2 (Γ± ; |ω ·ν|dq) norm of f deﬁned on Γ± . This implies that aε is coercive on W with constant of coercivity independent of ε. The constant of coercivity also depends on the scattering kernel only through the constants α and σ0−1 . It is also straightforward to observe that aε (u, v) is continuous on W ×W , though with a continuity constant that depends on ε. In order to obtain a constant of continuity independent of ε, the Hilbert space may be equipped with a norm · Wε deﬁned as the (square root of the) left-hand side of (20); this is in essence the norm introduced in [24]. On Wε (i.e., W equipped with · Wε ) the form aε is coercive and continuous with constants of coercivity and continuity independent of ε (see also Sect. 2.7). The source term is also bounded on Wε with a constant independent of ε, and bounded on W with a constant that depends on ε: ε 1 ω · ∇uε q + ¯ q ω · ∇uε + quε + uε|Γ+ b gb . α εσa1 (21) The notation a b stands for a ≤ Cb for some constant C independent of ε. We deduce from the constraints on aε and Lε that the unique solution uε of the variational formulation (15) satisﬁes the (a priori) estimate |Lε (uε )|

√ 1 ω · ∇uε + ω · ∇uε + uε q + εgb . ε

(22)

This implies that |Lε (uε )| q2 + εg2b ,

(23)

so that from (20), uε − u ¯ε L2 (X)

√

α(εq + ε3/2 gb ) .

To summarize we have the following result:

(24)

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G. Bal

Lemma 1. Provided that (19) holds, the unique solution uε of (15) veriﬁes that √ 1 1 1 uε L2 (X) + √ uε|Γ+ b q+ εgb ω·∇uε +uε + ω · ∇uε + uε −¯ ε ε ε (25) provided that q ∈ L2 (X) and g ∈ L2 (Γ− ; |ω · ν|dq). We recognize the above left-hand side as uε Wε . Least-Square Formulation Let us introduce the operator Lε = (εGε )−1/2 (ω · ∇ + Gε ) .

(26)

Using the divergence theorem on ∇ · (uvω) = (ω · ∇u)v + uω · ∇v, we verify that 1 (27) aε (u, v) = (Lε u, Lε v) + u, v , ε where (·, ·) is the usual inner product on L2 (X) and ·, · is the inner product on L2 (Γ− , |ω · ν|dq). We also have Lε (v) = (qε , Lε v) + g, v ,

qε = ε1/2 G−1/2 (q) . ε

(28)

Since Gε , whence aε , is symmetric, the variational formulation (15) is equivalent to minimizing the functional: 1 arg min aε (v, v) − Lε (v) , v∈W 2

(29)

which is also equivalent to minimizing the following least-square problem arg min u∈W

1 1 Lε u − qε 2L2 (X) + u − εg2L2 (Γ− ;|ω·ν |dq) , 2 2ε

(30)

as can easily be veriﬁed. We have thus recast solving the transport solution as minimizing the leastsquare problem associated to a variational form aε that is W -coercive with constant of coercivity independent of ε. The boundary conditions are also accounted for in a variational sense as the functions in the space W need not satisfy the boundary conditions exactly. These very important properties are a central element in the numerical methods developed in [24, 25], which are based on Galerkin projections; see also Sect. 2.7. Note that the scaling operator S used in the above references is replaced in our analysis by (εGε )−1/2 . The method developed in this paper may thus be seen as a case of ﬁrst-order system least-squares (FOSLS) [24, 25]. Because of (20), Galerkin methods, which are orthogonal projections onto subspaces of W with respect to the bilinear form aε , provide lowerdimensional approximations that are expected to be valid both in the transport regime (ε ∼ 1) and the diﬀusive regime (ε 1) as in [24, 25]. We now

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consider two Galerkin methods, which are not discretizations of the transport solution, but rather projections onto smaller subsets of W that are physically relevant in the diﬀusive regime. The ﬁrst method consists of projecting the transport solution onto functions that only depend on space and not on the angular variable. The resulting diﬀusion approximation is analyzed in the rest of this section. In the next section we project the transport solution onto functions that depend on the spatial variable only in parts of the domain. This allows us to couple the diﬀusion approximation to the full transport solution in regions where diﬀusion may not be valid. 2.3 Diﬀusion by Orthogonal Projection We want to use the above variational formulation to deduce the limit of the transport solution in the diﬀusion limit. Diﬀusion is characterized by high scattering and small absorption. High scattering implies that the initial directional content of the particles is quickly lost through interactions. It is therefore reasonable to assume that u(x, ω) does not depend on ω in a ﬁrst approximation. Such a condition is easy to implement in a variational setting: we orthogonally (with respect to aε (·, ·)) project the solution u of (4) onto functions that depend only on x. Let us deﬁne WD = f ∈ W, f = f (x) ≡ H 1 (Ω), ⊥ (31) = f ∈ W, aε (f, v) = 0 ∀v ∈ WD . WDε ⊥ We verify that W = WD ⊕ WDε since aε (·, ·) is an inner product on W [5]. Then the orthogonal projection Πε of W to WD for the inner product aε (u, u) allows us to deﬁne the “diﬀusion” solution

Uε = Πε uε ,

(32)

where uε is the solution to (15). By deﬁnition, we have that Uε is the solution in WD of v ∈ WD . (33) aε (Uε , v) = Lε (v), We may recast the above equation as (Dε ∇U · ∇V + σa U V )dx + Ω

= Ω

S n−1

1 cn U V dσ(x) ε ∂Ω −1 Gε (q)ωdµ(ω) · ∇V + qV dx +

gV |ω · ν| dq ,

Γ−

for all V ∈ WD , where

(εGε )−1 (ω) ⊗ ωdµ(ω)

Dε = S n−1

1 cn = 2

(34) |ω · ν|dµ(ω) .

S n−1

382

G. Bal

We verify that c2 = π2 and that c3 = 12 . After classical integrations by parts, this means that Uε is the solution of the following diﬀusion equation: −1 −∇ · Dε ∇Uε + σa Uε = q − ∇ · Gε (q)ωdµ(ω) S n−1 (35) cn ν · Dε ∇Uε + Uε = J(x) ≡ g(x, ω)|ω · ν|dµ(ω) . ε ω·ν <0 Here we assume that q vanishes at the domain boundary ∂Ω to simplify. This is consistent with our choice of incoming conditions of size εg as O(1) source terms at the domain boundary result in boundary layer analyses that we do not want to dwell into here; see [12]. It remains to understand whether the solution Uε , which is obviously uniquely deﬁned, is uniformly bounded in H 1 (Ω) and to obtain an error estimate for uε − Uε and for similar diﬀusion approximations of uε . Let us ﬁrst consider the simpler case where scattering is isotropic, which implies that k0 ¯) + εσa u , (36) Gε u = (u − u ε where u ¯ = S n−1 u(ω)dµ(ω) and where k0 (x) = k(x, ω − ω ). In this context we verify that G−1 ε u=

1 k0 ε u ¯ + u, ε σa σε σε

σε = k0 + ε2 σa .

(37)

Assuming that q = q¯ and that g ≡ 0 to simplify, this implies that the transport equation takes the form 1 1 k0 k0 ω · ∇u + 2 ω · ∇u ω · ∇v + 2 (u − u ¯)v + σa uv dp σε ε σa σε ε X (38) ω · ∇v 1 uvω · ν dq = q + v dp . + ε Γ+ εσa X Upon choosing u and v in WD in the above equation, we ﬁnd the variational formulation for Uε : 1 1 ∇U · ∇V + σa U V dx + cn U V dσ = qV dx . (39) nσε ε Ω ∂Ω Ω This is nothing but the variational formulation of (35) with the diﬀusion coeﬃcient given by Dε = (nσε )−1 as usual and the right-hand side given by q(x). We thus obtain the classical diﬀusion equation [19]. 2.4 Harmonic Decomposition and Diﬀusion Approximation The generalization of the derivation of the above diﬀusion solution to arbican be trary scattering kernels is relatively straightforward as long as G−1 ε

Transport Approximations in Partially Diﬀusive Media

383

explicitly characterized. We restrict ourselves here to the two-dimensional case with scattering kernel of convolution type and use explicitly the variational form written in the harmonic basis. Within this context and in the diﬀusive regime, Gε is given by −1 k0 − kn + εσa F . Gε = F (40) ε We recall that the Fourier transform is considered in the angular variable only. The inverse of Gε is then −1 G−1 ε =F

ε ε F ≡ F −1 F k0 − kn + ε2 σa σε − kn

(41)

where σε = k0 + ε2 σa , This yields (37) when kn = 0 for |n| ≥ 1 and (17) in general, where Hε u is a bounded operator in L2 (X). Let us introduce the spectral gap of Q: α=

min (k0 (x) − kn (x)) > 0 ,

(42)

σ0 = min σε (x) > 0 .

(43)

n≥1,x∈Ω

and the minimum of σε : x∈Ω

We then verify that the coercivity constraints (19) are veriﬁed. With this explicit form for Gε , the terms of the variational equation (15) now take the form 1 (ω · ∇u)n (ω · ∇v)−n aε (u, v) = 2 k − k 0 n +ε σa Ω n∈Z k0 − kn −1 + + σa u uvω · ν dq, ˆn vˆ−n dx + ε ε2 Γ+ · ∇v)−n εˆ qn (ω Lε (v) = + q ˆ v ˆ gv|ω · ν| dq . (44) dx + n −n k0 − kn + ε2 σa Ω Γ− n∈Z

The generalization of (39) to more general scattering terms may easily be carried out thanks to the above formulae. Indeed, U and V belong to WD if ˆn and Vˆn vanish for |n| ≥ 1. As a and only if all their Fourier coeﬃcients U result, U is the unique solution to the following variational problem ˆ ¯ˆ ˆ0 ∂ Vˆ0 ∂ U0 ∂ V0 + ∂¯U c2 ˆ ˆ U V + σ U V dσ dx + a 0 0 k0 − k1 + ε2 σa ε ∂Ω Ω qˆ1 ∂¯Vˆ0 + qˆ−1 ∂ Vˆ0 ˆ = + qˆ0 V0 dx ε k0 − k1 + ε2 σa Ω V g(x, ω)|ω · ν|dµ(ω) dσ . (45) + ∂Ω ω·ν <0

384

G. Bal

We have used here the symmetry of G implying that k−1 = k1 . We denote by qˆ−1 = 12 (qx + iqy ) and identify it with the vector q = (qx , qy ) in Cartesian coordinates. The above variational formulation shows that U is the (weak) solution in WD of the following diﬀusion equation: −∇ · Dε ∇U + σa U = q − ε∇ · Dε q, Dε

c2 ∂U + U = J, ∂n ε

Ω (46)

∂Ω .

The diﬀusion coeﬃcient is given by Dε =

1 1 = . 2(σε − k1 ) 2(k0 − k1 + ε2 σa )

(47)

This is the usual expression for the diﬀusion coeﬃcient. Remark 1. The property that the constant of coercivity of aε is independent of the mean free path ε is very important to obtain the diﬀusion solution by orthogonal projection. Consider for instance the transport problem Tε u ε ≡

1 1 ω · ∇uε + Gε uε = q(x) ε ε

in X,

uε = 0

on Γ− ,

(48)

and the variational formulation: ﬁnd uε such that a ˜ε (uε , v) = (wε Tε uε , Tε v) = (q, wε Tε v),

∀v ∈ W .

(49)

Here wε (x) is a weight function that could take the value 1 as in the introduction in [24] or σε−1 as in [7]. The above problem is equivalent to (48) for wε uniformly bounded from above and below by positive constants as is shown in [7]. This results from the fact that a ˜ε (uε , v) is coercive on W . However the constant of coercivity is not independent of ε. The equivalence between (48) and (49) thus somewhat degrades as ε → 0. The consequence is that (49) becomes inaccurate in the diﬀusive regime with dire consequences when it comes to discretizations as pointed out in [24]. The same consequence arises ˜ ε of uε onto WD is considered. Aswhen the natural orthogonal projection Π ˜ε = Π ˜ ε uε suming that Gε is isotropic as in (36) to simplify, we obtain that U solves the equation −∇ ·

wε ˜ ˜ε = wε σa q, ∇Uε + wε σa2 U n

in Ω .

(50)

Here n = 2, 3 is the spatial dimension. Although the local equilibrium ˜ε ≈ σa−1 q is veriﬁed in the limit of very strong absorption and sources, U the diﬀusion tensor is not correct, independently of the choice of the weight wε . This indicates that variational formulations of the form (49) should not be used in the diﬀusive regime and should be replaced by (4) or by rescaled formulations as in [24].

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2.5 Convergence Result and Error Estimates Let uε (x, ω) be the solution of the transport equation (15) and Uε = Πε uε the diﬀusion approximation solution of (33). Deﬁne δuε = uε − Uε . We ﬁrst show that δuε converges to 0 in W as ε → 0. The proof is similar to that in [5]. ⊥ . Thanks to the orthogonal projection, we ﬁrst obtain that δuε ∈ WDε Consequently, we have aε (δuε , δuε ) = aε (uε , δuε ) = Lε (δuε ) . We then deduce from (20) and (21) that √ 1 δuε W + δuε − δuε q + εgb . ε Since δuε is uniformly bounded in W , which is a Hilbert space with a unit ball compact for the weak topology, we deduce the existence of a subsequence of δuε converging to w. However the subsequence of δuε converges to the same limit so that w = w ∈ WD . This implies that aε (δuε , w) = 0 . Upon passing to the limit in the above expression, we ﬁnd that 2 1 2 (ω · ∇w)1 + σa |w| dx = 0 k0 − k1 Ω and that w|∂Ω = 0. This implies that w = 0, whence that δuε converges to 0. Note that all we have used to get this convergence result is that q and gb are bounded (in the L2 −sense). In order to obtain error estimates for δuε , additional regularity of the solutions and source terms is required. Let us recall (18): 1 Gε uv dp ω · ∇u ω · ∇v + Hε (ω · ∇u)ω · ∇v + aε (u, v) = 2 ε X ε σa 1 + uvω · ν dq , (51) ε Γ+ q Lε (v) = − εω · ∇Hε (q) v dp + gvω · ν dq , q−ω·∇ εσa X Γ− assuming that q vanishes on ∂Ω to simplify. We split the source term as Lε = L0 + L1 ,

qv dp −

L0 (v) = X

X

−ω · ∇

L1 (v) = X

εω · ∇Hε (q)v dp +

q v dp . εσa

gvω · ν dq , Γ−

(52)

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G. Bal

The term appearing in L1 (v) is of order ε−1 and cannot be estimated from the variational formulation. In order to estimate it explicitly, we deﬁne W1 as the subspace of W of functions u(x, ω) such that u ˆn (x) = 0 for |n| = 1. This is thus the Hilbert space of functions linear in ω. Let Π1ε be the orthogonal projection of W onto W1 with respect to aε (·, ·). We deﬁne U1ε = Π1ε uε , i.e., ∀V1 ∈ W1 .

aε (U1ε , V1 ) = Lε (V1 ),

(53)

Thanks to the two terms of order ε−2 in aε in (51), we verify that U1ε + ω · ∇U1ε εqW + ε3/2 gb .

(54)

In what follows, we need a similar estimate for the following solution: ˜1ε , V1 ) = L1 (V1 ), aε (U

∀V1 ∈ W1 .

(55)

We verify as above that ˜1ε + ω · ∇U ˜1ε ε∇q + ε3/2 gb . U

(56)

In order to estimate the error coming from the source term L0 (v), we deﬁne ψ2ε as ((εGε )ψ2ε , v) = L0 (v) − aε (Uε , v),

∀v ∈ W (57)

ψ2ε = 0 . Here (·, ·) is the usual inner product of L2 (X). We verify that ψ2ε is well-posed and is given by ψ2ε (x, ω) = Hε ω · ∇Hε (ω) · ∇Uε − ∇ · Dε ∇Uε + εω · ∇Hε (q) − 2Dε ε∇ · q . (58) The term within brackets in the above expression is mean zero by construction so that all equalities in (57) are veriﬁed. Let us now decompose the exact solution as uε = u0ε + u1ε ,

aε (ukε , v) = Lk (v),

∀v ∈ W, k = 0, 1 .

(59)

We introduce the following decompositions: u0ε = Uε + ε2 ψ2ε + δ0ε ,

˜1ε + δ1ε , u1ε = U

(60)

and estimate both terms δkε for k = 0, 1. We start with k = 0 and obtain from the transport equation aε (u0ε , δ0ε ) − L0 (δ0ε ) = 0 that

Transport Approximations in Partially Diﬀusive Media

387

1 aε (δ0ε , δ0ε ) = −ε2 aε (ψ2ε , δ0ε ) − 2 (εGε )ψ2ε , δ0ε ε 1 2 = −ε ω · ∇ψ2ε ω · ∇δ0ε + Hε (ω · ∇ψ2ε )ω · ∇δ0ε dp ε2 σ a X 1 ψ2ε δ0ε ω · ν dq . + ε Γ+ By the Cauchy-Schwarz inequality, this implies that aε (δ0ε , δ0ε ) ω · ∇ψ2ε ω · ∇δ0ε + ε2 ω · ∇ψ2ε ω · ∇δ0ε + εψ2ε|Γ+ b δ0ε|Γ+ b . We verify that ω · ∇ψ2ε ε thanks to its expression in (58). This shows that provided that q and Uε , whence ψ2ε , are suﬃciently regular, we obtain that (61) aε1/2 (δ0ε , δ0ε ) εψ2ε|Γ+ b + ε2 ω · ∇ψ2ε + ε−1 ω · ∇ψ2ε . This implies for instance that u0ε − Uε W ε .

(62)

˜1ε + δ1ε . We verify that It remains to address the term u1ε = U

1 Gε ˜ ˜ ˜ U1ε δ1ε dp ω · ∇U1ε ω · ∇δ1ε + Hε (ω · ∇U1ε )ω · ∇δ1ε + ε2 σa ε X 1 ˜1ε δ1ε ω · ν dq = L1 (δ1ε ) + ˜1ε )ω · ∇δ1ε dp U Hε (ω · ∇U + ε Γ+ X ˜1ε )ω · ∇δ1ε dp , Hε (ω · ∇U = aε (u1ε , δ1ε ) +

˜1ε , δ1ε ) = aε (U

X

˜1ε δ1ε = Π1ε (U ˜1ε δ1ε ). Now however, because ω · ∇v depends only on vˆ±1 and U ˜1ε ) ε thanks to (56). This implies that Hε (ω · ∇U aε (δ1ε , δ1ε ) εδ1ε W ε2 .

(63)

In summary we have proved using a variational approach that: Theorem 1. Let uε and Uε be the transport and diﬀusion solutions. Then we have (64) uε − Uε W ε , provided that the source term q and the solution U are suﬃciently regular. How regular the solution U and source term q need to be can be explicitly read oﬀ the previous formulae. We do not dwell on the details.

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G. Bal

2.6 Better Error Estimates in Inﬁnite Domains The accuracy of order O(ε) cannot be improved in general; see e.g. [12]. Indeed, we know that in the presence of boundaries, boundary layer terms need be accounted for by other means than volume asymptotic expansions and diﬀusion-like approximations. We refer e.g. to [3] for two-dimensional numerical simulations quantifying the role of the boundary layers. In the absence of boundaries however, i.e., when Ω = R2 , the above method provides more accurate approximations of the transport solution than O(ε). Indeed we verify from (61) that u0ε − Uε W ε2 .

(65)

The same type of variational arguments (based on explicit test functions similar to ψ2ε ) allows us to show that ˜1ε W ε2 u1ε − U as well, which accounts for the source term −(εσa )−1 ω · ∇q. This type of estimates hinges on the fact that u0ε only involves polynomials that are even in θ (in the sense that uˆ0εn = 0 for n odd) whereas u1ε only involves odd polynomials in θ as can been seen from the variational formulations (12) and (44). We ﬁnally observe, as is done in the Appendix, that ˜1ε (x, ω) = −εHε (ω) · ∇Uε (x) , U

(66)

up to terms that can be estimated to be of order O(ε2 ). In the absence of boundaries, whence of boundary layers, we therefore obtain the following classical result: uε (x, ω) = Uε (x) − εHε (ω) · ∇Uε (x) + O(ε2 ) ,

(67)

which can be shown to be equivalent to the expression uε (x, ω) = Uε (x) −

ε ω · ∇Uε (x) + O(ε2 ) . k0 − k1

(68)

Here the error terms are O(ε2 ) for the · W norm for instance. 2.7 A Remark on Discretizations in the Diﬀusive Regime The variational formulation (15) for transport allows us to easily construct discretizations of the transport solution uε by Galerkin approximation. Indeed let Wh be a discrete subspace of W . We may then deﬁne Πh as the orthogonal projection onto Wh for the inner product aε (·, ·). Denoting by uh = Πh uε , we have the equivalent characterization: aε (uh , v) = Lε (v),

∀v ∈ Wh .

(69)

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389

Let us equip W with the norm ·Wε whose square is deﬁned on the left-hand side of (20). For this norm, aε is not only coercive but also continuous with constants independent of ε as we noted earlier. C´ea’s lemma [9] then implies that (70) uε − uh Wε ≤ C min uε − vWε , v∈Wh

where C is independent of ε. The error estimate then becomes an approximation theory problem as in [24,25], where the main diﬃculty arises because the Wε norm depends on ε. Although it was more convenient to equip W with its natural norm in the analysis of diﬀusion approximations, it is natural to introduce the norm · Wε in the analysis of discretizations (as in [24, 25]). In the numerical solution of transport problems, it is very often useful to obtain discretizations that behave well in the diﬀusive regime. This is not always satisﬁed and is usually characterized by ensuring that the diﬀusive limit of the discretized transport equation is indeed a consistent discretization of the diﬀusion equation [15, 20, 21]. We note here the following corollary of orthogonal projections: (71) Πh Πε = Πε Πh . This asserts that discretizing and taking the diﬀusive limit are commuting operations so that indeed the discretization of the diﬀusion limit is indeed the same as the ε → 0 limit of the discretized transport equation.

3 Transport-Diﬀusion Coupling We want to generalize the results obtained in the preceding section to the physical situation where the diﬀusive regime is valid in large parts of the domain but not everywhere. We refer to [5] for possible applications, which typically include large domains compared to the mean free path so that the diﬀusion approximation holds everywhere except for localized areas where the scattering or absorption coeﬃcients may vary too fast. 3.1 Orthogonal Projection A plausible solution to this issue consists of solving the diﬀusion equation where it is valid and the transport equation elsewhere. It thus remains to ﬁnd a method that couples both equations at the interface separating their domains of deﬁnition. As was shown in [5] in the simpliﬁed setting of the even-parity formulation of the transport equation, orthogonal projection is a natural approach when a variational formulation is available. Let Ω be the physical domain and Ωdi and Ωtr a non-overlapping partition of Ω. We denote by γ the common interface shared by Ωdi and Ωtr and assume that it does not overlap with ∂Ω. We denote by ν di (x) the outward normal to Ωdi at x ∈ γ. Since the diﬀusion approximation is valid on Ωdi by assumption,

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G. Bal

we expect the transport solution u(x, ω) to depend only on x on Ωdi , but to depend on the full phase-space variables x, ω on Ωtr . This justiﬁes the introduction of the following spaces ⊥ WCε = {f ∈ W, aε (f, v) = 0 ∀v ∈ WC } . (72) Let now ΠCε be the orthogonal (for aε ) projector onto the Hilbert subspace WC of W . For uε the solution of the transport solution of (15) we then deﬁne the coupled transport-diﬀusion solution as

WC = {f ∈ W, f = f (x) on Ωdi },

uCε = ΠCε uε .

(73)

aε (uCε , v) = Lε (v) ∀v ∈ WC .

(74)

Variationally, this means that

Because aε is an inner product on W , whence on WC , the above solution is uniquely deﬁned. 3.2 Local Equations The solution of (74) can directly be estimated numerically by Galerkin (orthogonal) projection onto ﬁnite dimensional subspaces. This is one of the main advantages of variational formulations [24–26]. It turns out that (74) does not seem to admit any simple “local” representation involving ﬁrst-order and diﬀusion equations. In order to better exhibit the nature of the coupling at the interface γ, we introduce the notation utr (ω, x) = uCε|Ωtr (ω, x),

udi (x) = uCε|Ωdi (x)

(75)

for the solution uCε of (74). Because uCε ∈ W , we directly obtain the continuity relation utr (ω, x) = udi (x) for x ∈ γ. The other relations are obtained as follows. We ﬁrst observe that we may recast aε (u, v) − Lε (v) as 1 (ω · ∇u + Gε u − εq) (I + G−1 aε (u, v) − Lε (v) = ε ω · ∇)vdp ε X u + g− vω · νdq ε Γ− 1 (I − ω · ∇G−1 = ε )(ω · ∇u + Gε u − εq)vdp ε (Ωdi ∪Ωtr )×V + (εGε )−1 (ω · ∇ + Gε )(udi − utr ))vω · ν di dq γ×V

1 (ω · ∇u + Gε u−εq)ω · νvdq + ε ∂X u + g− vω · νdq . ε Γ− (76)

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Upon restricting the support of v on Ωtr and Ωdi , we ﬁnd that utr and udi satisfy the following equations (I − ω · ∇G−1 ε )(ω · ∇utr + Gε utr − εq) = 0, −∇ · Dε ∇udi + σa udi = q − ε∇ · Dε q,

Ωtr × V Ωdi .

(77)

Once the volume source terms in (76) have vanished, we obtain the following boundary conditions on ∂Ω: (ω · ∇utr + Gε utr − εq) + (g − utr ) = 0, ω · ∇utr + Gε utr − εq = 0, cn ν di · Dε ∇udi + udi = J(x), ε

Γ− ∩ (∂Ωtr × V ) Γ+ ∩ (∂Ωtr × V )

(78)

∂Ω ∩ ∂Ωdi .

We observe that Gε (utr −udi ) = 0 on γ since uεC ∈ W . The coupling condition on γ is then equivalent to: ω · ∇utr ω · ν di dµ(ω) = ν di · Dε ∇udi on γ . (79) V

In the absence of coupling, i.e., when Ωtr = Ω, the above equations may be somewhat simpliﬁed on Ωtr as follows (see also [7]). Let ϕ ∈ L2 (Ωtr × V ) be an arbitrary test function and let us deﬁne v ∈ W (Ωtr ) as the solution to ω · ∇v + Gε v = G−1 ε ϕ, v = 0,

Ωtr × V Γ− (Ωtr ) ,

(80)

where Γ± (Ωtr ) and W (Ωtr ) are deﬁned as Γ± and W with Ω replaced by Ωtr . Classical transport theories [12] show that v is uniquely deﬁned and belongs to W (Ωtr ). Upon multiplying the ﬁrst equation in (77) by v and integrating by parts, we obtain using the second equation in (78) that (ω·∇utr+Gε utr −εq)ϕdp+ G−1 ε (ω·∇utr+Gε utr −εq)vω·ν di dq = 0. Ωtr ×V

γ×V

(81)

When Ωtr = Ω so that γ = ∅, the above formulation implies that ω · ∇utr + Gε utr − εq = 0,

Ωtr × V .

(82)

The above equality holds in the L2 (Ωtr ×V )-sense, which implies that it holds almost everywhere. However, the variational formulation does not allow us to obtain that it also holds at the boundary of the domain, so that the ﬁrst equation in (78) does not simplify. Only when regularity of the solution can be obtained, so that (82) holds in a stronger sense implying that it still holds at the domain boundary can one conclude that u = g on Γ− . In the transport-diﬀusion coupling, the situation is more complicated as ω · ∇utr + Gε utr − εq has no reason to vanish on Γ+ (Ωtr ). The ﬁrst-order

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transport equation (82) then no longer holds and needs to be replaced by the equation for utr in (77). As in the even-parity formulation considered in [5], we obtain the coupling of the diﬀusion equation with the second-order transport equation, not the ﬁrst-order transport equation. This further exempliﬁes the importance of the boundary conditions (or of the interface conditions in the transport-diﬀusion coupling) when using second-order variational formulations of ﬁrst-order transport equations (see also [25, 26]). 3.3 Convergence and Error Estimates The convergence results and error estimates are very similar to those for the diﬀusion approximation and the even-parity formulation developed in [5]. We outline the diﬀerences. Let uε be the transport solution, uCε the coupled transport-diﬀusion solution, and δuε = uε − uCε . We obtain as before that ⊥ so that δuε ∈ WCε aε (δuε , δuε ) = aε (uε , δuε ) = Lε (δuε ) . From (20) and (21), we still obtain that √ 1 δuε W + δuε − δuε q + εgb . ε This implies the convergence of δuε to w (after possible extraction of a subsequence) and w = w so that w ∈ WC ; whence aε (δuε , w) = 0. Passing to the limit ε → 0 in the above variational formulation yields that w ≡ 0. In order to obtain convergence results, we still split Lε as in (52). The contribution coming from L1 is of order O(ε) as before and we thus concentrate on the contribution from L0 . Let us deﬁne ψ2ε as ((εGε )ψ2ε , v) = L0 (v) − aε (uCε , v),

∀v ∈ W (83)

ψ2ε = 0 . We verify that ψ2ε is given by ψ2ε (x, ω) = χdi (x) Hε ω · ∇Hε (ω) · ∇uCε − ∇ · Dε ∇uCε + εω · ∇Hε (q) − 2Dε ε∇ · q .

(84)

This is the main diﬀerence with respect to the diﬀusion case. Because the (second-order) transport solution is calculated on Ωtr , the correction term only involves errors made on Ωdi . So when the coeﬃcients σa and kn wildly oscillate or do not have the correct behavior to justify the diﬀusion approximation on Ωtr , they will generate errors only through the behavior of uCε ,

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whence ψ2ε , on Ωdi , where the validity of the diﬀusion approximation renders these errors much smaller. The rest of the derivation is then as in Sect. 2.5. We obtain that δuε is of order ε in W provided that ψ2ε is suﬃciently regular on Ωdi . Let us deﬁne ˆn (x) = 0 for W1C as the subspace of W of functions u(x, ω) such that u |n| = 1 on Ωdi . When the boundary conditions are treated with the transport equation so that ψ2ε|Γ+ , and the corrector u1Cε given by the orthogonal projection of uε onto W1C is added to uCε , we obtain an approximation of order ε2 provided that ψ2ε is suﬃciently regular as in [5].

4 Generalized Diﬀusion Models When the diﬀusion approximation does not hold in an area relatively large compared to the transport mean free path but still small compared to the overall size of the domain, the transport-diﬀusion coupling presented above may be the only alternative to the more costly full transport solution. There are cases however where the area of invalidity of diﬀusion is suﬃciently speciﬁc so that more macroscopic models may be deﬁned. An example is the treatment of clear layers in optical tomography, where diﬀusion does not hold locally although modiﬁed (generalized) diﬀusion models can still be used eﬃciently. We refer to [4, 6] for the derivation of such a model and to [2, 3, 13, 17, 29] for additional references on the problem. It is not clear how to derive the generalized diﬀusion models presented in [6] by purely variational means and orthogonal projections. Our objective in this section is rather to extend the models developed in [4,6] to more general geometries of non-scattering inclusions. From the application’s viewpoint, the main novelty of the following derivation is the treatment of narrow nonscattering tubes or ﬁlaments in three-dimensional geometry surrounded by highly scattering media. This may have applications in radiation problems in astrophysics and atmospheric cloud modeling. We consider non-scattering and non-absorbing inclusions to simplify the presentation although the results can be generalized to weakly scattering and absorbing media as in [4]. The framework we consider here is the following. Let Ω be a smooth compact domain in Rn and Ωε a smooth non-scattering subset of Ω. We consider the transport problem: ﬁnd uε ∈ W such that ω · ∇uε + Gε uε = εq(x), ω · ∇uε = 0, uε = 0,

in Ω\Ωε × V in Ωε × V on Γ− .

(85)

It is understood that the jump of uε across Σε = ∂Ωε vanishes since uε ∈ W . We consider the case where Gε is isotropic and given by (36) to simplify. The diﬀusion approximation holds in Ω\Ωε but not in Ωε . Because scattering and absorption are supposed to vanish in Ωε , the variational formulations deﬁned

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in earlier sections need to be regularized (see [26] for more details on this problem). Yet independently of this issue, we claim that for speciﬁc forms of Ωε , there are simpler methods to approximate uε than the transport-diﬀusion coupling introduced in the previous section. 4.1 A Non-Local Diﬀusion Equation We model Ωε as follows. Let Σ be a smooth (non self-intersecting) closed (to simplify) surface of co-dimension d in the n-dimensional domain Ω. Then Ωε is the subset of Ω of points that are suﬃciently close to Σ: Ωε = {x ∈ Ω;

d(x, Σ) < Lε } ,

(86)

where d(x, Σ) is the Euclidean distance from x to Σ and Lε is a constant that depends on ε. We may thus parameterize Ωε as Σ × BLε , where BLε is the d-dimensional ball of radius Lε , at least for suﬃciently small Lε . Let Tx Σ be the n − d dimensional vector space of vectors tangent to Σ at x ∈ Σ and Nx Σ the d dimensional vector space of vectors normal to Σ at x ∈ Σ. The tangent and normal bundles T Σ and N Σ are as usual the unions of Tx Σ and Nx Σ, respectively, where x runs over Σ. We also deﬁne N as the subset (x, n(x)) ∈ N Σ such that |n| = 1 and Nx as the subset n ∈ Nx Σ such that |n| = 1. The latter set is isomorphic to the sphere S d−1 . It is the unit circle when Σ is a curve in three dimensions and is restricted to two points when Σ is a surface in three dimensions or a curve in two dimensions. We then realize that ∂Ωε = Σ + Lε N is a smooth co-dimension one surface for suﬃciently small Lε . When Lε is a positive constant independent of ε, it is shown in [4] that uε converges as ε → 0 to the solution U of a diﬀusion equation on Ω\Ωε (which is in fact independent of ε) with the boundary condition on ∂Ωε that U is constant on ∂Ωε and the average of ∂U ∂n over ∂Ωε vanishes. This essentially means that nothing much happens inside Ωε . Because no scattering hampers the propagation of particles within Ωε , an equilibrium is reached which stipulates that uε is approximately constant in Ωε . A more interesting regime may be obtained when Lε is allowed to depend on ε. We assume that Lε converges to 0 with ε. If Lε converges too slowly to 0, then we are back to the case where the transport solution equilibrates to a constant inside Ωε . If Lε converges too fast to 0, then the non-scattering inclusion is too small to have any eﬀect and the approximate solution Uε of uε becomes the solution of a diﬀusion equation with no inclusion. There is an intermediate regime where the physics is richer. Because Lε 1 in the regime of interest, we can assume that Uε becomes constant on the d-dimensional cross-section Bx = x + τ Lε Nx for 0 ≤ τ ≤ 1 as in [4]. This means that Uε (y) on ∂Ωε depends only on x ∈ Σ, where |x − y| = Lε and generalizes the condition that the jump of Uε across a co-dimension one surface vanishes as in [4].

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395

Let us now consider the response operator Rε on the non-scattering inclusion Ωε , which maps u|Γ− (∂Ωε ) to u|Γ+ (∂Ωε ) solution of ω · ∇u = 0 in Ωε . Here, Γ± (∂Ωε ) = {(x, ω) ∈ ∂Ωε × V, ±ω · ν(x) > 0}. We also deﬁne Γ± (x) = {ω ∈ V, ±ω · ν(x) > 0} for x ∈ ∂Ωε and note that ∂Bx for x ∈ Σ is the subset of ∂Ωε at a distance Lε from x. We decompose the response operator applied to functions Uε as Rε = R0 + εR1 , where R0 maps Uε (x) on Γ− (∂Ωε ) to Uε (x) on Γ+ (∂Ωε ). The correction εR1 will be of order O(ε) provided that Lε is suitably chosen. Following the same procedure as in [4], we obtain that an approximation Uε of order O(ε) of uε satisﬁes the following equation −∇ · Dε ∇Uε + σa Uε = q, Dε

Ω\Ωε

cn ∂Uε + Uε = 0, ∂n ε

∂Ω (τ, n) ∈ (0, 1) × Nx ,

Uε (x + τ Lε n) = Uε (x), Dε ∂Bx

∂Uε dσ = ∂n

x∈Σ

ω · ν(R1 Uε )dµ(ω)dσ,

∂Bx

(87)

x∈Σ.

Γ+ (x)

The ﬁrst two equations are the usual diﬀusion equation with boundary conditions on Ω\Ωε where the diﬀusion approximation is valid. The third equation indicates that the solution Uε is constant on Bx for all x ∈ Σ. The fourth equation, which is necessary to evaluate the latter constant, ensures the conservation of the particle current through the interface ∂Bx for each x ∈ Σ. The left-hand side models ε−1 times the current coming into the non-scattering inclusion, while the right-hand side is ε−1 times the current going out of the non-scattering inclusion: ω · νvε dq = ω · νUε dq + ω · νRε Uε dq , ∂Bx ×V

∂Bx

Γ− (x)

∂Bx

Γ+ (x)

where vε solves ω · ∇vε = 0 in Ωε and vε = Uε on Γ− (∂Ωε ). Because R0 is the identity operator on functions of the form Uε (x), ε−1 times the transport current takes the form given on the right-hand side of the fourth equation in (87). We thus obtain an equation for Uε (x), which is much less expensive to solve numerically than the full transport solution uε . Note however that the boundary conditions in the fourth equation of (87) are non-local and require us to estimate R1 explicitly. In the case where Σ is a co-dimension one surface, then ∂Bx and Nx reduce to two points for x ∈ Σ and it is shown in [4] that (87) admits a unique solution for ε suﬃciently small.

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4.2 Generalized Diﬀusion Equation It remains to ﬁnd the value of Lε for which R1 is indeed an O(1) operator and to see whether the non-local conditions in (87) can be localized. Both questions are answered by the same asymptotic expansions as follows. We ﬁrst need to evaluate R1 Uε (x + Lε n, ω). For each (x + Lε n, ω) ∈ Γ+ (x), we have a unique y(x, n, ω) ∈ ∂Ωε \{x+Lε n} such that x+Lε n−y = |x + Lε n − y|ω, by following the characteristics of the operator ω · ∇. Let us ¯ (x, n, ω) as the closest point to y on Σ. Then we have deﬁne x R1 Uε (x + Lε n, ω) =

1 (Uε (¯ x(x, n, ω)) − Uε (x)) . ε

(88)

Assuming to simplify that Σ has positive curvature (in the sense that each ¯ | 1 when Lε 1. Let us curve in Σ has positive curvature), then |x − x ¯ is on the (unique) geodesic deﬁne the tangent vector τ (ω) ∈ Tx Σ such that x ¯ ) be the geodesic distance (on starting at x with direction τ (ω), and let d(x, x ¯ . Geodesics are meant here with respect to the induced Σ) between x and x metric on Σ seen as a submanifold of Rn equipped with the Euclidean metric. Then we verify by Taylor expansion that 2 ¯) d (x, x ¯ )) . (89) τ (ω) · ∇Σ Uε (x) + O(d3 (x, x Uε (¯ x) − Uε (x) = τ (ω) · ∇Σ 2 Here ∇Σ is the restriction (projection) of ∇ to Tx Σ. We also deﬁne ∇⊥ Σ as ¯ )), the restriction of ∇ to Nx Σ. Neglecting the smaller-order term O(d3 (x, x we see that the operator R1 is now local in x. Moreover, it will be of order O(1) provided Lε is chosen so that 2 ¯) d (x, x 1 τ (ω) · ∇Σ Uε (x)dµ(ω)dσ ω · ντ (ω) · ∇Σ ε ∂Bx Γ+ (x) 2 = ∇Σ · DΣ (x)∇Σ Uε (x) = O(1) .

(90)

In other words, we want the (positive deﬁnite) second-order tensor DΣ (x), deﬁned explicitly in (90) for a given geometry, to be of order O(1). Tedious calculations similar to those in [4] show that for an interface Σ of co-dimension d ≥ 1 in a n = 2, 3 dimensional domain Ω, the domain Ωε must be characterized by a radius Lε such that Ld+1 | ln Lε | = O(ε) . ε

(91)

The result probably holds for larger values of n although this was not considered in detail. In the physically interesting case√ n = 3, we therefore obtain that clear layers (where d = 1) of thickness Lε ≈ ε (neglecting logarithmic terms) and tubes (where d = 2) of radius Lε ≈ ε1/3 will have an order O(1) eﬀect on the diﬀusion solution. The case of clear layers is treated in [4, 6]. In

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397

the case of a non-scattering tube, the limiting equation for Uε as the thickness of the tube tends to 0 is thus given by the local generalized diﬀusion model: −∇ · Dε ∇Uε + σa Uε = q,

Ω\Σ

∂U cn + Un = 0, ∂n ε θ · ∇⊥ Dε (x) lim+ Σ Uε (x + ηθ)dµ(θ) = −∇Σ · DΣ (x)∇Σ Uε , Dε

η→0

∂Ω Σ.

Nx

(92) It may be easier to understand the solution of the above equation by recasting 1 (Ω) such that for it in a variational form: Find the unique solution U ∈ HΣ 1 all V ∈ HΣ (Ω), we have: cn (Dε ∇U · ∇V + σa U V )dx + U V dσ(x) ε ∂Ω Ω DΣ ∇Σ U ∇Σ V dl(x) = V gdx . (93) + Σ

Ω

When n = 3 and d = 2, then l(x) is the (one-dimensional) arclength along ∂ 1 . The Hilbert space HΣ (Ω) is deﬁned the curve Σ and ∇Σ is nothing but ∂l as the completion of the pre-Hilbert space of functions of class C 1 on Ω such that U, U Σ < ∞, where the inner product ·, · Σ is deﬁned as: U, V Σ = (∇U · ∇V + U V )dx + ∇Σ U ∇Σ V dl(x) . (94) Ω

Σ

1 HΣ (Ω)

We know that is a Hilbert space [28] and that thanks to the uniform positiveness of Dε , σa , and DΣ on their domains of deﬁnition, the variational formulation (93) admits a unique solution by the Lax Milgram theory. Note that functions in H 1 (Ω), deﬁned in (31) as the natural space for the classical diﬀusion approximation, do not necessarily admit traces on onedimensional curves (though functions in H 1+δ (Ω) do for all δ > 0). This renders the use of the above completion argument necessary to construct 1 (Ω). This issue does not arise for (co-dimension one) surfaces as functions HΣ in H 1 (Ω) indeed admit traces on surfaces. In any event, from the numerical viewpoint, we observe that the non-scattering ﬁlament is simply modeled by one additional integration over Σ in the variational formulation (93). This renders its numerical simulation rather straightforward and much less costly computationally than the full transport equation or the non-local diﬀusion model (87).

Acknowledgments I would like to thank Frank Graziani for inviting me to the very nice conference on computational methods in transport organized at the Granlibakken

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conference center in September of 2004. I enjoyed many stimulating discussions during my stay there. This research was funded in part by NSF Grant DMS-0239097 and an Alfred P. Sloan fellowship.

A Local Second-Order Equation and Linear Corrector This appendix provides some useful calculations related to the variational formulations introduced in the text and sketches the derivation of the expression for the corrector of order ε. We ﬁrst verify that by choosing a test function with support in Ω and by integrations by parts that the solution uε of (15) also solves (I − ω · ∇G−1 ε )(ω · ∇u + Gε u − εq) = 0 .

(95)

Let us introduce the shift operators S± = F −1 Sˆ± F,

(Sˆ± u ˆ)n = u ˆn±1 .

(96)

We then verify that ¯ +. ω · ∇ = ∂S− + ∂S Thanks to (40) and the above equalities, we deduce that (95) is equivalent to: ∂u ˆn ∂¯u ˆn ∂¯u ˆn+2 ∂u ˆn−2 σε − kn ¯ ¯ +∂ +∂ +∂ u ˆn + − ∂ σε − kn−1 σε − kn+1 σε − kn−1 σε − kn+1 ε2 qˆn−1 qˆn+1 = qˆn − ε ∂ + ∂¯ . (97) σε − kn−1 σε − kn+1 Let u be the transport solution. The equation for its orthogonal projection over linear functions in the angular variable Π1 u is thus ∂u ˆ1 ∂¯u ˆ1 ∂u ˆ−1 σε − k1 ¯ +∂ +∂ u ˆ1 − ∂ + σε − k0 σε − k2 σε − k0 ε2 qˆ0 qˆ2 = qˆ1 − ε ∂ + ∂¯ . σε − k0 σε − k2 Up to terms of smaller order that can be estimated, this is ∂¯u ˆ1 qˆ0 σε − k1 ∂u ˆ−1 +∂ u ˆ1 = −∂ . + − ∂ σε − k0 σε − k0 ε2 εσa We check that this is also the equation veriﬁed by U1 = −G−1 ε (ω · ∇Uε ) , up to smaller-order terms, which may be written in the Fourier domain as

Transport Approximations in Partially Diﬀusive Media

(G−1 ε (ω · ∇Uε )1 = ε

399

ˆε ∂U , σε − k1

where Uε is the diﬀusion approximation. Therefore, the expression for the corrector is: (98) U1 (x, ω) = −εHε (ω) · ∇Uε (x) . This is the usual expression for the ﬁrst-order corrector to the transport solution in the absence of boundaries.

References 1. R. T. Ackroyd, Completely boundary free minimum and maximum principles for neutron transport and their least-squares and galerkin equivalents, Ann. Nucl. Energy, 9 (1981), p. 95. 2. S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, The ﬁnite element model for the propagation of light in scattering media: A direct method for domains with nonscattering regions, Med. Phys., 27(1) (2000), pp. 252–264. 3. G. Bal, Particle transport through scattering regions with clear layers and inclusions, J. Comp. Phys., 180(2) (2002), pp. 659–685. , Transport through diﬀusive and non-diﬀusive regions, embedded objects, 4. and clear layers, SIAM J. Appl. Math., 62(5) (2002), pp. 1677–1697. 5. G. Bal and Y. Maday, Coupling of transport and diﬀusion models in linear transport theory, M2AN Math. Model. Numer. Anal., 36(1) (2002), pp. 69–86. 6. G. Bal and K. Ren, Generalized diﬀusion model in optical tomography with clear layers, J. Opt. Soc. Amer. A, 20(12) (2003), pp. 2355–2364. 7. L. Bourhrara, New variational formulations for the neutron transport equation, Transport Theory Statist. Phys., 33(2) (2004), pp. 93–124. 8. P. S. Brantley and E. W. Larsen, The simpliﬁed P3 approximation, Nucl. Sci. Eng, 134 (2000), pp. 1–21. 9. S. C. Brenner and L. R. Scott, The mathematical theory of ﬁnite element methods, Springer verlag, New York, 2002. 10. S. Chandrasekhar, Radiative Transfer, Dover Publications, New York, 1960. 11. M. Clark and G. C. Pomraning, The Variational Method Applied to the Monoenergetic Boltzmann Equation, Part I, Nucl. Sci. Eng., 16 (1963), pp. 147– 154. 12. R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol.6, Springer Verlag, Berlin, 1993. 13. H. Dehghani, D. T. Delpy, and S. R. Arridge, Photon migration in nonscattering tissue and the eﬀects on image reconstruction, Phys. Med. Biol., 44 (1999), pp. 2897–2906. 14. J. J. Duderstadt and W. R. Martin, Transport Theory, Wiley-Interscience, New York, 1979. 15. F. Golse, S. Jin, and C. D. Levermore, The convergence of numerical transfer schemes in diﬀusive regimes. I. Discrete-ordinate method., SIAM J. Numer. Anal., 36 (1999), pp. 1333–1369. 16. F. Golse, S. Jin, and C. D. Levermore, A domain decomposition analysis for a two-scale linear transport problem, M2AN Math. Model. Numer. Anal., 37(6) (2003).

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17. A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, Comparison of ﬁnite-diﬀerence transport and diﬀusion calculations for photon migration in homogeneous and heterogeneous tissues, Phys. Med. Biol., 43 (1998), pp. 1285– 1302. 18. A. Ishimaru, Wave Propagation and Scattering in Random Media, IEEE Press, New York, 1997. 19. E. W. Larsen and J. B. Keller, Asymptotic solution of neutron transport problems for small mean free paths, J. Math. Phys., 15 (1974), pp. 75–81. 20. E. W. Larsen and J. E. Morel, Asymptotic Solutions of Numerical Transport Problems in Optically Thick Diﬀusive Regimes II, J. Comp. Phys., 83 (1989), p. 212. 21. E. W. Larsen, J. E. Morel, and W. F. Miller Jr., Asymptotic solutions of numerical transport problems in optically thick, diﬀusive regimes, J. Comp. Phys., 69 (1987), pp. 283–324. 22. C. D. Levermore, W. J. Morokoff, and B. T. Nadiga, Moment realizability and the validity of the Navier-Stokes equations for rareﬁed gas dynamics, Phys. Fluids, 10(12) (1998), pp. 3214–3226. 23. E. E. Lewis and W. F. Miller Jr., Computational Methods of Neutron Transport, J. Wiley and sons, New York, 1984. 24. T. A. Manteuffel and K. Ressel, Least-squares ﬁnite-element solution of the neutron transport in diﬀusive regimes, SIAM J. Numer. Anal., 35(2) (1998), pp. 806–853. 25. T. A. Manteuffel, K. Ressel, and G. Starke, A boundary functional for the least-squares ﬁnite-element solution of the neutron transport equation, SIAM J. Numer. Anal., 37(2) (2000), pp. 556–586. 26. J. E. Morel and J. M. McGhee, A self-adjoint angular ﬂux equation, Nucl. Sci. Eng., 132 (1999), pp. 312–325. 27. J. Planchard, M´ethodes math´ ematiques en neutronique (in French), Collection de la Direction des Etudes et Recherches d’EDF, Eyrolles, 1995. 28. M. Reed and B. Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, Inc., New York, second ed., 1980. 29. J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, Boundary conditions for light propagation in diﬀuse media with non-scattering regions, J. Opt. Soc. Amer. A, 17(9) (2000), pp. 1671–1681. 30. R. P. Rulko, D. Tomasevic, and E. W. Larsen, Variational P1 Approximations of General-Geometry Multigroup Transport Problems, Nucl. Sci. Eng., 121 (1995), p. 393. 31. H. Sato and M. C. Fehler, Seismic wave propagation and scattering in the heterogeneous earth, AIP series in modern acoustics and signal processing, AIP Press, Springer, New York, 1998. 32. W. M. Stacey, Variational Methods in Nuclear Reactor Physics, Academic Press, New York, 1974. 33. M. D. Tidriri, Asymptotic analysis of a coupled system of kinetic equations, C.R. Acad. Sci. Paris, t.328, S´erie I Math., (1999), pp. 637–642. , Rigorous derivation and analysis of coupling of kinetic equations and 34. their hydrodynamic limits for a simpliﬁed boltzmann model, J. Statist. Phys., 104 (2001).

High Order Finite Volume Nonlinear Schemes for the Boltzmann Transport Equation Barna L. Bihari1 and Peter N. Brown2 1

2

Center for Applied Scientiﬁc Computing, Lawrence Livermore National Laboratory, Livermore, California 94550 [email protected] Center for Applied Scientiﬁc Computing, Lawrence Livermore National Laboratory, Livermore, California 94550 [email protected]

Abstract. We apply the nonlinear WENO (Weighted Essentially Nonoscillatory) scheme to the spatial discretization of the Boltzmann Transport Equation modeling linear particle transport. The method is a ﬁnite volume scheme which ensures not only conservation, but also provides for a more natural handling of boundary conditions, material properties and source terms, as well as an easier parallel implementation and post processing. It is nonlinear in the sense that the stencil depends on the solution at each time step or iteration level. By biasing the gradient calculation towards the stencil with smaller derivatives, the scheme eliminates the Gibb’s phenomenon with oscillations of size O(1) and recudes them to O(hr ), where h is the mesh size and r is the order of accuracy. Our current implementation is three- dimensional, generalized for unequally spaced meshes, fully parallelized, and up to ﬁfth order accurate (“WENO5”) in space. For unsteady problems, the resulting nonlinear spatial discretization yields a set of ODE’s in time, which in turn is solved via high order implicit time-stepping with error control. For the steady-state case, we need to solve the non-linear system, typically by Newton-Krylov iterations. There are several numerical examples presented to demonstrate the accuracy, nonoscillatory nature and eﬃciency of these high order methods, in comparison with other ﬁxed-stencil schemes.

1 Introduction In the deterministic modeling of uncharged particle transport, physical processes are described by the Boltzmann Transport Equation (BTE) which is a linear integro-diﬀerential equation to be solved for the scalar unknown Ψ , usually called the particle ﬂux. During the solution process, iterative methods are routinely used to solve the large systems of equations resulting from various discretizations, especially when it comes to solving the steady-state problem. As noted in [FaMa89], such approaches can be viewed as iterative solutions of a matrix equation Ax = b

(1)

This work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract W-7405-Eng-48.

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for the unknown vector x representing scalar ﬂuxes Ψ , energy densities, or other integrated quantities of interest. The spatial discretization has been traditionally done by some ﬁxed stencil ﬁnite diﬀerence or ﬁnite element method typically solving for point values of the solution, thereby resulting in a “node-centered” scheme. Finite volume schemes, on the other hand, were invented for the numerical solution of high speed ﬂuid dynamics where conservation is crucial. Since they solve for the cell average of the unknown instead of its point values, they are sometimes refered to as “cell-centered” schemes as well. Application of these schemes to the BTE ensures not only conservation, but also provides for a more natural handling of boundary conditions, material properties and source terms, as well as an easier parallel implementation and post processing. The ﬁnite volume scheme also lends itself to an eﬃcient implementation of high order spatial discretizations. Material interfaces and time-dependent large source terms can introduce severe oscillations even with second order ﬁxed stencil schemes. Slope limiting, or essentially nonoscillatory (ENO) spatial interpolations eliminate these oscillations, and make higher-than-second-order spatial accuracies possible. A newer variation of these nonlinear schemes is the Weighted ENO (WENO) scheme that makes the stencil transition less abrupt and boosts the accuracy in smooth regions. For unsteady problems, the resulting nonlinear spatial discretization yields a set of ODE’s in time, which in turn is solved via high order implicit time-stepping with error control. For the steady-state case, we need to solve the non-linear system, typically by Newton-Krylov iterations. Both of these approaches would ideally expect a preconditioner in order to obtain a reasonable rate of convergence. We will discuss the advantages of using an ENO/WENO method, as well as the various issues introduced by such nonlinear methods originally designed for computing shocked ﬂuid ﬂows. There will be several 1-D, 2-D, and 3-D numerical examples presented to demonstrate the accuracy, non-oscillatory nature and eﬃciency of these high order methods, in comparison with other ﬁxed-stencil schemes. The paper is organized as follows. In Sect. 2, we describe the continuous problem from which the discrete linear system (1) is derived. In Sect. 3, the exact form of (1) is established via the introduction of the discrete ordinate angular discretization together with the nonlinear spatial discretization. Although these discretizations are described elsewhere in the literature, we present them in some detail for the sake of clarity and completeness. In Sect. 4 we show several numerical results, and close with concluding remarks in Sect. 5.

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2 Background In this section we ﬁrst introduce the linear time-dependent BTE in three dimensional box geometry with general scattering [Pom73]. The spatial domain is the box D ≡ {r = (x, y, z)|ax ≤ x ≤ bx , ay ≤ y ≤ by , and az ≤ z ≤ bz }, the direction variable is Ω ∈ S 2 , the unit sphere in R3 , the energy variable is E ∈ (0, ∞), time is t, and the equation in the ﬂux ψ = ψ(r, Ω, E, t) is given by 1 ∂ψ (r, Ω, E, t) + Ω · ∇ψ(r, Ω, E, t) + σ(r, E)ψ(r, Ω, E, t) v(E) ∂t

∞

= 0

S2

(2) ψ(r, Ω , E , t)σs (r, Ω · Ω , E → E)dΩ dE + q(r, Ω, E, t) ,

with initial condition ψ(r, Ω, E, t0 ) = ψ 0 (r, Ω, E)

(3)

where ∇ψ ≡ (∂ψ/∂x, ∂ψ/∂y, ∂ψ/∂z), v(E) is the particle speed, and ψ 0 is the initial state at time t = t0 . The code employs a general algorithm that solves for multiple energy groups via a semi-discretization using a ﬁnite number of energy “bins”. However, for simplicity of notation and clarity of exposition we shall assume a single energy group E for the rest of this paper, and thereby remove the energy-dependency from ψ, v, σ and q in (2). The scattering integral on the right hand side of (2) is handled by expanding the ﬂux ψ(r, Ω, t) in surface harmonics according to ψ(r, Ω) =

n ∞

m φm n (r, t)Yn (Ω) .

n=0 m=−n

Here, Ynm (Ω) is a surface harmonic deﬁned by |m| Ynm (Ω) = am n Pn (ξ)τm (ϕ) , |m|

where Ω = (µ, η, ξ) = (sin θ cos ϕ, sin θ sin ϕ, cos θ), Pn Legendre polynomial [Lib80], and τm (ϕ) =

cos mϕ, if m ≥ 0, and sin |m|ϕ, if m < 0 .

The constants am n are deﬁned by

am n

(2n + 1)(n − |m|)! = 2(1 + δm0 )π(n + |m|)!

1/2 ,

is an associated

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where δn,n is the Kronecker delta, and (r) ≡ ψ(r, Ω)Ynm (Ω)dΩ, φm n S2

is the (n, m)th moment of ψ. Similarly, the source q is expanded as q(r, Ω, t) =

n ∞

qnm (r, t)Ynm (Ω) ,

n=0 m=−n

where qnm (r, t)

≡

S2

q(r, Ω, t)Ynm (Ω)dΩ .

For ease of exposition in what follows, we have elected to use real-valued surface harmonics, all scaled to have unit norm in L2 (S 2 ). Given ψ in the above form, one is able to rewrite the scattering integral in the form ∞ n m σs (r, Ω · Ω )ψ(r, Ω , t)dΩ = σs,n (r) φm n (r, t)Yn (Ω), S2

n=0

m=−n

where the σs,n are given by σs,n (r) ≡ 2π

1

−1

σs (r, µ0 )Pn (µ0 )dµ0 ,

and where µ0 is the cosine of the scattering angle. The total cross section σ is given by 1 σs (r, µ0 )dµ0 = σa (r) + σs,0 (r) , σ(r) ≡ σa (r) + 2π −1

where σa is the absorption cross section. Substituting the appropriate terms into (2), the equation to be solved now becomes: 1 ∂ψ (r, Ω, t) + Ω · ∇ψ(r, Ω, t) + σ(r)ψ(r, Ω, t) v ∂t (4) ∞ n m m = σs,n (r) φn (r, t)Yn (Ω) + q(r, Ω, t) , n=0

m=−n

Boundary conditions must also be speciﬁed so as to make (4) well-posed. Various options include a reﬂecting condition on a face, or a Dirichlet condition in which the incident ﬂux is speciﬁed on a face. For simplicity, we will consider only the latter case. Namely, we will consider boundary conditions of the form ψ(r, Ω) = g(r, Ω) for all r ∈ ∂D and Ω ∈ S 2 with n(r) · Ω < 0 , where n(r) is the outward pointing unit normal at r ∈ ∂D.

(5)

High Order Schemes for BTE

405

3 Discretization of the 3-D Problem We now turn to the angular, spatial and temporal discretizations of (4). In previous work [AsBr90], we derived a matrix version of the well-known diamond diﬀerence discretization scheme for the 1-D slab problem analogous to (4)–(5). We extend that development here to 3-D problems. The ﬁrst subsection deals with the quadrature rules for approximating integrals on S 2 , the next describes the spatial discretization, and the ﬁnal subsection considers the discrete-ordinates method, i.e., the combined discretization of problem (4)–(5). 3.1 Quadrature Rules The quadrature rules to approximate integrals on S 2 use the standard symmetry assumptions. Following Carlson and Lathrop [CaLa68], the quadrature rules we consider are of the form S2

ψ(Ω)dΩ ≈

L

w ψ(Ω ) ,

(6)

=1

where Ω ≡ (µ , η , ξ ), for all = 1, . . . , L, with L = ν(ν + 2) and ν is the number of direction cosines (ν = 2, 4, 6, . . .). Since Ω ∈ S 2 for all , we have µ2 + η2 + ξ2 = 1 for all .

(7)

With full symmetry, latitudinal point arrangement, and ν direction cosines, (7) becomes 2 =1 ξi2 + ξj2 + ξν/2+2−i−j for i = 1, 2, . . . , ν/2 and j = 1, 2, . . . , ν/2 − i + 1. This last equation can be solved to give 2(1 − 3ξ12 ) , (8) ξi2 = ξ12 + (i − 1) ν−2 for i = 1, . . . , ν/2 and 0 < ξ12 ≤ 1/3. For the weights, ﬁrst note that a constant function on S 2 must be integrated exactly for all ν, and so we must have 4π = S2

1 · dΩ =

L

w .

=1

For ν = 2 or 4, by requiring that weights be invariant under 90◦ rotations of the Ω coordinate system, it is easily seen that each weight must be the same. For ν > 4 there are ν/2 distinct weights, and for these sets one can determine the weights by requiring that

406

B.L. Bihari and P.N. Brown L

w ξ2n =

=1

4π , 2n + 1

(9)

for n = 0, . . . , ν/2 − 1. These conditions guarantee that as many even powers of ξ as possible are integrated exactly by the quadrature rule. Note that due to the symmetrical placement of the ξ along the ξ axis, all odd powers of ξ are integrated exactly. For ν ≥ 22, demanding that (9) holds for all n = 0, . . . , ν/2 − 1 leads to negative weights. As an alternative, one could simply demand that the weights are all equal, in which case (9) always holds for n = 0 and 1. For either type of quadrature rule, we only require that (9) holds with n = 0 and 1, and that all the weights are positive. Finally, it also follows from the symmetrical placement of the direction cosines that the following additional results hold: L =1

w µ = 0,

L

w η = 0, and

=1

L

w ξ = 0 .

(10)

=1

3.2 Finite Volume Spatial Discretization In the formulation of the ﬁnite volume method, we ﬁrst discretize D into cells (also called “zones”). Introduce the spatial grids ax ≡ x 21 < · · · < xi− 12 < xi+ 12 < · · · < xM + 12 ≡ bx , ay ≡ y 12 < · · · < yj− 12 < yj+ 12 < · · · < yJ+ 12 ≡ by , and az ≡ z 12 < · · · < zk− 12 < zk+ 12 < · · · < zK+ 12 ≡ bz ,

and deﬁne rijk = (xi , yj , zk ). Next, deﬁne ∆xi = xi+ 12 − xi− 12 for i = 1, . . . , M ∆yj = yj+ 12 − yj− 12 for j = 1, . . . , J, and

∆zk = zk+ 12 − zk− 12 for k = 1, . . . , K .

Also deﬁne ∆rijk ≡ ∆xi ∆yj ∆zk . The {rijk } are referred to as nodes (or grid points), and function values at these points are called nodal values. Assume that σ and σs,n have constant values on each cell Zijk ≡ r|xi− 12 < x < xi+ 12 , yj− 12 < y < yj+ 12 , zk− 12 < z < zk+ 12 , ijk denoted by σijk and σs,n , respectively. Function values that are constant on cells will be referred to as cell-centered values. We use ψijk to denote the approximation to ψ(rijk ), the true solution at rijk . To obtain the ﬁnite volume method, average equation (4) over cell Zijk . We then have a semi-discrete form

High Order Schemes for BTE 1 ¯ v ∂ ψijk

∂t

=

(Ω, t) +

∞ n=0

ijk σ ¯s,n

1 ∆rijk

n

407

Zijk

Ω · ∇ψ(r, Ω, t)dr + σ ¯ijk ψ¯ijk (Ω, t) (11)

m ¯ijk (Ω, t) , φ¯m n,ijk (t)Yn (Ω) + q

m=−n

where the quantities marked with a bar are “cell-averaged” (in essence integral) quantities, and not point values. In particular: 1 ψ¯i,j,k (Ω, t) = ψ(r, Ω, t)dr , ∆rijk Zijk φ¯i,j,k (t) =

1 ∆rijk

q¯i,j,k (Ω, t) =

φ(r, t)dr , Zijk

1 ∆rijk

(12)

q(r, Ω, t)dr , Zijk

For the ﬁrst integral in (11), use Green’s Theorem to obtain 1 µ ¯ Ω · ∇ψ(r, Ω, t)dr = (ψ 1 − ψ¯i− 12 ,j,k ) ∆rijk Zijk ∆xi i+ 2 ,j,k η ¯ 1 + (ψ − ψ¯i,j− 12 ,k ) ∆yj i,j+ 2 ,k ξ ¯ ¯ 1 − ψ + (ψ i,j,k− 12 ) . ∆zk i,j,k+ 2

(13)

The averaged values with half-indexed subscripts now denote face values over each respective cell face. These too are averaged values, but over faces instead of cells: y 1 z 1 j+ k+ 1 2 2 ¯ ψ(xi+ 12 , y, z, Ω, t) dy dz , (14) ψi+ 12 ,j,k = ∆yj ∆zk y 1 z 1 j−

2

k−

2

and similarly for the other face values. Note that with the substitution of (13) into (11) we have both cellaveraged and face-averaged quantities. To close the system, we express the face values in terms of the cell averages, often called the reconstruction procedure in the literature. This is the step where the spatial accuracy is determined in any ﬁnite volume scheme. For example, if we simply take , if µ > 0 ψ¯ (15) ψ¯i+ 12 ,j,k = ¯i,j,k ψi+1,j,k , if µ < 0 we get the simple upwind, or “step” method. By replacing the face value by the average of the cell values on each side of the face we arrive at the diamond diﬀerence approximation:

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B.L. Bihari and P.N. Brown

1 ψ¯i+ 12 ,j,k = ψ¯i,j,k + ψ¯i+1,j,k 2

(16)

If the interpolation scheme that relates the face- and cell-averages to each other is higher order, the spatial accuracy will be (at least formally) higher order as well. With linear interpolation schemes, however, oscillations will occur near discontinuities in the solution, in the source term, or in the material interfaces, or even at smooth, but steep gradients in the solution. The WENO (“Weighted Essentially Nonoscillatory”) method is in fact an interpolation scheme where the weights used in the linear combination of the interpolation points vary with the size of the derivative. The larger the gradient is near an interpolation point, the smaller the weight will be for that point. The idea goes back to the TVD (Total Variation Diminishing) method [Har83] which eliminated oscillations by choosing the left or right biased slope for reconstruction. The ENO (Essentially Nonoscillatory) [Har87] scheme raised the inherently second order accuracy of the TVD scheme by relaxing the TVD property to “essentially nonoscillatory” and choosing the interpolation stencil matching the order of accuracy from a set of available ones. With ENO the oscillations are not completely ruled out, but their size is reduced from O(1) to O(hr ), where r is the order of accuracy. That is, in essence, oscillations were eliminated. The WENO scheme improved on this concept by not choosing, but rather, weighting the stencils by their smoothness [Shu97] . 3.3 The Weighted Essentially Nonoscillatory Interpolation In this section we give a brief overview of the speciﬁc WENO method used in the numerical experiments. The formulae come mostly from [Shu97], but for completeness we include them here. For clarity, we present the 1-D version of the interpolation, which can be readily used in a direction-by-direction fashion to reconstruct all three spatial dimensions in (13). Given a cell averaged grid function {¯ vj }N j=1 on a set of grid cells {xj } J corresponding to a grid {xj− 12 }j=1 , we approximate vj+ 12 at the cell faces via a weighted linear combination of all possible interpolations: vj+ 12 =

k−1 r=0

(r)

wr vj+ 1 2

(r)

where typically k = 2 (for WENO3) or k = 3 (for WENO5). The vj+ 1 are 2 the various interpolated values using polynomials corresponding to stencil r. The interpolation used for each stencil r can, and in our code it does, take into account variable grid sizes, so no assumption of equally spaced grids is made.

High Order Schemes for BTE

409

The weights wr are given by: αr wr = k−1 s=0

αs

for r = 0, ..., k − 1

,

αr =

dr ( + βr )2

where is a small positive number. For the two most commonly used WENO schemes we have: (i) if k = 2: 2 3 1 d1 = 3 vi+1 − v¯i ) β0 = (¯

d0 =

(17)

β1 = (¯ vi − v¯i−1 ) (ii) if k = 3: d0 = d1 = d2 = β0 = β1 = β2 =

3 10 3 5 1 10 13 (¯ vi − 2¯ vi+1 + v¯i+2 )2 + 12 13 (¯ vi−1 − 2¯ vi + v¯i+1 )2 + 12 13 (¯ vi−2 − 2¯ vi−1 + v¯i )2 + 12

1 (3¯ vi − 4¯ vi+1 + v¯i+2 )2 4 1 (¯ vi−1 − v¯i+1 )2 4 1 (¯ vi−2 − 4¯ vi−1 + 3¯ vi )2 4

(18)

3.4 Boundary Conditions For the boundary conditions in (5), when x = x0 , the normal n(r 12 ,j,k ) = (−1, 0, 0) for all j, k. Hence, n(r 12 ,j,k ) · Ω = −µ, and for µ > 0 we have ψ¯ 12 ,j,k = g 12 ,j,k (≡ g(r 12 ,j,k )) .

(19)

For y = y0 , n(ri, 12 ,k ) = (0, −1, 0) for all i, k, and so n(ri, 12 ,k ) · Ω = −η, and for η > 0 we have (20) ψ¯i, 12 ,k = gi, 12 ,k . For z = z0 , n(ri,j, 12 ) = (0, 0, −1) for all i, j, and so n(ri,j, 12 ) · Ω = −ξ, and for ξ > 0 we have

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B.L. Bihari and P.N. Brown

ψ¯i,j, 12 = gi,j, 12 .

(21)

Here g 12 ,j,k , gi, 12 ,k , and gi,j, 12 are face averages as deﬁned by (14). The other three cases are handled similarly. Of course, for a given Ω = Ω only three of the above six cases can hold. (The quadrature rules deﬁned above guarantee that no component of Ω is ever zero.) 3.5 Temporal Discretization The semidiscrete form (11) can be written as a matrix operation on a solution vector Ψ = (ψi,j,k,l )T : ˙ + T(Ψ)Ψ − F = 0 V−1 Ψ ˙ = ∂Ψ is the temporal derivative, T(Ψ) represents the semidiswhere: Ψ ∂t cretization in space and direction that is nonlinear in the sense that it depends on the solution Ψ, and F includes source and boundary terms. In compact notation, we may write it as a system of ODE’s. Note that the node centered version of the scheme would become a diﬀerential- algebraic equation (DAE) system ˙ =0 F (t, Ψ, Ψ) because the boundary values would have to be solved for simultaneously with the interior values. Since the code is general enough to have some node-centered spatial discretizations (such as Petrov-Galerkin), we use the same temporal solver for all spatial options. Hence for uniformity the time integration is accomplished via the IDA (Inexact Newton Diﬀerential/Algebraic Equation) package. It uses backward diﬀerencing methods which are variable in order (up to ﬁfth order in time) and stepsize, and are also implicit.

4 Numerical Experiments We now present several numerical results in 1-, 2-, and 3-D, using the same three-dimensional code, but simulating the lower dimensional problems by creating a very large (on the order of 106 ) single grid cell in the irrelevant dimension. The initial condition in (3) is ψ 0 (r, Ω, E) = 0 for t = 0 in all cases below. Also, the boundary conditions used were all Dirichlet BC and (5) was set via g = 0 at all incident boundary faces. 4.1 1-D Examples We now present some time-dependent problems in 1-D slab geometry pointing out the salient features of the WENO method when compared to the more traditional, linear spatial discretizations.

High Order Schemes for BTE

411

(1) Single material, steady source. In this case the cross sections were set to σ = σ(r) = 0.1, σs = σs (r, Ω · Ω) = 0.01 with the source term deﬁned by q=

10, 0,

if x ∈ [0.4, 0.6] otherwise

We used M = 50 grid cells and compared WENO3 to the Petrov-Galerkin ﬁnite element (node centered) scheme and the ﬁrst order upwind scheme of (15). As it is illustrated in Fig. 1, the third order WENO method can capture sharp transitions and corners signiﬁcantly better than the ﬁrst order accurate upwind method, while neither one produces oscillations. When compared to Petrov-Galerkin, it is clear that P-G creates large oscillations (and therefore negative ﬂuxes) especially at the beginning stages of the development of the proﬁle. This problem actually has a non-zero steady-state solution, to which the WENO3 method seems to converge faster than either of the other methods, as shown on the last ﬁgure of the series. In explaining this superior performance in convergence, we conjecture that the higher spatial accuracy is the explanation. Recall that the upwind method is ﬁrst-, P-G is second-, and WENO3 is third-order accurate in smooth regions. It is also interesting to compare the tested methods to a very ﬁne grid upwind solution, which, in some sense, should be the most “reliable” in converging to the correct physical solution under grid reﬁnement. In Fig. 2 we show the temporal behavior of a single point located at x = 0.6 and include a 1000 cell upwind solution for reference. Note how the WENO is extremely close to the “super-ﬁne” upwind proﬁle, while the other two methods are oﬀ by about 10%. (2) Single material, steady point-source. In order to test the method for a very narrowly supported pseudo point-source, we ran the same problem as in case (1) above, but with a source deﬁned as q=

10, 0,

if x ∈ [0.48, 0.52] otherwise

where now the source is nonzero in only two grid cells. This is a challenge for spatial discretizations which are higher than ﬁrst order, because the jumps caused by the source are closer to each other than the stencil width itself. We now included WENO5 in the comparison to push the limits of the scheme for the WENO5 scheme uses a stencil that is 5 cells wide. As shown on Figs. 3 and 4, the behavior of the WENO schemes is still acceptable. They produce no oscillations at the base of the source, but they both overshoot the “best” solution, the super-ﬁne upwind. The expectation is that as the grid is reﬁned the behavior will be identical to that of the previous case, since the distance between locations of the large gradients will then be several grid cells wide.

412

B.L. Bihari and P.N. Brown Time = 1e−08

Time = 6e−08

0.2

0.9 WENO3 Petrov−Galerkin Upwind

WENO3 Petrov−Galerkin Upwind

0.8

0.15

0.7

0.6

2

2

φ (n/cm −s)

0.5

φ (n/cm −s)

0.1

0.4

0.05

0.3

0.2

0

0.1

0

−0.05

0

0.1

0.2

0.3

0.4

0.5 x (cm)

0.6

0.7

0.8

0.9

−0.1

1

0

0.1

0.2

0.3

0.4

Time = 1.5e−07

0.5 x (cm)

0.6

0.7

0.8

0.9

1

Time = 2.3e−07

2.5

3.5 WENO3 Petrov−Galerkin Upwind

WENO3 Petrov−Galerkin Upwind

3

2 2.5 1.5

2

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(3) Thin/thick materials, large unsteady source. We now test the code on a two-material problem deﬁned by: 0.0001, if x ∈ [0, 0.1] σ(x) = 100, otherwise σs = 0 and the source, localized by x ∈ [0, 0.1], changes in time as: 8 × 1012 , if t ∈ [10−9 , 4 × 10−9 ] q(t) = 0, otherwise The suite of methods tested now includes not just WENO3 and WENO5, upwind, and P-G, but the Simple Corner Balance (SCB) [Ad97] and the Diamond-Diﬀerence with negative ﬂux ﬁx-up (D-D) [LM93] methods as well; both of the latter are cell-centered schemes. We compare all these methods on a grid of M = 200 cells with a very ﬁne grid upwind solution on 5000 cells. The latter is again guaranteed to converge to the right solution under grid reﬁnement.

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An examination of Fig. 5 reveals the two salient features of the WENO scheme: (i) its nonoscillatory nature and (ii) high order of accuracy in smooth regions. The latter property is more pronounced when the solution has areas of smooth variation. In Fig. 5 we show close-ups of the highly varying regions on the right side of each snapshot. These conﬁrm our expectation of the WENO scheme coming very close to the superﬁne solution, while the lower order schemes produce large errors near extrema and/or oscillations near steep gradients. To avoid having to include a large number of plots, we chose to use the integrated quantity φ in Fig. 5. (4) Thick/thin materials, large unsteady source. If we reverse the two material properties, but leave the source term as in Case (3), we can test the code on a “thick-to-thin” problem. Thus set: 100, if x ∈ [0, 0.1] σ(x) = 0.0001. otherwise σs = 0 In this case the high order of accuracy oﬀered by the WENO method is quite pronounced near the corner regions of the proﬁles. The WENO3 and

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WENO5 schemes seem to give the same accuracy as the upwind scheme on a very ﬁne grid - but with 1/25th of the number of cells. 4.2 2-D Test Problem The method is now tested in 2-D mode using the same code, but now the (single) cell in the z-direction is very large to simulate a 2-D slab. Initial and boundary conditions are again set to be nil, and the grid is a 300 × 300 Cartesian mesh in a [0, 100] × [0, 100] square domain. Cross sections were taken to be σ = 0.001 and σs = 0.001, and the source was deﬁned by: 1, if (x, y) ∈ [33.25, 66.75] × [33.25, 66.75] q= 0, otherwise This was actually a steady-state simulation where we used the KINSOL package as our nonlinear solver. The same results were arrived at running the unsteady option out to steady-state. On the top portion of Fig. 7 we show a contour plot of the ﬂux when both direction cosines are positive. We also ran the upwind and SCB methods for comparison. The latter showed some oscillations and severe inaccuracy in capturing the middle part of the proﬁle, even at steady-state. The upwind method, due to its dissipative nature, “smoothed” out the corners. Both WENO3 and WENO5 performed well, but except for the sharp left corner, there was little diﬀerence between the two. 4.3 3-D Test Problem We ﬁnally ran the code in fully 3-D mode, on a domain [0, 1] × [0, 1] × [0, 1] with 40 × 40 × 40 grid cells. Cartesian mesh in a [0, 100] × [0, 100] × [0, 100] square domain. In this case we set σ = 10.0 and σs = 0. The source term was again in the middle: 1, if(x, y, z) ∈ [.4, .6] × [.4, .6] × [.4, .6] q= 0, otherwise This case was again run in steady-state mode and then veriﬁed by the unsteady option. On Fig. 8 we show the two WENO options, Petrov-Galerkin and upwind on two diﬀerent grids (40 × 40 × 40 and 100 × 100 × 100) at the middle cut of y = 0.5, z = 0.5. The ﬁner upwind solution was not run on a grid that is orders of magnitude ﬁner than the others, hence it should not be taken now as a benchmark solution. It is included merely to show the diﬀerence grid reﬁnement makes in the accuracy of the solution. The WENO3 and WENO5 solutions are expected to be better, and in fact they are somewhat diﬀerent from each other as well. The Petrov-Galerkin method, on the other hand, exhibits sizeable oscillations/negative ﬂuxes near the proﬁle corners.

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5 Discussion We have presented a new application of the WENO scheme for the spatial discretization of the Boltzmann Transport Equation. Developed for highly nonlinear systems of partial diﬀerential equations, such as the Euler and Navier-Stokes equations, the scheme was designed to accurately compute shocked ﬂuid ﬂow. As such, it was an unlikely candidate for application to a scalar and linear integro-diﬀerential equation. Indeed, the notion of “slopelimiting,” and in general, of nonlinear schemes has been known for over two decades, yet unutilized (to our knowledge) in the transport arena. However, once the realization is made that it is merely an interpolation technique, it is a natural ﬁt for any problem where the problem is linear, but the coeﬃcients or the source terms are discontinuous and thus give rise to steep gradients in the solution. We hope to have demonstrated that the WENO scheme can be very useful for many problems, especially unsteady ones. While they do not guarantee positivity, by reducing the size of oscillations to O(hr ), in practice they give positive ﬂuxes in an overwhelming majority of the cases where other linear schemes fail to do so. We have shown that the high order of spatial accuracy

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on a relatively coarse grid can be interchangeable with a very ﬁne grid spacing using a low order method. This translates into an often sought-after, and now possible trade-oﬀ between processor power and memory capacity. The WENO method seems to be a good choice especially for cases where the solution simultaneously has steep gradients/discontinuities and large areas of smooth variations. For future work, we intend to further improve on the essentially nonoscillatory property to ensure positivity by lowering the order of accuracy in those rare regions where small negative ﬂuxes still remain. We also plan to develop better sweep-preconditioners that mimic the behavior of the nonlinear discretization used in the WENO scheme itself. Furthermore we need to conduct rigorous grid reﬁnement studies to verify the order of accuracy at least in the ideal situation where the source term is smooth and an exact solution exists (no scattering). Finally, more testing is necessary on large 3-D problems with multiple material and diﬀerent source terms.

Acknowledgements The authors wish to acknowledge IPAM (Institute for Pure and Applied Mathematics) at UCLA for co-sponsoring the ﬁrst (summer) project in this subject area. In particular we thank Richard Tsai, Filip Matejka, David Stevens, Nicholas Kridler and Rodney Chan for their initial contributions in summer of 2002.

References [Ad97]

Adams, M.: Subcell balance methods for radiative transfer on arbitrary grids. Transport Theory Stat. Phys., 26, 385–431 (1997) [AsBr90] Ashby, S.F., Brown, P.N., Dorr, M.R., Hindmarsh, A.C.: Preconditioned iterative methods for discretized transport equations. UCRL-JC-104901, Lawrence Livermore National Laboratory (1990) [CaLa68] Carlson, B.G, Lathrop, K.D.: Transport Theory: The method of discrete ordinates. H. Greenspan et al (eds) Computing Methods in Reactor Physics. Gordon and Breach, New York, 166–266 (1968) [FaMa89] Faber, V.T., Manteuﬀel, T.A.: A Look at Transport Theory from the point of view of linear algebra. In: P. Nelson et al (eds) Transport Theory, Invariant Imbedding, and Integral Equations. Marcel Dekker, New York, 37–61 (1989) [Har83] Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys., 49, 357 (1983) [Har87] Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniform high order accurate essentially non-oscillatory schemes III. J. Comput. Phys., 71, 231 (1987) [LM93] Lewis, E.E.,Miller Jr., W.F: Computational Methods of Neutron Transport. American Nuclear Society, La Grange Park, Illinois (1993)

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Liboﬀ, R.L.: Introductory Quantum Mechanics. Holden-Day, Inc.,San Francisco (1980) [Shu97] Shu, C.-W.: Essentially non-oscillatory and weighted essentially nonoscillatory schemes for hyperbolic conservation laws. ICASE Report No. 97–65, NASA Langley Research Center (1997) [Pom73] Pomraning, G.C.: The Equations of Radiation Hydrodynamics. Pergamon Press, Oxford (1973)

Obtaining Identical Results on Varying Numbers of Processors in Domain Decomposed Particle Monte Carlo Simulations N.A. Gentile1 , Malvin Kalos1 and Thomas A. Brunner2 1

2

University of California, Lawrence Livermore National Laboratory∗ , Livermore California 94550 [email protected],[email protected] Sandia national Laboratories† , Albuquerque New Mexico 87185 [email protected]

Abstract. Domain decomposed Monte Carlo codes, like other domain-decomposed codes, are diﬃcult to debug. Domain decomposition is prone to error, and interactions between the domain decomposition code and the rest of the algorithm often produces subtle bugs. These bugs are particularly diﬃcult to ﬁnd in a Monte Carlo algorithm, in which the results have statistical noise. Variations in the results due to statistical noise can mask errors when comparing the results to other simulations or analytic results. If a code can get the same result on one domain as on many, debugging the whole code is easier. This reproducibility property is also desirable when comparing results done on diﬀerent numbers of processors and domains. We describe how reproducibility, to machine precision, is obtained on diﬀerent numbers of domains in an Implicit Monte Carlo photonics code.

1 Description of the Problem There are two main issues that can cause a code to get diﬀerent results when run on diﬀerent numbers of domains. The ﬁrst is that domain decomposition can cause the code to use a diﬀerent sequence of pseudo-random numbers. The second, which also applies to deterministic codes which do not employ pseudo-random numbers, is that the order of operations on ﬂoating point numbers can change, leading to diﬀerent results. We will examine both of these problems and describe solutions. In the method we describe, problems are broken up into spatial domains. Computational work on each domain is performed by a single processor. Thus ∗

This work was performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract No. W-7405-ENG-48. † Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000.

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the number of domains and the number of processors are the same. We have not considered having multiple processors work on a single domain, as could occur when threads are used. Because we have done this work in the context of an Implicit Monte Carlo code, we will brieﬂy describe that algorithm. The algorithm simulates the time-dependent interaction of photons and matter. It does this by creating, tracking, and destroying particles whose behavior models that of real photons in matter. This requires calculating probabilities for physical events such as emission and scattering. The behavior of each photon is determined by using a pseudo-random number to pick one of the behaviors. This is repeated until the photon is completely absorbed by the matter, leaves the domain, or reaches the end of the time step. The behavior of matter is simulated on grid of zones, each with diﬀerent material properties, such as diﬀerent temperature and opacity. These zones lose energy when the emit particles, and gain energy when particles pass through them. Particles can visit a zone more than once (for example, by leaving and scattering back in from another zone.) In that case, a particle will deposit energy in the zone more than once. Details of the algorithm can be found in [FC71]. This Implicit Monte Carlo program is used in the KULL [GKR98] and ALEGRA [BM04] inertial conﬁnement fusion simulation codes. Parallel domain decomposition was accomplished using the algorithm described in [BUEG04]. Domain decomposition is necessary when the grid is too large to ﬁt on the memory of one processor. It is also done to make problems run faster by bringing more computation resources to bear. To domain decompose the problem, we partition the grid and put parts on diﬀerent processors. This necessitates moving particles between domains when they are tracked to domain boundaries. In order to debug this code, and have conﬁdence in the results, we desire that a problem run on several domains (i.e., processors) get the same answer as when we run it on one domain (one processor). The two issues that impact reproducibility are illustrated by considering the behavior of particles that cross a domain boundary. If the problem is run using one processor, there is (of necessity) one domain. With two or more processors, some zones will be on a domain boundary. Let Zone 1 and Zone 2 abut each other across a domain boundary, as shown in Fig. 1. Let Particle A be emitted in Zone 1 and enter Zone 2, scattering back into Zone 1. Let Particle B be emitted later in Zone 1 and stay there, executing a scatter. When the problem is run on one processor, computations on Particle A continue until it is terminated (e.g., it leaves the problem through a transmitting boundary.) Only then are computations on Particle B begun. When it is run on two processors, Particle A is followed to the domain boundary and passed oﬀ to the processor on which Zone 2 resides. Then Particle B is followed until it is terminated.

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Scatter B Particle B Particle A Scatter A

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Two things happen diﬀerently in the one domain case than in the two domain case. First, the order of scatters of Particle A and Particle B is reversed. Scattering events use pseudo-random numbers to determine their outcomes (scattering angle, etc.) Hence the scatters will result in diﬀerent behavior unless we ensure that the particles use the same pseudo-random numbers independent of the order in which those numbers are accessed. Second, the order of energy deposition in Zone 1 is diﬀerent. Particle A deposits energy twice in Zone 1 before Particle B does when one processor is used. When two processors are used, Particle B deposits between the two deposition events involving Particle A. Addition in ﬂoating point arithmetic is not always exactly commutative: (x + y) + z can diﬀer from x + (y + z) by a small amount on the order of roundoﬀ. Hence, the ﬁnal energy in Zone 1 may be slightly diﬀerent in the two domain case than the one domain case. It might be thought that this small diﬀerence in results, on the order of roundoﬀ, could be tolerated. We will demonstrate that it can have large eﬀects on the result of a calculation by aﬀecting the behavior of particles in subsequent time steps. We will now discuss our solutions to these two issues.

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2 Ensuring the Invariance of the Pseudo-Random Number Stream Employed by Each Particle Domain decomposition alters the order in which particle event take place. The results for events which employ pseudo-random numbers will be diﬀerent unless we ensure that each particle draws the same stream of pseudorandom numbers independent of the order in which it is simulated. The way we accomplish this is to give each particle its own pseudo-random number generator by giving each particle its own pseudo-random number generator state. An example will illustrate this. A simple pseudo-random number generator is sn = a · sn−1 + b

(1)

rn = d · sn

(2)

Here a, b, and s are 64 bit unsigned integers, with a = 2862933555777941757, b = 3037000493, and sn is the nth state of the generator; d and rn are double precision numbers, with d = 5.4210108624275222 · 10−20 ≈ 1/264 and rn is the nth pseudo-random number. Applying the ﬁrst part of this step maps the 64 bit unsigned integer s into another 64 bit unsigned integer. Since the maximum value of a 64 bit integer is 264 , multiplying by the inverse of this number results in a value for rn in the range [0, 1]. In this pseudo-random number generator, s is referred to as the state. The current value of s (along with the constants a and b) completely determines the next value, and so it completely determines the subsequent stream of pseudo-random numbers. If each particle has its own value of s, it will sample the same stream of pseudo-random numbers independent of the order of particle calculations. That is, for example, the ﬁfth pseudo-random number used by Particle A will be the same, independent of which processor it is being simulated by, or how many other particles have been involved in computations since Particle A used its forth pseudo-random number. Although we have illustrated the algorithm with a very simple pseudorandom number generator with a single integer as a state, it will work with more complicated pseudo-random number generators with larger states. The SPRNG library [SPRNG] has several pseudo-random number generators that work well in the context of this algorithm. In order for this procedure to work, the ﬁrst value of the state s, called the seed, will have to be determined in a manner that is independent of the domain decomposition. Its value will have to be determined from values that are invariant under domain decomposition. Some examples are global zone numbers, zone position, and the number of particles that have already been created in a given zone.

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Using zone position is safer than using global zone numbers, because a code may not produce those. However, the position may not be invariant, because the code producing the grid positions may give slightly diﬀerent (i.e., “jittery”) positions with diﬀerent numbers of domains. We will now describe an algorithm that gives invariant seeds from zone positions, provided that there is not too much jitter in these positions. The ﬁrst step is to ﬁnd the minimum and maximum values, in each spatial dimension, of the locations of the zone centers on each processor. That is, we get the minimum and maximum values of x, y, and z on each domain. Then we ﬁnd the minimum and maximum over all domains by using the MPI Allreduce command. Since these global values may have some jitter, we shave oﬀ the lower order bits. This is done by a “shaving” algorithm given in the appendix. Now we have three double precision numbers for the grid that are invariant over the number of domains. Next, we loop over each zone in the grid and scale its position by the minimum and maximum values for the grid. That is, we calculate (xzone − xmin )/(xmax − xmin ), for x, y, and z. Because xzone may also contain some jitter on diﬀerent numbers of processors, we apply the shaving algorithm to these three numbers. This gives us three numbers in [0, 1] for each zone that are invariant over the number of domains. At least one of these numbers will be diﬀerent for each zone. (Equality could occur for one or two of the numbers because, for example, zones in a one dimensional problem would have the same x and y values if the grid only had extent in the z direction.) Then, we multiply these three numbers by 1.8446744 · 1019 ≈ 264 , which maps them into a 64 bit unsigned integer, and we add the three numbers together. This yields a 64 bit integer number that is diﬀerent for each zone. To reduce correlations between zones, we then change this value into a new unique 64 bit unsigned integer by subjecting it to the DES hash algorithm [PTVF02]. This yields one 64 bit unsigned integer for every zone that is invariant over the number of domains. To get initial seeds for each particle in the zone, we increment the zone value by one and apply the DES hashing. This gives each particle a unique seed that is independent of the number of domains. Thus each particle accesses the same pseudo-random number stream, independent of the number of domains.

3 Ensuring That Addition is Commutative Using the algorithm described above, we can ensure that all particles access the same pseudo-random number stream independent of the order in which they are simulated. They will still, however, deposit energy in the zones in a diﬀerent order when the number of domains is changed. Because ﬂoating point addition is not exactly commutative, there will small diﬀerences in the total energy deposited in each zone at the end of the time step.

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These diﬀerences are on the order of roundoﬀ. However, we cannot tolerate them because they will eventually have macroscopic consequences. This is because the energy deposited in the zone will aﬀect the temperature and opacity of the zone in the next time step. The opacity can be a nonlinear function of the temperature, and so small diﬀerences in the temperature can be magniﬁed. Diﬀerences in the value of the opacity will cause diﬀerences in the deposition of energy by every photon that enters the zone. This in turn will eﬀect the creation and behavior of particles in the zone. Eventually, some particle will behave diﬀerently (e.g., not scatter) because of slight diﬀerences in the values of temperature and opacity. This diﬀerent behavior will have a macroscopic eﬀect on the problem, which will aﬀect other particles in subsequent time steps. Soon, the diﬀerence between the two cases will be as large as if diﬀerent random number streams were used. This eﬀect is illustrated in Fig. 2. This plot shows the diﬀerence in the temperature in the ﬁrst zone of three diﬀerent simulations of a test problem 0

10

-2

Temperature Difference, |T1 -T2|

10

-4

10

-6

10

-8

10

-10

10

-12

10

| Tf-Tr | | Tf-Ta |

-14

10

-16

10

0

50

100

200 150 Time Step

250

300

350

Fig. 2. The absolute value of the diﬀerence in temperature in the ﬁrst zone of three simulations vs. time step. Two simulations used the same pseudo-random number stream but diﬀerent particle order. (These are denoted the “forward”, Tf , and “reverse”, Tr , simulations.) The third simulation used a diﬀerent pseudo-random number stream (the “alternate”, Ta , simulation.) |Tf −Tr | is depicted with a dotted line and |Tf −Ta | with a solid line. |Tf −Ta | is large at the beginning, and ﬂuctuates but does not increase with time. |Tf − Tr | begins at a value near the roundoﬀ error for double precision arithmetic, but grows and eventually jumps to same level as the diﬀerence between the “forward” and “alternate” simulations

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from [FC71]. The opacity is given by (2) in [FC71]: σ=

1 27 (1 − eν/T ) cm ν 3 T 3/2

(3)

with ν and T in keV. The simulations used 100 zones with ∆x = 0.4. It was run for 400 time steps with ∆t ﬁxed at 2.0·10−12 sec. A temperature source with Ts = 1.0 keV was applied at x = 0. The initial temperature was 0.01 keV. The equation of state had a constant heat capacity of Cv = 8.11829 · 109 erg/(cm3 keV) = 0.5917aTs3 , where a is the radiation constant. Each simulation used 1000 particles in each time step. All three simulations were run on one processor, and all used the pseudorandom number described above. Two simulations employed the algorithm described above which ensured that particles got the same pseudo-random number stream in each case. The only diﬀerence between these simulations was that the particles were run in a diﬀerent order. After new particles were created at the beginning of each time step, the order of the list was reversed before they were tracked. Thus any diﬀerence in the results of these two simulations is due to diﬀerences in the order of ﬂoating point addition when the particles deposit energy. The third simulation used a diﬀerent pseudo-random number stream entirely. This was accomplished by adding 78654092354 to the initial value of s in each zone in the simulation. Figure 2 plots |Tf −Tr | and |Tf −Ta | versus the time step in the simulation. Here Tf is the temperature in the ﬁrst zone when the particles are run in the usual order, Tr is the temperature in that zone when the particle order was reversed, and Ta is the temperature in the run which used a diﬀerent pseudo-random number stream. The diﬀerence between runs using diﬀerent pseudo-random number streams is ﬂuctuates between 0.01 and 0.1 throughout the simulation. (The value of the temperature in the ﬁrst zone quickly becomes approximately equal to the Ts = 1.0.) The diﬀerence between runs using the same pseudorandom number stream is initially very small. The diﬀerence in the ﬁrst time step is O(10−14 ), which is the size of roundoﬀ errors. This causes a diﬀerence in the opacity in the zone, which means that photons in the zone will deposit slightly diﬀerent amounts of energy in the two simulations. The diﬀerence in temperature grows with each time step. However, the diﬀerence remains small, and global measures like the total number of particles simulated remain the same until time step 304. At time step 304, the accumulated diﬀerence in temperature is large enough to change the large-scale behavior of the code. A diﬀerent number of particles are created in the two simulations in this time step. After that, the diﬀerences quickly grow until they are of the same order as the diﬀerence we ﬁnd compared to the simulation with a completely diﬀerent pseudo-random number stream.

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The cure for this problem is relatively simple. Floating point addition is not commutative, but integer addition is. We eliminate diﬀerences in results by mapping the energy to a 64 bit unsigned integer before doing the addition. This scaling is a simple multiplication that maps the range of photon energy in the problem into the range that a 64 bit unsigned integer can hold, which is [0, 264 − 1]. This number changes with each time step, and is calculated at the beginning of each time step. The multiplier is S = 263 /(Ecensus + Esource ) .

(4)

Here Ecensus is the sum of the energy of photons present at the beginning of the time step, and Esource is the total amount of energy added to photons at the beginning of the time step (e.g., aT 4 Vzone ∆t for the thermal radiation emitted from a zone.) We have used 263 rather than 264 − 1, the maximum value of an unsigned 64 bit integer, as a safety factor. Note that S is a double precision value, not an integer. To calculate the energy deposited into a zone, we sum the 64 bit unsigned integers obtained from scaling the energy deposited by each photon: int = integer(Ephoton × S + 0.5) Ephoton int Edep

=

int Edep

+

int Ephoton

(5) (6)

int is an unsigned 64 bit integer, and integer() represents a cast Here Edep from a double precision value to a 64 bit unsigned integer. The 0.5 is added to round to the closest integer instead of simply truncating the fractional part of the value. At the end of the time step, when deposition is complete, we get the total energy deposited in each zone by reversing the multiplication: int /S (7) Edep = double Edep

Here, Edep is of type double precision, and double() represents a cast from a 64 bit unsigned integer to double precision.

4 Results Here we demonstrate that the algorithm outlined above will give the same results for simulations using diﬀerent numbers of processors. The results shown are for the analytic transport benchmark of Su and Olson [SO97]. This test problem has an initially cold slab with constant opacity which is heated by an isotropic source of photons. Interactions with the radiation ﬁeld heat the matter and cause it to eventually reach thermal equilibrium. The results depicted in Fig. 3 are for the case κa = 1, κs = 0, at τ = 0.3, in the notation of [SO97].

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0.8 Tmat 1 domain Trad 1 domain Tmat 4 domains Trad 4 domains

0.7

Normalized Temperature

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 Normalized Distance

0.7

0.8

0.9

1

Fig. 3. Matter and radiation temperature in the Su Olson test problem vs. spatial coordinate. Solid lines depict the results of a simulation run on one domain. Symbols depict the results for every tenth zone of a simulation run on four domains. The four domain results are identical to the one domain results, even reproducing the details of statistical ﬂuctuations

Figure 3 shows results for matter and radiation temperature for one and four domain runs. Simulations using one domain are solid lines. Every tenth point for the four domain runs is plotted with a symbol. The four domain results exactly reproduce the one domain results. Even the statistical noise in the Monte Carlo solution is reproduced. As part of the KULL regression suite, this test is run weekly. The results for the temperature in every zone for one, two, and four domain runs are printed to sixteen places, and the ﬁles are compared to ensure that the results are identical. This reproducibility has been demonstrated for larger problems, up to 1024 processors.

5 Conclusions We have described algorithms that can be used to make domain decomposed Monte Carlo photonics code produce the same answer, bit for bit, on various numbers of domains. Two main issues are introduced when the number of domains can vary. The ﬁrst is that the particles may not get the same pseudo-random number

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stream. The second is that the order of operations can change, causing differences in the results arising from the fact that ﬂoating point addition is not commutative. The ﬁrst issue is eliminated with by giving each particle its own random number generator state, and seeding these states in a manner that is independent of the number of domains. The second issue is eliminated by using integers, which are commutative, to do addition. We have demonstrated that we can achieve bit for bit agreement on problems when the number of domains varies from one to one thousand.

Appendix: Shave Algorithm This algorithm takes a double precision number x and truncates all but the highest Nd base-ten digits. We use Nd = 7 in our application. shave( x, Nd ): if x == 0.0: return 0.0 Store the magnitude and sign of x. xsign = sign(x) xabs = abs(x) Truncate base-ten exponent to get magnitude of x. integer n = integer(log10 (xabs )) Scale xabs to be between 10Nd and 10Nd+1 . double s = 10n−Nd double xscaled = xabs /s Shave oﬀ digits by casting to integer. integer xshaved scaled = integer(xscaled ) Restore correct magnitude and sign. return s · xsign · xshaved scaled

References [FC71]

[GKR98]

[BM04]

Fleck, Jr., J. A., Cummings, J. D. : An Implicit Monte Carlo Scheme for Calculating time and frequency dependent nonlinear radiation transport. J. Comp Phys., 31, 313–342 (1971) Gentile, N. A., Keen, N., Rathkopf, J.: The KULL IMC Package, Tech. Rep. UCRL-JC-132743, Lawrence Livermore National Laboratory, Livermore CA (1998) Brunner, T. A., Mehlhorn, T. A.: A User’s Guide to Radiation Transport in ALEGRA-HEDPP, Tech. Rep. SAND-2004-5799, Sandia National Laboratories, Albuquerque, NM (2004)

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[BUEG04] Brunner, T. A., Urbatsch, T. J., Evans, T. M., Gentile, N. A., Comparison of Four Parallel Algorithms for Domain Decomposed Implict Monte Carlo, Tech. Rep. SAND-2004-6694J (2004) Submitted to J. Comp. Phys. [SPRNG] http://sprng.cs.fsu.edu/ [PTVF02] Press, W. Tuekolsky, S.A., Vetterling, W. T., Flannery, B. P.: Numerical Recipies in C++. Cambridge University Press, United Kingdom (2002) [SO97] Su, B., Olson, G. L.: An analytical benchmark for non-equilibrium radiative transfer in an isotropically scattering medium. Ann. Nucl. Energy, 24, No. 13 (1035–1055 (1997)

KM-Method of Iteration Convergence Acceleration for Solving a 2D Time-Dependent Multiple-Group Transport Equation and its Modiﬁcations A.V. Gichuk, L.P. Fedotova, R.M. Shagaliev One of the main diﬃculties during ﬁnite-diﬀerence simulation of spectral 2D problems of particle transport and interaction with medium is to ﬁnd cost-eﬃcient solutions to rather large systems of interconnected diﬀerence equations. The approach based on the method of simple iterations combined with various acceleration algorithms is very often used to solve numerically both single-group and multiple-group diﬀerence transport equations. Note that conceptually many various acceleration methods are close to the method based on introduction of an approximate operator (which is easy-to-use) along with an operator of the original divergent equation. Following this path, a number of eﬃcient methods for solving 2D transport problems have been oﬀered. KM-method is one of them. Finite-diﬀerence approximation using the KM-method has a number of features owing to which it becomes highly eﬀective in practice. First of all, one of such features is a combination of explicit calculation of a collision integral with stability and conservativeness, this provides simultaneous satisfaction of the two balance correlations typical for transport equation: for particle transport and for the number of particles resulted from one collision event. The scheme stability in time has been proved by analytical studies of some partial cases and the experience of using KM-method in practice. The approaches used to construct the KM-method found their further development in MKM- and KM3-methods.

1 Statement of a 2D Transport Problem In the kinetic multiple-group model describing the transport processes a 2D time-dependent transport equation written in the classic divergent form looks like, as follows: N 1 1 ∂εi (0) + Lεi + αi εi = βij εj , i = 1, . . . , N , vi ∂t 2π j=1

(1)

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R

0

Z

Fig. 1. The coordinate system

1 ∂ " 1 ∂ " ∂εi + r 1 − µ2 cos ϕ εi − 1 − µ2 sin ϕ εi . (2) ∂z r ∂r r ∂ϕ 4 5 The equation has to be solved in axially symmetric region D = (r, z) ∈ ,

Lεi = µ

where is the cross section of the body of revolution by a plane coming across Z-axis (Fig. 1). The following notations are used to write equations: r, z are cylindrical coordinates of a particle; Ω (µ, ϕ)–is a unit vector in the particle ﬁght direction;

→

µ = cos (θ) ,

−1 ≤ µ ≤ 1 ;

→

θ is an angle between vector Ω and Z axis of symmetry; → ϕ is an angle between vector Ω projection to the plane coming through point (r, z) at right angle to Z and the vector connecting points (0, z) and (r, z), 0 ≤ ϕ ≤ π; N is the number of energy groups; εi = εi (t, r, z, µ, ϕ) is the mean ﬂux of particles in group “i” in the direction →

determined by vector Ω ; π 1 (0) εi = 0 −1 εi dµdϕ is the particle ﬂux density in group “i”; αi = αi (r, z) is full cross-section in group “i”; βij = βij (r, z) is secondary particle array. System (1) is supplemented with initial and boundary conditions. Finite-diﬀerence approximation to the system of (1) in space and angular variables should meet the following requirements. First of all, it should be

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437

conservative relative to collision-less particle transport with a possibility of consistently approximating the collision term and the right-hand side of the system of group kinetic equations. Secondary, it should be possible to reduce the diﬀerence operator to the triangular form. In case of rectangular spatial grids, DS n -type schemes meet the requirements above. The particular approximation form is insigniﬁcant for further description and, for this reason, we only give the equation written in the form (1), where operator L means a diﬀerence transport operator and function εI and equation factors are the grid values introduced in accordance with the diﬀerence template in use.

2 KM-method The KM scheme is an iterative two-phase scheme. During the ﬁrst phase (predictor) the diﬀerence equation is solved using the sweep (point-to-point) computation algorithm with the known right-hand side. During the second phase (corrector), the solution found is corrected so as to provide satisfaction of balance correlations for particle transport and particle interaction. The values obtained are put into the right-hand sides of the ﬁrst phase equations and the process is repeated up to the desired precision achieved. To approximate the original system of equations in time, write the KMmethod in the following form: Phase I s+1/2

N s s+1/2 s+1/2 −εni 1 1 εn+1 (0) n+γ i + L εn+γ + α ε = β ε i ij j , i i vi ∆t 2π j=1 s+1/2 εn+γ i

=γ

s+1/2 εn+1 + (1 i

(3)

− γ) εni , i = 1, . . . , N .

Phase II s+1/2 N N s s+1 s+1 s+1 1 ∆εn+1 1 (0) (0) n+γ n+γ n+γ i + L ∆εi + αi ∆εi − βij ∆εj = βij εj − εj , vi ∆t 2π j=1 j=1 s+1

s+1

s+1

s+1/2

s+1

s+1

∆εn+γ = εn+γ − εn+γ , ∆εn+γ = γ ∆εn+1 , i = 1, . . . , N . i i i i i (4) Here, γ is weighting parameter of time approximation, n is a time step number, s is an iteration number. In the scheme above, the second phase is written in the corrector form. The (4) also allows a record relative to the mean ﬂux function value, however, such a variant requires storing a large amount of interim data. Adding the (4) to the (3) and summarizing the results for the angular grid directions we obtain that in the grid solution KM-method strictly follows the correlation between the number of collided particles per unit volume

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and the number of particles resultant from such collisions. With regard to the transport operator approximation conservativeness, we obtain that the proposed scheme is balanced both relative to particle transport and relative to particle collisions. Of great importance, from viewpoint of substantiation and application, is the problem of the KM-method stability in time. Such stability can be roved analytically for some partial cases.

3 MKM-method In spite of some evident advantages of the KM method, it also has certain disadvantages, namely, there is a signiﬁcant increase in the number of iterations when numerically solving 2D problems with grid fragmentation in optically dense physical regions. One of the main assumptions made for the (4) of the corrector phase is the approximate substitution: N j=1

s+1 (0)

βij ∆εj

≈ 2π

N

s+1

βij ∆εj ,

(5)

j=1

that is valid (to a satisfactory precision) for processes with isotropic or almost isotropic distribution of particles. In essentially anisotropic cases, the (5) has to be speciﬁed. The implications above caused the development of the MKMmethod. This method also has two phases, the ﬁrst of them (predictor) coincides with (3). The diﬀerence between the two methods is in the corrector phase, →

→−

where two opposite directions of particle ﬂight ( Ω and Ω ) are considered simultaneously. The second phase equations are: s+1 s+1 s+1 1 ∆εn+1 i + L ∆εn+γ + αi ∆εn+γ i i vi ∆t

⎛ ⎞ s+1/2 N N s s+1 1 1 (0) (0) (0) = βij ∆εj + βij ⎝ εj − εj ⎠ , 2π j=1 2π j=1

(6)

s+1

s+1 s+1 1 ∆ε−n+1 i + L− ∆ε−n+γ + αi ∆ε−n+γ i i vi ∆t ⎛ ⎞ s+1/2 N N s s+1 1 1 (0) (0) (0) = βij ∆εj + βij ⎝ εj − εj ⎠ . 2π j=1 2π j=1 →

(7) →−

Here, corrections ∆ε and ∆ε correspond to directions Ω and Ω , the same is true for diﬀerence transport operators. The mean ﬂux of particles

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in the (7) can be expressed via the right-hand side and the incoming ﬂuxes through the illuminated faces of a phase space cell. Set the incoming ﬂuxes equal to zero for the correction function (corrector), this corresponds to the assumption that ﬂuxes found during the ﬁrst half-step are used in this phase. Using the approximate substitution (0)

∆εj

≈ π ∆εj + ∆ε− j

(8)

from the (6) and (7) and equating the right-hand sides give us the correlation factors ∆ε− i = Ai ∆εi + Bi , the resultant formula (8) now looks like (0)

∆εj

≈ π ((Aj + 1) ∆εj + Bj ) .

(9)

Substitution (9) in the (6) gives us the ﬁnal form of the second phase (corrector) of the MKM-method: s+1

s+1 s+1 s+1 1 ∆εn+1 1 i + L ∆εn+γ + αi ∆εn+γ − βij (Aj + 1) ∆εn+γ i i j vi ∆t 2 j=1 ⎛ ⎞ s+1/2 N N s 1 1 (0) (0) = βij Bj + βij ⎝ εj − εj ⎠ , 2 j=1 2π j=1 N

(10)

where Aj and Bj can be found by equating the left-hand sides of the (6) and (7). The values from the previous iteration are used in the oﬀered version of → the MKM-method to obtain the expression (9) for direction Ω , so the method becomes non-conservative. There is a way to bypass the problem by solving the (6) and (7) using the sweep method, however, this way leads to signiﬁcant complication the computational algorithm.

4 KM3-method As it was mentioned above, the practice of using the KM-method shows a signiﬁcant increase of the number of iterations during numerical solution of some classes of problems. For this reason, it becomes necessary to decrease (subdivide) time steps in order to solve time-depend problems. For the KM-method eﬃciency enhancement and further extension of the range of multidimensional problems that can be solved using this method, its new version called “KM3-method” is oﬀered in the paper. The main features of the standard KM-method are preserved in the new scheme construction.

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The KM3-method’s speciﬁc feature is that it allows solving a three-phase system of equations that provides signiﬁcant enhancement of its cost-eﬃciency and less dependence of the number of iterations on the time step value. Similar to KM- and MKM-methods, KM3-method is constructed using the predictor-corrector scheme. The (3) is solved in the ﬁrst phase. The second phase (corrector) has a more complicated structure, it includes two equations in which the internal sub-iteration cycle is arranged, namely: s+1,ν+1/2 s+1,ν+1/2

s+1,ν+1/2

1 ∆εn+1 i + L ∆εn+γ i vi ∆t

+ αi ∆εn+γ −l i

N

s+1,ν+1/2

βij ∆εn+γ j

j=1

⎛

⎞ s+1/2 s+1,ν N N s 1 1 (0) (0) (0) = (1 − l) βij ∆εj + βij ⎝ εj − εj ⎠ , 2π j=1 2π j=1 s+1/2

s+1,0

∆εn+γ = ∆εn+γ , i i

(11)

s+1,ν+1

N s+1,ν+1 s+1,ν+1 s+1,ν+1 1 ∆2 εn+1 2 n+γ 2 n+γ i + L ∆2 εn+γ + α ∆ ε − β ∆ εj i ij i i vi ∆t j=1

= (1 − l) s+1,0

s+1/2

s+1,ν+1/2 s+1,ν N N 1 1 (0) (0) βij ∆εj − (1 − l) βij ∆εj , 2π j=1 2π j=1 s+1

s+1/2

s+1,ν+1/2

∆εi = ∆εi , εi = εi +

∆εi

s+1,ν+1

+ ∆2 εi .

(12)

Here, ν is a sub-iteration number with respect to the corrector phase steps, l is parameter that may be dependent on the space grid’s cell geometry and the energy group (generalizations to this approach are also possible, where the angular dependence is assumed as well). If l = const, l ∈ (0; 1). Iterations in ν can be performed either up to the speciﬁed precision achievement (∼10+ 20%), or a certain number of times (∼3–5).

5 Test Computation Results All the methods described above have been veriﬁed for radiation transport problems. The corresponding transport equation system can be easily reduced to the form (1) as a result of Planck function linearization with respect to temperature using the energy equation. KM-method To demonstrate the method eﬃciency, the parameters of multiple-group Fleck problem computations are given below. The problem geometry is shown in Fig. 2.

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R 104 16 x 3 102.4 102

16 x 3

20 x 3

100 T=1 keV 0

1

Z

Fig. 2. The spectral Fleck problem geometry

The material density in the system is ρ = 1, the material energy dependence on temperature is described by equation E = T . Cross-sections: χsi = 0 is scattering cross-sections; i /T )) (optically transparent regions) and χai = 27(1−exp(−ω ω3 i

i /T )) χai = 10000(1−exp(−ω (optically dense region) are absorption crossωi3 sections.

The boundary condition “mirror reﬂection” is speciﬁed on the left {Z = 0; 100 ≤ R ≤ 104} and the right {Z = 1; 100 ≤ R ≤ 104} ends. The angular grid is S8 quadrature. The mean numbers of iterations in computations with various time steps during the problem solution using the method of source iterations and the KM-method are given in Table 1. The radiation temperature distribution for ct = 30 is shown in Fig. 3. Table 1. The mean number of iterations per 100 steps c∆t, cm

Source Iterations

KM-method

0.03 0.15 0.30

166.92 1907.53 3961.35

11.91 24.82 40.48

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Fig. 3. The radiation temperature distribution, ct = 30 cm

R Region 2 (10 100)

1.2 1

Region 1 (10 100)

0

10

Z

Fig. 4. The geometry of the test problem with tube

MKM-method The method was veriﬁed using the axially symmetric two-region one-group problem of a tube which geometry is shown in Fig. 4. The material density in the system is ρ = 1, the material energy dependence on temperature is described by equation E = 0.81 T . Photon absorption is taken into account in computations, with the photon absorption cross-section being speciﬁed by the formula: χa = A/T 3 . The optically dense (region 1) and the optically transparent (region 2) regions are speciﬁed by various values of A factor. A = 50.89 in region 1 (optically dense). A = 0.1374 in region 2 (optically transparent). The absorption cross-section was set equal to zero in computations (χs = 0). The following boundary conditions with respect to radiation are speciﬁed on the left end of the rectangular region {Z = 0; 1 ≤ R ≤ 1.2}: “mirror reﬂection” for the boundary section belonging to the dense casing and the incoming isotropic radiation ﬂux corresponding to temperature T = 1 for the boundary section belonging to the transparent region. The incoming radiation ﬂux was set equal to zero on the upper boundary and the right end. The results of computations using the standard KM-method and the MKM-method are compared (see Table 2).

KM-Method of Iteration Convergence Acceleration

443

Table 2. The number of iterations in regions Number of Iterations c∆t = 0.3 cm Step No. 5 10 15 20 25 30 35 40 45 50

KM 13 12 11 15 10 8 8 8 8 7

23 26 26 25 25 26 29 31 29 22

c∆t = 1.5 cm

MKM 13 12 11 11 10 10 9 8 8 8

25 26 27 26 27 25 24 24 24 23

KM 31 26 23 14 12 12 12 11 11 11

100 100 93 89 85 84 77 74 70 66

c∆t = 3 cm

MKM 36 31 27 24 21 18 15 12 11 9

93 100 94 83 77 73 69 64 60 55

KM 49 30 18 13 11 10 8 7 6 5

MKM

100 100 100 100 100 100 84 73 55 49

61 51 41 30 20 14 10 7 6 5

100 100 100 100 100 91 79 67 58 46

Table 3. The Number of Iterations in Regions c∆t = 0.3 cm

c∆t = 1.5 cm

c∆t = 3 cm

Step No.

SI

KM

KM3

SI

KM

KM3

SI

KM

KM3

1 5 10

19108 3332 1559

14 24 27

14 20 20

28696 6363 8477

26 121 101

26 88 75

26454 7960 9146

35 238 179

35 174 135

KM3-method To demonstrate the method serviceability, the table below (Table 3) gives the tube problem computation parameters for computations using the method of source iterations (SI), KM- and KM3-methods. The test problem description is given above.

A Regularized Boltzmann Scattering Operator for Highly Forward Peaked Scattering Anil K. Prinja1 and Brian C. Franke2 1

2

University of New Mexico, Chemical and Nuclear Engineering Department, Albuquerque, NM 87131, USA [email protected] Sandia National Laboratories, P.O. Box 5800, MS 1179, Albuquerque, NM 87185 [email protected]

1 Introduction Extremely short collision mean free paths and near-singular elastic and inelastic diﬀerential cross sections (DCS) make analog Monte Carlo and deterministic computational approaches impractical for charged particle transport. The widely used alternative, the condensed history method, while eﬃcient, also suﬀers from several limitations arising from the use of precomputed inﬁnite medium distributions for sampling particle directions and energies. Accordingly, considerable attention has recently focused on the development of computationally eﬃcient algorithms that implement the correct transport mechanics. Fokker-Planck [JEM81] and Boltzmann FokkerPlanck [CL83] approximations have historically proved very useful in handling highly peaked scattering in certain classes of problems but these approaches are limited in the accuracy they can ultimately deliver. A more general methodology that allows accuracy to be systematically increased with practically no enhancement of algorithmic complexity has become possible with the advent of recently proposed higher order Fokker-Planck expansions [GCP96] and their implementation in so-called Generalized FokkerPlanck models [LL01,PP01,PKH02]. The goal of these newer approaches is to approximate the analog transport problem by one which is characterized by longer or stretched mean free paths and nonsingular collision operators but which can be solved numerically with considerably less eﬀort than the analog problem and whose accuracy and eﬃciency can be readily adapted to a broad class of problems. One such implementation that has proved particularly efﬁcient uses purely discrete scattering angle and hybrid discrete-continuous scattering angle representations [FPKL1, FPKL2]. Moreover, generalizations of these methodologies to describe energy-loss straggling have been successfully demonstrated [PKH02]. Here we describe an angular moment-preserving procedure to regularize the near-singular collision integral for electron elastic scattering by

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approximating and then re-summing what we refer to as a generalized Fermi expansion. It’s diﬀerence from the generalized Fokker-Planck expansion of Leakeas and Larsen [LL01] will become apparent in the ensuing but we note here that the advantage of using the formulation developed below is that the resulting renormalized scattering kernel has a simple and explicit form which lends itself to a convenient implementation in Monte Carlo simulations. Furthermore, unlike our recent discrete scattering angle formulation [FPKL1,FPKL2], the present approach is continuous in scattering angle and consequently is free of ray artifacts. The generalized Fermi expansion is developed in the next section, followed by a presentation of the regularization procedure. Numerical results are then presented and the paper closes with some concluding remarks.

2 Generalized Fermi Expansion Our approach consists of a modiﬁed use of earlier proposed higher-order Fokker-Planck expansions to describe scattering that is strongly anisotropic but not suﬃciently forward peaked that a strictly Fokker-Planck approximation can be justiﬁed [GCP96, LL01, PP01]. We note that this is invariably the situation for realistic electron scattering interactions such as described by the screened Rutherford cross section and its variants, but which nevertheless are near-singular at zero deﬂection. As will become apparent shortly, our implementation amounts to approximating the angular diﬀusion FokkerPlanck operator, which is just the Laplacian on the unit sphere, by a planar Laplacian in this higher order expansion and it generalizes a result ﬁrst due to Fermi [RG41]. We begin by writing the analog transport equation for the angular ﬂux ψ(r, Ω) of electrons at spatial position r along direction Ω as Ω · ∇ψ(r, Ω) = J[ψ],

ψ(rs , Ω) = δ(1 − Ω0 · Ω), n · Ω < 0 ,

(1)

where n is the outward directed unit normal at surface point rs and Ω0 is the incident beam direction. Also in (1), J[ψ], a linear functional of the angular ﬂux, is the elastic scattering collision integral, and is given by: σs (Ω · Ω) ψ(Ω )dΩ − σs0 ψ(Ω) , (2) J[ψ] = 4π

where σs (Ω · Ω) is the diﬀerential elastic scattering cross section (DCS), µ0 = Ω · Ω is the cosine of the scattering angle, and σs0 the corresponding inverse scattering mean free path. Energy losses are neglected in the present investigation but it has been demonstrated before that singular inelastic energy-loss operators can also be regularized [PKH02]. Since charged particle interactions are mediated by long range Coulomb forces, the DCS σs (Ω · Ω) falls very sharply away from Ω · Ω = 1, decreasing

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in a continuous manner by several orders of magnitude over a scattering angle range of only a few degrees. This has the consequence that an initially peaked distribution will diﬀuse slowly in angle with increasing depth into the material and it enables us to introduce two simplifying approximations. First, we write Ω · Ω ≈ 1 −

1 (η − η )2 + (ξ − ξ )2 , 2

(3)

where η and ξ are the x and y direction cosine components of Ω. Second, we extend the domain of η and ξ to the open interval (−∞, ∞), a reasonable step since the angular ﬂux decays rapidly away from the initial direction. Using these results in (2) gives after some rearrangement ∞ ∞ σ ˆs (η 2 + ξ 2 ) ψ(η − η , ξ − ξ )dη dξ − σs0 ψ(η, ξ) , (4) J[ψ] = −∞

−∞

where σ ˆs (η 2 + ξ 2 ) ≡ σs [1 − (η 2 + ξ 2 )/2]. Our goal is to represent (4) as a diﬀerential expansion and to that end we introduce the two dimensional Fourier transform pair ∞ ∞ ¯ l) = 1 (5) e−iK·α ψ(η, ξ)dηdξ , ψ(k, 2π −∞ −∞ ∞ ∞ 1 ¯ l)dkdl , ψ(η, ξ) = eiK·α ψ(k, (6) 2π −∞ −∞ where α(η, ξ) and K(k, l) are angle and transform vectors, respectively, and K · α = (kη + lξ). Noting the convolution structure of (4), the latter readily transforms to ¯ l) , (7) J¯ = σ ¯ (K 2 ) − σs0 ψ(k, where K 2 = K · K. Here and in the ensuing, appropriate behaviour of ψ and its derivatives is assumed in order to justify the use of integral transforms. We further ﬁnd it useful to express the Fourier transform of the diﬀerential cross section σ ¯ (K 2 ) as a Hankel transform, namely ∞ dλ λJ0 (Kλ)ˆ σs (λ2 ) , (8) σ ¯ (K 2 ) = 0

where J0 (·) is the zeroth order Bessel function of the ﬁrst kind. Proceeding, ¯ as an exthe inﬁnite series representation of J0 is used in (8) to express σ pansion in K 2 and the result inserted into (7). We next take a formal inverse Fourier transform of this equation to get, after some considerable algebra, the desired diﬀerential representation of the collision integral J[ψ] =

∞ n=1

an σn Ln ψ(η, ξ) = a 1 σ1 Lψ + a2 σ2 L2 ψ + O(a3 σ3 ) .

(9)

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In the above expression, the an are positive numerical coeﬃcients, L is the 2D planar Laplacian deﬁned by L≡

∂2 ∂2 + 2 , 2 ∂η ∂ξ

(10)

and we have introduced momentum transfer moments deﬁned by 1 ∞ σn = 2π (1 − µ0 )n σs (µ0 ) dµ0 = 2−n 2π dλ λ2n+1 σ ˆs (λ2 ) . −1

(11)

0

We note in passing that (9), with L replaced by the Laplacian on the unit sphere, is related, but not identical, to the higher order Fokker-Planck expansion introduced by Pomraning [GCP96] and subsequently used by Leakeas and Larsen [LL01] to develop a generalized Fokker-Planck approximation for the scattering integral. If scattering is suﬃciently forward peaked, the σn will form a rapidly decreasing sequence and truncation of the expansion in (9) is suggested. It is especially interesting to consider the approximation to the collision integral that results from retaining only the ﬁrst term in this sequence, namely σtr ∂ 2 ψ ∂ 2 ψ + , (12) J[ψ] ≈ 2 ∂η 2 ∂ξ 2 where σtr ≡ σ1 is the familiar transport cross section. If, additionally, a thin target is assumed, throughout which the angular distribution remains nearly collimated, the resulting transport equation may be simpliﬁed to read ∂ψ ∂ψ σtr ∂ 2 ψ ∂ 2 ψ ∂ψ +η +ξ = + , (13) ∂z ∂x ∂y 2 ∂η 2 ∂ξ 2 with boundary condition ψ(x, y, 0, η, ξ) = δ(x)δ(y)δ(η)δ(ξ) .

(14)

This is the celebrated Fermi model, originally derived on physical grounds in the context of cosmic ray penetration of the atmosphere [RG41] and which has the following closed form joint Gaussian solution 3 ψ(x, y, z, η, ξ) = 2 2 4 π σtr z ! 3(x2 + y 2 ) 3(xη + yξ) (η 2 + ξ 2 ) 2 × exp − − + . σtr z3 z2 z

(15)

An energy dependent generalization of this result in the continuous slowing down approximation forms the basis of Fermi-Eyges theory that is widely used but particularly so in electron dose calculations in radiotherapy [HMA81]. The simplicity of this solution is undoubtedly a very attractive

Regularized Scattering Operator

449

feature, but its accuracy is strictly limited to a small region around the pencil beam axis. The error increases dramatically with increasing lateral distance away from the axis and this deﬁciency in the solution can be traced to the neglect of large angle scattering which is manifested through the neglect of all momentum transfer moments higher than the ﬁrst. While this conclusion suggests that greater accuracy may be achieved with the retention of additional terms in the expansion, more detailed theoretical analysis shows that in fact all truncations beyond the ﬁrst term lead to unstable or unbounded solutions, in then sense of transforms at least. A similar conclusion was reached with the use of higher order Fokker-Planck expansions [LL01, PP01]. The infeasibility of adopting higher order truncations eliminates the possibility of systematically obtaining closed form generalizations of the Fermi solution (however, see [PP98]). It is necessary to then consider approximations which require numerical solution methods but prove more eﬃcient to solve than the analog problem. The considerations of this section are still very relevant, however, for the equivalence of the Boltzmann and diﬀerential forms of the collision operator, namely (2) and (9), strongly suggests that approximate models which correctly reproduce a certain number of the cross section angular moments deﬁned in (11) are likely to be more accurate than those that do not. This principle underlies the moment-preserving approach described in the next section.

3 Regularized Collision Operator Consider the following functional of the angular ﬂux G[ψ] = A exp(βL) ψ − Aψ ,

(16)

where L is the 2D planar Laplacian deﬁned above and A and β are unspeciﬁed constants. Expanding the exponential up to third order yields G[ψ] = A β Lψ +

A β2 2 L ψ + O(A β 3 ) . 2

(17)

Comparing (9) and (17), it follows that the J- and G-functionals can be made formally equivalent through second order if A and β are chosen to satisfy A=2

σ12 , σ2

β=

1 σ2 , 4 σ1

(18)

so that we can write J = G + O(σ22 /32σ1 + σ3 /288) .

(19)

Thus, the two formulations of the scattering integral give identical ﬁrst and second momentum transfer moments. The higher order terms will diﬀer, of

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A.K. Prinja and B.C. Franke

course, but if scattering is suﬃciently forward peaked (σ1 σ2 σ3 · · · ) the G-functional should represent an accurate approximation to the analog scattering integral. In particular, under circumstances where the Fokker-Planck approximation is accurate, the G-functional should also be accurate. Moreover, because the latter retains all higher order moments, although approximately, it has the potential of being more accurate than the Fokker-Planck approximation. We now demonstrate the signiﬁcance of the G-functional formulation. Consider the quantity χ(s, η, ξ) = exp(s L) ψ(η, ξ) ,

(20)

where s is a continuous non-negative real variable. With this deﬁnition, (16) can alternatively be written as G[ψ] = Aχ(β, η, ξ) − Aψ .

(21)

Now, diﬀerentiating (20) with respect to s it is not diﬃcult to show that χ satisﬁes the following “time dependent, inﬁnite medium heat conduction” equation ∂2χ ∂2χ ∂χ = + 2 , s ≥ 0, −∞ < η, ξ < ∞ , (22) ∂s ∂η 2 ∂ξ with auxiliary data χ(0, η, ξ) = ψ(η, ξ),

lim

|η|,|ξ|→∞

χ(s, η, ξ) = 0 .

(23)

This initial-boundary value problem can be readily solved to obtain ∞ ∞ (η − η )2 + (ξ − ξ )2 1 ψ(η , ξ )dη dξ . χ(s, η, ξ) = exp − 4πs −∞ −∞ 4s (24) Finally, setting s = β in (24) and inserting the result into (21) yields the following alternate representation of the G-functional ∞ ∞ (η − η )2 + (ξ − ξ )2 A ψ(η , ξ )dη dξ exp − J[ψ] ≈ G[ψ] = 4πβ −∞ −∞ 4β − A ψ(η, ξ) .

(25)

For obvious reasons, we refer to the above representation of the scattering integral as a moment-preserving approximation. It can form the basis for generalizing Fermi-Eyges theory to account for the eﬀects of large angle scattering on electron dose deposition, and we hope to report on this in a future communication. Before we comment further on this result, however, we ﬁrst note that (25) may be expressed in a form that is more practical for numerical implementation by restoring the domain of the direction variable to the unit sphere. This immediately follows upon using (3), getting

Regularized Scattering Operator

J[ψ] =

A 4πβ

4π

(1 − µ0 ) exp − ψ(Ω )dΩ − A ψ(Ω) . 2β

451

(26)

It should be noted that in so restricting the domain an error of O(exp(−1/β)) is introduced in the cross section moments. Although the coeﬃcients can be rescaled to restore the exact moments, this is not necessary since β is typically a small number. That is, the error is exponentially small and has no practical consequence. There are a number of attractive features about the approximate collision integral obtained above and its generalization described below. First, we observe that the scattering operator has been regularized, i.e., the singularity at forward directions has been eliminated, and there is a concomitant reduction in the magnitude of the total scattering cross section. This smoothed approximation of the transport process greatly improves prospects for application of standard deterministic and Monte Carlo solution techniques. Also, the direct use of existing neutral particle transport codes becomes a distinct possibility since the algorithmic modiﬁcations necessary when diﬀerential Fokker-Planck approximations are employed are not required with the present formulation. Second, the approach presented here yields a univeral diﬀerential cross section that depends only on cross section angular moments. That is, the ﬁnal result is independent of the detailed form of the underlying analog cross section, making it feasible to handle empirical or semi-empirical cross section represenations with relative ease. Finally, we note that sampling deﬂection angles from the exponential kernel in a Monte Carlo implementation is straightforward. This is in contrast to the inﬁnite spherical harmonics expansion representation that results from the use of a generalized Fokker-Planck expansion [LL01]. Our approach can be readily extended to preserve higher moments of the analog DCS by generalizing (16) as follows N N An exp(βn L) ψ − An ψ . (27) G[ψ] = n=1

n=1

The 2N free parameters {An , βn , n = 1, 2 . . . N } can be selected by requiring the consolidated generalized Fermi expansion of (27) to be identical to that of the analog expansion up to the appropriate order, thereby ensuring that the ﬁrst 2N momentum transfer moments of the DCS are preserved. This procedure will yield a nonlinear system of algebraic equations for the expansion parameters which in general must be solved numerically. Retaining higher order terms enables larger scattering angle eﬀects to be accommodated although experience with other moment preserving strategies [FPKL1,FPKL2] indicates that in practice two terms (four moments) is usually suﬃcient to accurately capture angular distributions even for relatively thin targets. However, a more eﬀective way of incorporating large scattering angle eﬀects is to ﬁrst extract from the analog diﬀerential cross section a component that is

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not forward peaked and is characterized by a long mean free path. By further requiring the shape and amplitude of this smooth cross section to be identical to or closely resemble the analog cross section, high order moments can be accurately reproduced and large angle scattering eﬀectively captured. The residual component will carry the singular part of the cross section and is more suitable for the regularization procedure described above. We have previously implemented this hybrid procedure with a discrete scattering angle representation of the residual cross section and observed the approach to be very accurate indeed [FPKL2].

4 Numerical Results Monte Carlo simulations have been performed for 1 MeV electrons incident on a gold target of thickness T = 3.85 × 10−3 cm [FPKL2], and we compare transmitted and reﬂected angular distributions obtained using our new regularized model, in pure and hybridized form, and our previous discrete scattering angle representation [FPKL1] against corresponding benchmark analog results. The screened Rutherford scattering cross section [FPKL1] was used for sampling angular deﬂections in the analog model and for computing the necessary momentum transfer moments. Energy dependence was neglected in these simulations so that the electrons were either transmitted or reﬂected from the inﬁnite slab. In Fig. 1, the relative diﬀerences with analog results are shown for the 1and 2-discrete angle model and the exponential kernel model. The absence of ray artifacts with our regularized model stands in stark contrast to the discrete models. The exponential model is less accurate than the 2-discrete angle model near grazing angles for reﬂection, but this can partly be attributed to the discrete model preserving two additional moments. Because of the greater expense of sampling from an exponential distribution, our new model is only slightly faster than the 2-discrete angle method. The discrete model shows ray eﬀects in the reﬂected angular distribution due to the presence of a discrete scattering angle in the backward direction while the exponential model yields a smooth distribution across all angles, as expected. Both methods perform well in calculating the transmitted distribution, with the discrete method performing better at large angles relative to the surface normal. The 2 discrete angle method accurately captures four angular scattering moments, while the exponential model accurately captures only two moments. Because of the relative size of the interaction cross section and the greater expense of sampling from an exponential distribution, the exponential model is only slightly faster than the 2 angle method. In Fig. 2, the relative diﬀerences with analog results are shown for the exponential model and the hybrid model. The hybrid model shows greatly improved accuracy for both transmitted and reﬂected distributions, a consequence of more accurately treating the higher order momentum transfer

Regularized Scattering Operator

453

(ψAnalog - ψApprox) / ψAnalog

0.1

0

-0.1

-0.2

0

1 Discrete Angle 2 Discrete Angles Exponential 30

60

90

Reflection Angle (Degrees)

(ψAnalog - ψApprox) / ψAnalog

0.2 0.1 0

-0.1 -0.2 1 Discrete Angle 2 Discrete Angles Exponential

-0.3 -0.4 -0.5 0

30

60

90

Transmission Angle (Degrees) Fig. 1. Error in Angular Escape Distributions Relative to Outward Normal for Electrons Incident on a Gold Slab for Discrete and Exponential Kernels

moments. However, this improved accuracy is achieved at a slightly increased computational expense, with the runtime for the hybrid model approximately two times greater than for the pure exponential model.

Conclusions We have demonstrated that a closed form regularized scattering kernel for electron transport, derived using an angular moment-preserving technique, yields accurate escape distributions. Unlike our previous discrete scattering

454

A.K. Prinja and B.C. Franke 0.1

(ψAnalog - ψApprox) / ψAnalog

Exponential Hybrid 0

-0.1

-0.2

0

30

60

90

Reflection Angle (Degrees) 0.1

(ψAnalog - ψApprox) / ψAnalog

Exponential Hybrid 0

-0.1

-0.2

0

30

60

90

Transmission Angle (Degrees) Fig. 2. Error in Angular Escape Distributions Relative to Outward Normal for Electrons Incident on a Gold Slab for Exponential Kernels

angle representation, the present model is free of ray artifacts yet it is computationally competitive. Moreover, the procedure described herein can be readily and systematically generalized to incorporate more accurate physics, withough increasing algorithmic complexity, by retaining successively higher order angular moments of the analog diﬀerential cross section. Finally, numerical testing shows that the best performance, from accuracy and computational eﬃciency considerations, is realized when the cross section is ﬁrst decomposed into smooth, analog-like and singular components and the regularization procedure applied to the latter.

Regularized Scattering Operator

455

Acknowledgement Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000. The work of the ﬁrst author (AKP) was supported in part by a grant from the US Department of Energy under the award DE-FG07-02ID14336.

References [JEM81] Morel, J.E.: Fokker-Planck Calculations Using Standard Discrete Ordinates Codes. Nucl. Sci. Eng., 79, 340 (1981) [CL83] Caro, M., Ligou, J.: Treatment of Scattering Anisotropy of Neutrons Through the Boltzmann Fokker-Planck Equation. Nucl. Sci. Eng., 83, 242 (1983) [GCP96] Pomraning, G.C.: Higher Order Fokker-Planck Operators. Nucl. Sci. Eng., 124, 390 (1996) [LL01] Leakeas, C.L., Larsen, E.W.: Generalized Fokker-Planck Approximations of Particle Transport with Highly Forward-Peaked Scattering. Nucl. Sci. Eng., 137, 236 (2001) [PP01] Prinja, A.K., Pomraning, G.C.: A Generalized Fokker-Planck Model for Transport of Collimated Beams. Nucl. Sci. Eng., 137, 227 (2001) [PKH02] Prinja, A.K., Klein V.M., Hughes, H.G.: Moment Based Eﬀective Transport Equations for Energy Straggling. Trans. Am. Nucl. Soc., 86, 204 (2002) [FPKL1] Franke, B.C., Prinja, A.K., Kensek, R.P., Lorence, L.J.: Discrete Scattering-Angle Model for Electron Pencil Beam Transport. Trans. Am. Nucl. Soc., 86, 206 (2002) [FPKL2] Franke, B.C., Prinja, A.K., Kensek, R.P., Lorence L.J.: Ray Eﬀect Mitigation for Electron Transport with Discrete Scattering-Angles. Trans. Am. Nucl. Soc., 87, 133 (2002) [RG41] Rossi, B., Greisen, K.: Cosmic Ray Theory. Rev. Modern Phys., 13, 240 (1941) [PP98] Pomraning, G.C., Prinja, A.K.: The Pencil Beam Problem for Screened Rutherford Scattering. Nucl. Sci. Eng., 130, 1 (1998) [HMA81] Hogstrom, K.R., Mills, M.D., Almond, P.R.: Electron Beam Dose Calculations. Phys. Med. Biol., 26, 445 (1981)

Implicit Riemann Solvers for the Pn Equations Ryan McClarren1 , James Paul Holloway1 , Thomas Brunner2 , and Thomas Mehlhorn2 1

2

Department of Nuclear Engineering and Radiological Sciences, College of Engineering, University of Michigan, 2355 Bonisteel Boulevard, Ann Arbor, MI 48109 2104, USA [email protected] Sandia National Laboratories∗∗ , PO Box 5800, Albuquerque, NM 87185-1186

Summary. The spherical harmonics (Pn ) approximation to the transport equation for time dependent problems has previously been treated using Riemann solvers and explicit time integration. Here we present an implicit time integration method for the Pn equations using Riemann solvers. Both ﬁrst-order and high-resolution spatial discretization schemes are detailed. One facet of the high-resolution scheme is that a system of nonlinear equations must be solved at each time step. This nonlinearity is the result of slope reconstruction techniques necessary to avoid the introduction of artiﬁcal extrema in the numerical solution. Results are presented that show auspicious agreement with analytical solutions using time steps well beyond the CFL limit.

1 Introduction Time dependent radiation transport is an important aspect in the problems of radiation hydrodynamics and subcritical accelerator-driven nuclear systems. In both of these problems small time scales cause transport eﬀects to be important in resolving macroscopic phenomena. In this article we present the ﬁrst implicit, Roe-type Riemann solver for the spherical harmonics (Pn ) approximation to the linear transport equation. In the past explicit [BH2001a, BH2001b, Bru2000] and steady state [EPO2003] time integration methods were implemented and shown to be eﬀective in capturing the propagation of sharp wavefronts. The most alluring characteristic of Riemann solvers is the fact that they respect the characteristics of hyperbolic equations by allowing information to travel only in the directions allowed by the model. For ﬁrst-order spatial accuracy, the Riemann solver is linear for the Pn equations, however, higher order methods must use nonlinear slope reconstruction. The implicit method we present uses the backward Euler method and BDF2 for time integration. For the nonlinear equations of the high resolution scheme a preconditioned inexact Newton method is used. The computational results obtained show that it is not necessary to resolve ∗∗

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000.

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the propagation of information across a spatial cell to produce numerical results tantamount to analytic solutions, that is, the CFL limit can be violated and eﬀective results still ensue.

2 Pn Equations The one-dimensional spherical harmonic approximation of order n to the linear Boltzmann equation is given by 1 ∂ ∂ ψ (x, t) + Q(x, t) . ψ (x, t) + A ψ (x, t) = −Sψ c ∂t ∂x

(1)

In the above, c is the particle speed, A is an n × n matrix, ψ is a vector of the moments of the angular ﬂux, Q(x, t) is the inhomogeneous source term, and S is a diagonal matrix that includes the collisional sources/sinks (namely scattering and absorption). These equations are hyperbolic, meaning that information propagates at ﬁnite speeds and that A has all real eigenvalues. We seek to cast this equation in terms of cell averaged quantities. First we prescribe a set of points, I = 1, 2, . . . , i, . . . , N , where each i denotes a spatial cell in our domain. Without loss of generality we assert that the spatial extent of a cell is constant, ∆x, such that ∆x = X/N , where X is the length of our spatial region of interest. Also, we shall designate the left and right edges of a general spatial cell, i, by i − 1/2, and i + 1/2, respectively. Over a general cell, (1) is integrated and divided by ∆x (i.e. spatially averaged). This yields 1 1 ∂ ψ i + Qi (t) , ψi + Fi+1/2 − Fi+1/2 = −Sψ c ∂t ∆x

(2)

where we have deﬁned Qi (t) =

1 ∆x

1 ψ i (t) = ∆x and

(i+1/2)∆x

Q(x, t)dx ,

(3)

ψ (x, t)dx ,

(4)

(i−1/2)∆x

(i+1/2)∆x

(i−1/2)∆x

ψ [i ± 1/2]∆x, t . Fi±1/2 = Aψ

(5)

The Fi±1/2 terms are called the “ﬂuxes” across a cell interface. In the most facile sense this is the ﬂow of information between cells. From (2) we see an immediate issue, the quantities Fi±1/2 are not known in terms of cell-average quantities. To express the F’s in these terms (and hence give us a numerical method) we shall examine the solution to a model problem that exactly gives the ﬂux between two cells–the Riemann problem.

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459

3 Solving the Riemann Problem The Riemann problem is to ﬁnd the ﬂux across a boundary, where the ﬂux is given by a matrix, R, times a vector of state variables, u; that is to say F = Ru. The state variables are governed by the equation ∂u ∂u +R =0, ∂t ∂x with the initial condition

u(x, 0) =

(6)

ul x < 0 , ur x > 0

(7)

where ul , ur are vectors of constants. Given the vectors lk , rk , the k th left and right eigenvectors of R, and λk , the k th eigenvalue of R, multiply (6) by lk ∂u ∂u + lk · R =0 ∂t ∂x ∂ak ∂u + λk l k · =0 ∂t ∂x ∂ak ∂ak + λk =0, ∂t ∂x

lk ·

(8)

where ak = lk · u. Similarly the initial conditions are multiplied by lk u ˆl,k x < 0 ak (x, 0) = , u ˆr,k x > 0 in the above u ˆl,k = lk · ul . The solution to (8) subject to IC (9) is u ˆl,k (x − λk t) x − λk t < 0 ak (x, t) = . u ˆr,k (x − λk t) x − λk t > 0

(9)

(10)

Next, the solution to (6) is reconstructed from the characteristic solutions (10) ak (x, t)rk = u ˆl,k (x − λk t)rk + u ˆr,k (x − λk t)rk . (11) u= k

x−λk t<0

At x = 0 u(0, t) =

x−λk t>0

u ˆl,k (−λk t)rk +

λk >0

The ﬂux at x = 0 is F(0, t) = Ru(0, t) = =

λk >0

u ˆr,k (−λk t)rk .

(12)

λk t<0

λk u ˆl,k rk +

λk >0

λk lk · ul rk +

λk <0

λk u ˆr,k rk

λk t<0

λk lk · ur rk .

(13)

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This ﬂux can be transformed to the standard form of the Riemann solver community by adding and subtracting λk >0 λk lk · (ur ) rk to (13) F(0, t) =

λk lk · ul rk +

λk >0

+

λk lk · ur rk

λk <0

λk lk · ur rk −

λk >0

λk lk · ur rk .

(14)

λk >0

Now looking at each term λk lk · ul rk − λk lk · ui rk = λk lk · [ul − ui ] rk λk >0

=−

λk >0

λk >0

λk lk · [ui − ul ] rk = −

λk >0

λk lk · [∆u] rk ,

(15)

λk >0

where ∆u = ur − ul . The other terms λk lk · ur rk + λk lk · ur rk = λk lk · ur rk λk <0

=

λk >0

λk

rk λk lk · ur = Rur .

(16)

λk

Combining these to get F(0, t) = Rur −

λk lk · ∆urk .

(17)

λk >0

Similarly, if we add and subtract

λk <0

F(0, t) = Rul +

λk lk · ul rk to (13) the result is

λk lk · ∆urk .

(18)

λk <0

Averaging (17) and (18) gives F(0, t) =

1 1 R(ur + ul ) − |λk |lk · ∆urk . 2 2

(19)

λk

The solution to the Riemann problem – the ﬂux between two cells – is written in (19) as a centered-diﬀerence scheme plus the minimum amount of dissipation to stabilize the scheme. However, the form in (13) shows us the real power of using this approach. In this expression it is easily seen that information is propagated upwind only, i.e. ul only contributes for positive eigenvalues and ur only for negative eigenvalues. We could immediately use this result in (2) by relabelling the l subscripts i − 1 and the r subscripts i.

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4 High Resolution Flux from Linear Reconstruction The preceding method for calculating the ﬂux between two cells is only ﬁrst order in space when used on a cell-averaged quantities in solving a diﬀerential equation. To create a scheme with greater spatial resolution we would like to perform an interpolation of cell averaged quantities to reconstruct a more precise value of ψ at the interface. A simple linear interpolation on each side of the cell to get ψ i±1/2 would be the obvious choice. However, Godunov’s Theorem [Lev1992] tells us that linear reconstruction would only be ﬁrst order in space if we prevent artiﬁcial oscillations (as we would be wont to do). Ergo, we will use a nonlinear slope calculation which prevents artiﬁcial maxima and minima creation (i.e. spurious oscillations). The method ﬁrst suggested by Van Leer [Lev1992] for ﬂuid dynamics applications and championed by Brunner [Bru2000] for particle transport is the harmonic mean limiter. In this approach the slope at either side of an interface is calculated using linear interpolation ui − ui−1 ∆x ui+1 − ui , m+ = ∆x m− =

(20) (21)

where ui is an element in the vector ui . Then we compute the harmonic mean of the neighboring slopes mi =

2m+ m− |m+ |m− + |m− |m+ = m+ + m − |m+ | + |m− |

(22)

If the sign of m− does not equal the sign of m+ then we need to set mi = 0 because cell i is an extremum and interpolation could create artiﬁcial extrema at its interfaces. Fortunately, the second form of the harmonic mean in (22) does this automatically and does not compel the use of conditional statements to check if each cell is an extreme point. In the case when both m+ and m− are of the same sign, then we set ui±1/2 = ui ±

∆x mi . 2

(23)

Withal, this method of reconstructing ui±1/2 ensures that ui±1/2 is always between ui−1 and ui+1 . This method has two new wrinkles compared to the ﬁrst-order scheme. In that low-resolution scheme the average value or ui was used as the value at the interfaces ui±1/2 . The high-resolution method interpolates a linear function to reconstruct the value of ui±1/2 . This reconstruction does not take place at extreme points in the values of ui ; at these cells the method is still ﬁrst-order. The other dissimilar facet of the high-resolution method is its nonlinearity, a diﬀerence that will become signiﬁcant when treating the time integration.

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The slope reconstruction method can be substituted into (19) ∆x ∆x 1 mi + ui−1 + mi−1 F(0, t) = R ui − 2 2 2 1 ∆x ∆x − mi − ui−1 − mi−1 rk . |λk |lk · ui − 2 2 2

(24)

λk

5 Time Integration Now we have two expressions for the F’s in (2), for both high-resolution and ﬁrst-order spatial schemes. There remains the time derivative component of (2) to handle. We will do this using implicit time integration methods in order to not be conﬁned by the Courant-Friedrichs-Lewy limit (c∆t/∆x < 1). The backward (or implicit) Euler method is a ﬁrst order method given by setting ψi (t + ∆t) − ψi (t) ψi n+1 − ψi n ∂ψi ≈ ≡ , ∂t ∆t ∆t

(25)

and presuming all other instances of time-dependent quantities to be known at time t + ∆t or time-step n + 1. This turns (2) into 1 n+1 − ψ ni 1 ψ n+1 i ψ n+1 + Qn+1 + , Fi+1/2 − Fn+1 i i+1/2 = −Sψ i c ∆t ∆x

(26)

Isolating the quantities at n + 1 ψ n+1 + i

c∆t n+1 ψ n+1 + c∆tSψ − c∆tQn+1 = ψ ni , Fi+1/2 − Fn+1 i i i+1/2 ∆x

(27)

which we write as ψ n+1 ) = ψ n . f (ψ

(28)

The problem given by (28) is a linear system in the case of the ﬁrst-order spatial scheme, and nonlinear in the high-resolution case. For the ﬁrst-order method, solving (28) involves solving a linear system of (N + 1)I equations. The nonlinear system for the high-resolution scheme has the same number of equations. A second-order time integration method, BDF2, was implemented as well. This method has 1 un−1 − un 3 un+1 − un ∂u ≈ + , ∂t 2 ∆x 2 ∆x

(29)

and the other terms of the equation evaluated at time n + 1. This method requires that the solution from the two previous time-steps be stored, a requirement that could be problematic for large problems.

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6 Implementation This method for solving the Pn equations was implemented in a C++ code named Implicit Riemann Object-Oriented Solver for the Transport of Energetic Radiation (Implicit ROOSTER). This code uses the Trilinos solver library, freely available from Sandia National Laboratory. For the solution of the linear systems that arise in the ﬁrst-order spatial scheme, a restarted GMRES solver is utilized (available in the AztecOO package of Trilinos). The slope reconstruction method, the nonlinear system of equations is treated using an inexact Newton method. The Newton iterations are deﬁned by (30) Jj xj+1 = Jj xj − (f (xj ) − x0 ) , where the subscripts denote the jth approximation to the solution of f (x) = x0 and J is the Jacobian of f . This Jacobian is computed using a ﬁnite diﬀerence method – meaning that we need not specify it explicitly. Furthermore, we have noticed that an eﬀectual preconditioner is the matrix from the ﬁrst-order linear system. This is a reasonable preconditioner because the high-resolution method is basically the ﬁrst-order method with a small correction term. Without this preconditioner the computational cost of even a small problem was signiﬁcant; with it the computation time is not much worse than the time for the ﬁrst-order method.

7 Results To test the method we look at a Green’s function problem involving an isotropic pulse of particles being emitted from a plane source in a 1D inﬁnite medium at time t = 0. The medium is purely scattering with Σs = 1 and c = 1. There is an analytic transport solution to this problem due to Ganapol [Gan1986]. An analytic P1 solution also exists for this probby a delta function of uncollem [Bru2000]. The P1 solution is characterized " lided particles travelling at speed ± 1/3. In Fig. 1 these exact solutions are shown. In Fig. 2 the numerical solutions 5 seconds after the pulse demonstrate a smoothing of the spikes in the analytic P1 solution. In this ﬁgure CFL = ∆x ∆t and CFL > 1 would be unstable for an explicit time integration method. Even time steps that allow particle to travel across ﬁfty spatial cells in one time unit show excellent agreement with the analytic P1 solution. It is clear that the solutions are converging to the P1 solution, as would be expected for a method solving that P1 equations in discretized form. Figure 3 shows an interesting coincidence, namely that lower time ﬁdelity (i.e. longer time steps) yields a solution nearer the transport solution. This is due to the fact that implicit methods move particles at diﬀerent speeds than the continuous time equations allow. In this problem, the P1 speeds are not the correct

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0.16 0.14

0.2

0.12 0.1

φ

φ

0.15

0.08

0.1

0.06

0 −6

0.04

Transport Analytic P1

0.05

−4

−2

0

Transport Analytic P1

0.02

2

4

0 −10

6

−5

0

5

10

x

x

Fig. 1. The analytic transport and P1 solutions to the test problem at T = 5 and T = 10 after the pulse 0.25 0.2

φ

0.15 dt = 2.5 CFL = 1250

0.1

dt = 1, CFL = 500 dt = 0.1, CFL = 50

0.05 0

Analytic P1 Transport

−3

−2

−1

0

1

2

3

x

Fig. 2. The high resolution P1 solutions between x = −3.5 and 3.5 at T = 5 0.16 0.14 0.12

φ

0.1 0.08 0.06 dt = 2, CFL = 1000

0.04

dt = 1, CFL = 500 dt = 0.1, CFL= 50

0.02 0

Analytic P1 Transport

−8

−6

−4

−2

0

2

4

6

8

x

Fig. 3. The high resolution P1 solutions at T = 10. The analytic P1 solution and the dt = 0.1 solution are coincident in the middle region

speeds for the transport solution. Consequently, by advecting particles at the wrong P1 speeds the numerical results happen to be closer to the transport solution than the P1 solution. However, more importantly the numerical scheme converged in time captures the analytic P1 solution. To test the time

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0.25 0.2

φ

0.15 0.1 0.05

dt = 0.5, CFL = 1000 dt = 0.1, CFL = 200 Analytic P1 Transport

0 −6

−4

−2

0

2

4

6

x

Fig. 4. The high resolution P1 solutions at T = 10 with small spatial cells, dx = 0.0005 and large CFL numbers

discretization, the spatial cells were shrunk to dx= 5 × 10−4 . In Fig. 4 the time integration error should dominate and the results show that it is still possible to get reasonable solutions with CFL numbers as high as 104 . At T = 10 after the pulse of particles the numerical P5 solution converged in time is suﬃcient to capture the transport solution. This is demonstrated in Fig. 5 where the φtransport − φP5 ≈ 10−4 for dt= 10−1 . The high resolution and ﬁrst order spatial scheme was compared on a series of tests using the P7 equations. The P7 solutions have a series of “spikes” representing the particles travelling at speeds characteristic of the P7 equations – much like the delta functions of the P1 equations. In the series of plots in Fig. 6, the high resolution method captures these features more accurately for coarse and ﬁne grids. This is advantageous insofar as the high resolution method does not do worse than the ﬁrst-order scheme even on these coarse grids. Finally, results from second-order time integration were 0.18

0.015

0.16

−φ

P5

0.12

0.005

φ

transport

0.1

0

φ

0.08 0.06 Transport dt = 0.1, CFL = 50 dt = 1, CFL = 500 dt = 2, CFL = 1000

0.04 0.02 0 −5

dt = 0.1 dt = 0.2 dt = 1 dt = 2

0.01

0.14

−4

−3

−2

−1

0

x

1

2

−0.005

3

4

5

−0.01 0

−2

−4

−6

−8

−10

x

Fig. 5. The P5 solution and the diﬀerence between the transport ﬂux and computed ﬂux at T = 10

R. McClarren et al. P7 at T=1, CFL = 0.25, Nx= 150

P7 at T=1, CFL = 0.25, Nx = 50

2

P7 atT=1, CFL = 0.25, Nx = 300

HR, Nx = 600 High Res 1st order

2

φ

1.5

φ

2.5

HR, Nx = 600 High Res 1st order

2.5

2

1.5

1.5

1

1

1

0.5

0.5

0.5

0 −1.5

−1

−0.5

0

x

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1.5

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−0.5

0

0.5

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2.5

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1.5

0 −1.5

−1

−0.5

0

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0.5

1

1.5

Fig. 6. Comparison of high resolution and ﬁrst-order solutions to the P7 equations with diﬀerent numbers of spatial grid points 1.5 1

φ

0.5 0 −0.5

BDF2, dt = 0.5 Backward Euler, dt = 0.5

−1 −1.5 −1.5

−1

−0.5

0

0.5

1

1.5

x Fig. 7. The BDF2, high resolution P7 solution at T = 1 has a negative scalar ﬂux

disheartening. Figure 7 shows the BDF2 solution compared with the backward Euler solution. The negative ﬂux is not desirable. Also, it is worth noting that in the previous ﬁgures the backward Euler method performed well for large time steps. Moreover, other computations performed with the BDF2 method demonstrated oscillations near extreme points in the solution.

8 Conclusion We have presented an implicit Riemann solver for the Pn equations. The implicit Riemann solver gives good agreement with the transport solution in regimes where the Pn approximation is valid, even for large CFL numbers. The high resolution method is substantially better in capturing both the sharp peaks in the solution and the smooth regions between. The test problem was solved to impressive accuracy well beyond the CFL limit for the P5 approximation. The implicit integration “smooths” the delta functions present in the analytic P1 solution, which serendipitously allows large time

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steps to agree with the transport solution better than smaller ones. Also, the second-order time integration methods did not provide better answers for time steps well beyond the CFL limit. Currently, the authors are investigating diﬀusion limits for these methods and how these Riemann solvers perform in the limit of optically thick cells. Furthermore, the extension to higher dimensions and unstructured grids will be pursued. The authors feel that the successes of implicit Riemann solvers in one dimension portends successful results in two and three dimensions.

References [BH2001a]

Brunner, T., Holloway, J.P: One-Dimensional Riemann Solvers and the Maximum Entropy Closure. Journal of Quantitative Spectroscopy and Radiative Transfer, 69, 543–566 (2001) [BH2001b] Brunner, T., Holloway, J.P: Two Dimensional Time Dependent Riemann Solvers for Neutron Transport. Proceedings of the 2001 ANS International Meeting on Mathematical Methods for Nuclear Applications, Salt Lake City, Utah, September 2001 (2001) [Bru2000] Brunner, T.A.: Riemann Solvers for Time-Dependent Transport Based on the Maximum Entropy and Spherical Harmonics Closures. PhD Thesis, University of Michigan, Ann Arbor (2000) [EPO2003] Eaton, M., Pain, C.C., de Oliveira, C.R.E., Goddard, A.: A High-Order Riemann Method for the Boltzmann Transport Equation. Nuclear Mathematical and Computational Sciences: A Century in Review; A Century Anew, Gatlinburg, Tennesee, April 6–11 2003, (2003) [Gan1986] Ganapol, B.D.: Solution of the One-Group Time-Dependent Neutron Transport Equation in an Inﬁnite Medium by Polynomial Reconstruction. Nuclear Science and Engineering, 92, 272–279 (1986) [Lev1992] Leveque, R.J.: Numerical Methods for Conservation Laws. Birkh¨auser Verlag, (1992)

The Solution of the Time–Dependent SN Equations on Parallel Architectures F. Douglas Swesty State University of New York at Stony Brook [email protected]

1 Introduction 1.1 Motivation The rapid growth of computing power, in the form of parallel architectures, over the last decade has provided the unprecedented capability for computational scientists and engineers to carry out large scale simulations of radiation transport and radiation-hydrodynamic phenomena. The development of massively parallel architectures on the scale of tens of thousands of processors provides, in principle, the rate of ﬂoating point operations needed to carry out multidimensional deterministic transport simulations involving multiple physical timescales. However, this new technological advance presents a tremendous challenge to the transport simulation developer in implementing a method for the parallel solution of the time-dependent discrete-ordinates Boltzmann equation on such platforms. Traditional iterative methods, such as source iteration, that have been developed in many research communities have undesirable features that present obstacles to eﬃcient parallelization. In this paper we present an alternative approach, the full linear system solution via Krylov subspace algorithms, that is more readily amenable to implementation on massively parallel architectures. 1.2 Requirements The ultimate goal of this work is to ﬁnd an eﬃcient method to solve the time-dependent discrete-ordinates transport equation on massively parallel distributed memory. However, there are several desiderata for such a method: • The solution should be highly scalable. The algorithm allow the eﬃcient use of the largest possible number of processors in order to reduce the wall clock time to solution for large long timescale problems that are often associated with radiation heating or cooling of material. Another way to state this criterion is that we desire a ﬁxed-size problem to scale well to large numbers of processors even though the work per processor is diminishing. We refer to such scaling as strong scaling. This is in contrast to ﬁxed-work-per-processor scaling which is much easier to achieve.

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• The overall method should not be sensitive to the particulars of the spatial, energy, or angular discretization schemes. A desirable solution method would also be able to be employed with curvilinear coordinate schemes as well as Cartesian coordinates. Ideally the method would also be extendable to complex meshes such as block structured AMR meshes or to unstructured meshes. • The method should work well in the context of radiation-hydrodynamic problems where the radiation is coupled to a ﬂuid. Ideally the method would use the same spatial domain decomposition schemes that are employed for numerical hydrodynamic simulations. • A solution method should be robust enough to handle scattering dominated problems, problems with anisotropic scattering, and multigroup problems with strong coupling between energy groups. • The method should exploit the sparsity of the linear systems arising from the implicitly discretized equations. The aforementioned list of desiderata may seem overly constraining but this list is compatible with the use of modern Krylov subspace algorithms to solve the linear systems that arise from the implicit discretization of the time-dependent Boltzmann equation.

2 A Brief Review of The Implicit Discrete Ordinates Discretization Method Before discussing approaches to implementing time-dependent Boltzmann transport on parallel architectures we brieﬂy review the implicit discrete ordinates method for solving the monochromatic Boltzmann equation [MM99]. 1 ∂I 1 t + Ω · ∇I + κ (ε)I − dΩ dε I(Ω , ε )κs (Ω , ε , Ω, ε) = S(ε) . (1) c ∂t c Throughout this paper we will utilize standard astrophysical nomenclature and notation. In (1) I = I(x, Ω, ε, t) is the radiation intensity which is a function of position x, direction Ω, energy ε, and time t. For readers more familiar with neutronics nomenclature can think of I as the ﬂux. The equation involves material properties through the total opacity κt (ε), the scattering opacity κs (Ω, ε, Ω , ε ), and the emissivity S(ε). In (1) c denotes the speed of light. Equation (1) describes the time evolution of particles traveling at the speed of light such as photons or massless neutrinos, but it may be generalized to describe massive particles such as neutrons by replacing c with the energy dependent velocity v(ε). This equation can be implicitly time-diﬀerenced as

Parallel Solution of the Time-Dependent SN Equations

1 n+1 I + Ω · ∇I n+1 + κt (ε)I n+1 − c∆t

dΩ

471

dε I n+1 κs (Ω , ε , Ω, ε) 1 In = S(ε) + (2) c ∆t

where we have suppressed the arguments of I in the integral term for clarity and where we use standard ﬁnite-diﬀerence notation of superscripts to denote the time at which the time-discretized analog of I is deﬁned: I n ≡ I(x, Ω, ε, tn ) .

(3)

We now consider the angular discretization of (2). For the sake of brevity we restrict ourselves to 1-D geometries such as Cartesian or spherical coordinates where the direction can be characterized by a single angle cosine µ and (2) simpliﬁes to 1 n+1 I + µ · ∇I n+1 + κt (ε)I n+1 − 2π dµ dε I n+1 κs (µ , ε , µ, ε) c∆t 1 In = S(ε) + . (4) c ∆t Angular discretization in multi-dimensions is straightforward and the reader is referred to [LM93] for details. In the discrete ordinates method the continuous range of the angular cosine, −1 ≤ µ ≤ 1, is replaced by a discrete set of angular ordinates {µk | k = 1, . . . , N } for which (4) holds true. For the direction µk the angular integral in (4) is approximated by a compatible quadrature

dµ I n+1 κs (µ , ε , µ, ε) ≈

N

w In+1 κsk, (ε , ε)

(5)

=1

where k and are subscripts denoting directions µk and µ . The energy integral can be dealt with similarly by discretizing the spectrum into a series of groups with energies at the group center given by {εg | g = 1, . . . , M } and the group widths given by {∆εg | g = 1, . . . , M }. For an energy εh and direction µk the double integral in (4) is approximated by

dµ

dε I n+1 κs (µ , ε , µ, ε) ≈

N M

∆εg w In+1 κsk,h,,g

(6)

=1 g=1

and the discrete analog of (4) becomes 1 n+1 n+1 n+1 Ih,k + µk · ∇Ih,k + κth Ih,k − 2π ∆εg w In+1 κsk,h,,g c∆t =1 g=1 n Ih,k 1 = Sh + . c ∆t N

M

(7)

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Equation (7) is referred to as the SN equation or the monochromatic, discreteordinates, Boltzmann equation. In order to numerically solve (7) some form of spatial discretization is applied to the streaming operator µk ·∇. This discretization may require some sort of closure relationship relating the intensities at adjacent spatial points. For a detailed description of a variety of spatial discretizations we refer the n+1 reader to [PA92]. For any spatial discretization scheme that is linear in Ih,k the set of equations (7) for all spatial zones, together with any linear equations describing boundary conditions on the computational domain, forms a large, n+1 sparse, linear system of equations that must be solved in order to obtain Ih,k at each timestep.

3 Iterative Approaches The linear systems described in Sect. 2 are sparse and, for multidimensional problems, extremely large. The sparsity suggests the use of iterative methods to solve the linear system. Direct solution methods for these linear systems have been applied in one dimensional problems, e.g. [MB93], but in multidimensional problems the size of the linear systems renders direct solution methods impractical in most situations. The iterative methods utilized have fallen into two categories. The ﬁrst are source iteration methods, which split the linear systems into parts, solving a portion of the system directly. This method has been widely utilized by the neutronics community for many years [LM93]. The second approach has been the use of Krylov subspace algorithms applied to the full linear system. This method has seen only limited use in both the astrophysical community [DM05] and the neutronics community [PH02]. 3.1 Source Iteration In this section we will brieﬂy describe the source iteration method in a linear algebraic context. An alternative, and perhaps more physical, description is provided in [LM93]. The linear system arising from the SN equations can be written in the form of Ax = b (8) n+1 and b is a vector made up where x is a vector containing the unknowns Ih,k of the right-hand-sides of (7). The full linear system matrix A can be split into parts arising from the various operators composing the left-hand-side of (7) A=D+T +S . (9)

In (9) D is a matrix arising from the time derivative and total opacity terms on the left-hand-side of (7) and S is matrix arising from the scattering term. The matrix T arises from the spatial discretization of the streaming operator.

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In a 1-D problem, with a lexicographic ordering of variables, these matrices may take on a very simple form where D may be diagonal or banded, T may be tridiagonal, and S could consist of blocks of size N M × N M located on, or adjacent to, the diagonal. In the general case D, T , and S may be more complex in structure. The strategy behind the source iteration scheme is to exploit the splitting to form an iterative scheme as follows: 1. An initial guess is made for the solution vector x 2. The equation (D + T )x = b − Sx

(10)

is solved for x using direct methods applied to the matrix D + T which is usually banded for simple geometries and meshes. 3. set x = x and repeat the procedure starting at step 1 until convergence is obtained. This simplistic iterative algorithm suﬀers from two major limitations that run counter to the desiderata listed in Sect. 2. The ﬁrst problem arises from the direct solve in step 2 of the source iteration algorithm. This solve usually exploits the banded structure arising from simple linear discretizations of the T operator by performing Gaussian elimination via forward and back substitution steps. A physical interpretation of this process is that one is sweeping in through the mesh in the direction of radiation travel updating the intensity in the downwind direction. However, this approach is recursive and is not well suited to parallelization. In simple Cartesian geometries with suﬃciently simple boundary conditions one can exploit wave-front parallelization techniques [KBA92] to implement this method on parallel architectures. However, complex boundary conditions and curvilinear geometries couple equations with diﬀering values of the angular cosines in a manner that makes such parallelization techniques diﬃcult. The second problem with source iteration is that the method converges poorly, or often fails to converge entirely, when the problems are scattering dominated. The reason for this is obvious from (10) where the dominant operator S is continually applied to the previous iterate on the right-handside. This problem is not insurmountable in many cases and the convergence can be hastened or obtained via the use of accelerators which provide better initial guesses for step 1 of the source iteration algorithm. There are two major beneﬁts to this algorithm in some circumstances. First, in optically thin situations where there is little or no scattering the solution of (10) provides a highly accurate in answer in very few iterations. The second beneﬁt is that the algorithm requires little in the way of storage especially if the scattering matrix can be applied in operator form or if it is trivial as in the case of isotropic scattering.

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3.2 Full Linear System Solution via Krylov Subspace Iteration An alternative approach to solving the linear systems arising from the SN equations is applying state-of-the-art Krylov subspace algorithms to obtain the solution of the full linear system (8). The derivations of, and a description of the features and limitations of various Krylov subspace algorithms are beyond the scope of this paper and the interested reader should consult [SA03] or [BA94] for details. However, we will brieﬂy describe aspects of the algorithms that are relevant to their use in solving the linear systems arising from the SN equations. There are several advantages to using nonstationary Krylov subspace algorithms for nonsymmetric problems such as GMRES [SS86] or Bi-CGSTAB [VO92] on the full linear system Ax = b: • Algorithms can be implemented in operator, or matrix-free, form where storage of the coeﬃcient matrix A is not required. The matrix-vector multiply operation required by these algorithms can be implemented by supplying a subprogram that evaluates the left-hand-side of the SN equation. • Curvilinear coordinate systems or complex boundary conditions are relatively easy to accommodate. • Parallelism derives from the parallel nature of the sparse linear algebraic operations involved in the Krylov subspace algorithm. • Algorithms require only a few basic linear algebraic operations which can usually be encoded using BLAS (Basic Linear Algebra Subprograms) operations or their equivalents. Some of these operations such as vector additions are embarrassingly parallel. Under a spatial domain decomposition scheme the matrix-vector multiplies require only nearest-neighbor communication within a process topology. Vector inner products can be accomplished via global reduction operations supplied by the message passing interface (MPI). There are also drawbacks associated with the use of nonstationary Krylov subspace algorithms. These algorithms require the use of temporary vectors to hold intermediate results. Each of these vectors are identical in size to the vector of unknowns x. Typically this means that the algorithms may require, at a minimum, 5–8 vectors worth of memory in addition to the memory associated with the vector of unknowns. Some algorithms such as GMRES require a number of vectors that grows as the iteration process progresses. This growth in memory utilization can be mitigated in some cases by restarting the iteration process anew (see [BA94] for details). For this reason we restrict ourselves to discussing the Bi-CGSTAB algorithm for the remainder of this paper. Another drawback that nonstationary Krylov subspace algorithms share with the source iteration process is that of non-convergence or slow

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convergence in some circumstances. We brieﬂy examine these issues in the next section.

4 Speeding Up and Obtaining Convergence There are techniques to improve the iterative convergence of both the source iteration method and the full linear system solution method. Since our focus for the remainder of this paper is on the use of the full linear system solution method we will only brieﬂy discuss techniques for improving the convergence of source iteration. In the case of the full linear system solution we will discuss one class of preconditioners that holds promise for use on parallel architectures. 4.1 Accelerators In the context of the source iteration approach improvements to iterative convergence are usually sought by means of accelerators which provide a better estimate for the vector of unknowns x for insertion into the right-hand-side of (10) on each iterative step. The calculation of this estimate is achieved via some algorithm that is usually referred to as an accelerator. There is an entire body of literature on the development of accelerators. Readers are urged to consult the excellent review article [AL02] for details. We conﬁne ourselves here to a brief discussion of one broadly utilized class of accelerators referred to as diﬀusion synthetic acceleration (DSA) techniques [AL77]. In these methods a discretized diﬀusion equation that is consistent with the SN equation is solved for the energy density which is then used to evaluate the scattering term on the right-hand-side of (10). This process accelerates the convergence of source iteration in optically thick or optically translucent situations. DSA schemes are usually derived for the case of isotropic scattering but the can be extended to the anisotropic case as well [MO82]. Another challenge to the construction of eﬀective accelerators is that it is often diﬃcult to construct a discretized diﬀusion equation that is consistent with the SN equation. Larsen [LA82] has developed a four-step process for constructing DSA accelerators but the construction can be complex for many schemes [PA92]. 4.2 Preconditioners The convergence of the full linear system solution can be improved through the use of preconditioners. The idea behind preconditioning is deceptively simple. A dominant factor in the convergence of a nonstationary Krylov subspace algorithm is the condition number of the coeﬃcient matrix A. Preconditioning seeks to improve the conditioning of the system by solving

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ML AMR−1 MR x = ML b

(11)

instead of solving (8). The matrices ML and MR are referred to as left and right preconditioners respectively. A clever choice of preconditioners may provide a coeﬃcient matrix ML AMR−1 that has a better condition number than does A. As with accelerators there exists a vast body of literature on preconditioners which is well beyond the scope of this paper. A recent review by Benzi [BE02] provides a fairly comprehensive bibliography for this subject. The desire for parallel scalability restricts the choice of preconditioners to those that are parallelizable. Many traditional preconditioning techniques that are used with Krylov subspace methods are inherently sequential. Nevertheless these preconditioning techniques can at times be utilized successfully in a parallel context [DM05]. Development of inherently parallel preconditioners is on ongoing area of research. For this reason we will only brieﬂy mention one particular class of preconditioners, sparse parallel approximate inverses (SPAI) which have desirable scalability properties. SPAI preconditioners have been utilized successfully on a variety of problems including radiation transport in the ﬂuxlimited diﬀusion approximation [SS04]. The idea behind SPAI preconditioners is quite simple. One wishes to exploit the sparsity pattern and the structure of the coeﬃcient matrix to ﬁnd an eﬀective approximation to the inverse that is sparse. The sparsity pattern of an example system arising from the diamond diﬀerence spatial closure is shown in Fig. 1 which depicts an 80 × 80 section of the upper left corner of the coeﬃcient matrix A. The values of the elements in this matrix section are shown as a grayscale map in Fig. 2. The coeﬃcient matrix A has been row scaled so that the diagonal elements have a value of unity. Figure 2 clearly shows that dominant oﬀ-diagonal elements lie in bands spaced a distance equal to the number of ordinates (N = 16 in this case) away from the diagonal. A pattern is chosen for the non-zero elements in the approximate inverse. The choice of pattern may or may not be inﬂuenced by the values of the elements in the true inverse A−1 . Ideally we would like to ﬁnd a left or right approximate inverse by choosing values for the non-zero elements such that ML ≈ A−1 or MR−1 ≈ A−1 . The values of the non-zero elements are in general over-determined but we can ﬁnd an optimal set of values by doing a least squares ﬁt to determine the values of the non-zero elements in each row of ML or column of MR−1 . For details of this procedure consult [BE02] and references therein. The inverse A−1 may have structure that can be exploited in forming the sparse approximate inverse. An example of this is illustrated in Fig. 3 where the grayscale image shows the value of each element of the upperleft 80×80 corner of the inverse of the matrix shown in Fig. 2. One can clearly see that the dominant elements of the inverse lie along the diagonal and in bands that are spaced a distance equal to integer multiples of the number of angular ordinates away from the diagonal. This inverse structure is typical for

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the inverse of matrices arising from partial diﬀerential equations with local spatial coupling [SM05]. The size of the oﬀ-diagonal elements decays slightly as one moves away from the diagonal. The time dependence of the Boltzmann equation also factors into structure of the inverse. In simulations where the time evolution is being carried out with small CFL number timesteps the D matrix will dominate the coeﬃcient matrix A. In this case the system is likely to be diagonally dominant. In the case of diagonal dominance of a matrix that can be formulated in a block tridiagonal form the inverse has elements that decay in size exponentially as one moves away from the diagonal (see [AX96] Sect. 8.6 for details ). Such a situation is ideal for approximate inverse preconditioners since the inverse can be eﬀectively approximated with only a few bands. If the timestep sizes are increased eventually the behavior of the Sn equation becomes more elliptic and the decay behavior in the inverse becomes less pronounced. In the inverse depicted in Fig. 3 the decay of the elements is slight as there is no time dependence in this problem. However, even in the large CFL number timestep limit simple SPAI preconditioners can continue to preform remarkably well. A major advantage of the procedure for forming the SPAI preconditioner is that under a spatial domain decomposition the least squares ﬁt of each row or column of the preconditioner can be performed in parallel with only nearest-neighbor communication if the sparsity pattern involves elements corresponding to only nearby spatial zones. In general for the SN equation this is true as the spatial discretization usually couples only a few neighboring zones. This localized structure of the linear system can thus be exploited to yield fairly eﬀective parallel preconditioners for the SN equation.

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4.3 A Brief Comparison of Approaches In this section we oﬀer a brief comparison of the source iteration and full linear system solution approaches on a series of 1-D test problems with and without reﬂective boundaries that illustrate a few of the points mentioned in Sects. 4.1 and 4.3. For the sake of simplicity we consider the diamonddiﬀerence scheme in Cartesian or slab geometry. The details of the source iteration method and the DSA accelerator for this case are fully documented in [AL77]. The full linear system solution is accomplished with the Bi-CGSTAB algorithm. We consider a SPAI preconditioner with ﬁve non-zero elements per row. The sparsity pattern for this preconditioner is shown in Fig. 4 where we show the upper-left 80 × 80 corner of A−1 . This preconditioner is easily implemented and the values of the elements need only be computed once and stored in an array. The storage cost of the vector is approximately the cost of 5 unknown vectors. The ﬁrst problem we consider is a 1-D translucent slab with incoming boundary conditions on either end. The problems is zoned into 1024 uniform

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Fig. 5. Number of Iterations versus the number of angular quadrature points for the slab geometry problem. Circles denote the unaccelerated source iteration method. Squares denote the unpreconditioned Bi-CGSTAB method. Triangles indicate the Bi-CGSTAB preconditioned with a simple SPAI preconditioner with 5 non-zero elements per row. Diamonds indicate source iteration with DSA acceleration

width zones and the opacity is such that each zone has an optical depth of 0.1. There are no sources and no time-dependence in the problem. The comparative performance of the source iteration and Bi-CGSTAB applied to the full linear system are shown in Fig. 5. The unpreconditioned BiCGSTAB converges in substantially fewer iterations than does the source iteration method. However, with the addition of a DSA accelerator the source iteration method converges in far fewer iterations than the Bi-CGSTAB+SPAI combination. Also, the behavior is clearly independent of the angular resolution. The second problem is almost identical to the ﬁrst with the exception that it has a reﬂective boundary on the right end of the computational domain. In this case results shown in Fig. 6 the Bi-CGSTAB+SPAI combination proves superior converging in substantially fewer iterations than the source iteration+DSA combination. 3500

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Fig. 6. Same as Fig. 5 but for the reﬂecting boundary problem

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The diﬀering results between ﬁgures clearly illustrate one important point that must be made when comparing iterative methods for the solution of the SN equations: deﬁnitive statements about which method is better can only be made about speciﬁc source-iteration/accelerator and Krylovalgorithm/preconditioner combinations applied to a speciﬁc problem. For a given problem it may always be possible to ﬁnd a better accelerator or preconditioner that shifts the balance in favor of one iterative approach over another. One ﬁnal note we wish to make regarding the lessons oﬀered by this example problem. In this simple case the best choice for source iteration method would be to arrange the order of the forward and back substitution sweeps associated with the direct solve so that the so that the left-to-right sweep was ﬁrst followed by the right-to-left sweep. The calculation of the reﬂected intensity, which will determine the boundary value for I for ordinates where µh < 0, ideally should be carried out after the left-to-right sweep is complete and before the right-to-left sweep begins. This allows the reﬂected boundary condition to be consistent with the values of I for µh > 0 that were computed in the left-to-right sweep. However, in the case of two reﬂective boundaries such an optimal ordering for the sweeps does not exist and the reﬂective boundary condition at one end cannot be computed self-consistently with the current value of the intensity I. This lack of self-consistency will normally slow or halt the iterative convergence of the source iteration method.

5 Parallel Implementation of the Full Linear System Approach Since the full linear system solution approach to solving the SN equation is relies only on a Krylov subspace algorithm to solve a linear system, the parallel implementation of this method, in some sense, reduces to the parallel implementation of the Krylov subspace algorithm. However, one of the desiderata for a scalable solution method was that it be compatible with hydrodynamic spatial domain decomposition schemes. For this reason we will not consider other domain decomposition strategies that could be used for pure transport simulations. 5.1 Parallelization of the Bi-CGSTAB Algorithm Unlike the direct solves associated with the source iteration method most nonstationary Krylov subspace algorithms do not make use of operations that are inherently sequential. In fact the operations required for the parallelization of the Bi-CGSTAB algorithm are relatively few. • A matrix-vector multiply operation. Under spatial domain decomposition schemes this requires only nearest-neighbor communication.

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• Vector addition and scalar-vector multiplication. Under spatial domain decomposition both of these operations are embarrassingly parallel. • Preconditioner creation and solves. For simple SPAI preconditioners this requires only nearest neighbor communication. • Vector inner products. Under spatial domain decomposition this operation requires a global summation across processors. In order to illustrate the simplicity of this method we write down the steps of the Bi-CGSTAB Algorithm: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

Provide initial guess: x(0) Calculate initial residual: r0 = b − Ax(0) Set ˜r = r0 For i = 1, 2, . . . i−1 = ˜rT r(i−1) if i−1 = 0 the method has failed if i = 1 p(i) = r(i−1) else βi−1 = (i−1 /i−2 )(αi−1 /ωi−1 ) p(i) = r(i−1) + βi−1 (p(i−1) − ωi−1 v(i−1) ) endif ˆ = p(i) precondition by solving M p (i) v = Aˆ p αi = i−1 /˜rT v(i) s = r(i−1) − αi v(i) if s < tolerance ˆ x(i) = x(i−1) + αi p stop endif precondition by solving Mˆs = s t = Aˆs ωi = tT s/tT t ˆ + ωiˆs x(i) = x(i−1) + αi p r(i) = s − ωi t if r(i) < tolerance stop endif endfor

The algorithm above can easily be implemented in less than 100 lines of F95 or C++ if BLAS subroutines or the equivalent are utilized for the basic vector operations. Implementing this algorithm under logically Cartesian spatial domain decomposition requires that elements corresponding to zones at the edge of a subdomain (hereafter referred to as ghost zones) be exchanged between

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processors at certain stages in each iteration. Immediately prior to step 2 the ghost zones of x(0) must be exchanged in order to evaluate Ax(0) . Similarly between steps 13 and 14 ghost zones of p must be exchanged and between steps 21 and 22 ghost zones of ˆs must be exchanged prior to performing the matrix-vector multiplies in steps 14 and 22. If a simple SPAI preconditioner is used in steps 13 and 21 then ghost zones exchanges are needed for the vectors p(i) immediately prior to step 13 and s immediately prior to step 21. These ghost zone exchanges can be accomplished via calls to MPI send and receive routines. Another parallel operation that is required is a global summation for vector inner products in steps 5, 15, and 23. These three steps require a total of four inner products. However, steps 17 and 26 require the evaluation of the norm of residual vectors which brings the total number of inner products to six per iteration. The global summations required for these inner products can be accomplished via MPI”7016GLOBAL”7016REDUCE calls. The overall ﬂoating-point operation time of the algorithm is usually dominated by the matrix-vector multiply and/or the preconditioning steps. However, the message passing time associated with the ghost-zone exchanges and the global reduction operations may be the prohibitive factor if the number of ﬂoating-point operations per processor in the matrix-vector multiply is small. Ultimately the diminishing ratio of computation time to communication time as one increases the number of processes in a parallel simulation of ﬁxed size will degrade the scalability.

6 Parallel Scalability of a 2-D Test Problem To examine the parallel scalability of the full linear system solution approach we consider the solution of a standard crooked pipe, a.k.a. “Top Hat” problem [GE01] using the simple corner balance spatial discretization scheme (see [PA92] for details). This problem involves radiation traveling down a optically translucent duct that makes several right angle turns. The material surrounding the duct is optically thick. Heating of the material is modeled using a standard heating opacity with a model absorption opacity and a blackbody emissivity. This problem involves very long timescales as radiation does not propagate around a turn in the duct until the material at the end of a leg is heated and photons are re-radiated in new directions. We zone this problem into 100 × 350 uniform spatial zones with 10 energy groups and an S8 angular quadrature with 48 cosine pairs. Since the simple corner balance scheme in 2-D discretizes each spatial zone into four corners with an SN equation to be solved in each corner the total number of equations to be solved is 100 × 350 × 10 × 48 × 4 ≈ 6.7 × 107 . The problem is domain decomposed into a 2-D Cartesian process topology with roughly equal numbers of zones per process to optimize parallel load balancing. For a ﬁxed number of processors the number of processes in each dimension are

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chosen to make process topology as square as possible. The code, V2D, that 10 Speedup

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Fig. 7. Parallel Speedup relative to 32 processors for the 100 × 350 SCB crooked pipe, a.k.a. “Top Hat” problem

implements the solver is written in F95 and uses MPI for all message passing. The parallel speedup relative to 32 processors for this ﬁxed-size problem obtained on the National Energy Research Supercomputing Center (NERSC) IBM-SP is shown in Fig. 7. The for this problem problem size algorithm scales quite well out to to 256 processors. As the number of processors is increased to 512 and 1024 the speedup levels oﬀ as the ratio of communication time to computation time increases. For example, at 1024 processors we have employed 32 × 32 process topology which means each process in the has roughly a 3 × 11 interior mesh surrounded by a roughly 32 ghost zones. Thus the ratio of ghost zones to interior mesh points is approximately unity. Of course, a larger problem could easily maintain reasonable parallel eﬃciency to larger numbers of processors but eventually communication costs will doom the scalability of any ﬁxed-size problem.

7 Conclusions and Future Directions The application of nonstationary Krylov subspace algorithms to timedependent discrete ordinates transport problems provides a successful avenue for implementing such problems on massively parallel architectures. This method does not suﬀer from the drawback of having spatially recursive sweeps which provide barriers to eﬀective parallelization in many cases. We have also found a set of parallelizable preconditioners that have proven effective in aiding convergence of Krylov subspace iteration for time-dependent SN equations. These preconditioners exploit the diagonal dominance of the coeﬃcient matrix that is often present in time dependent problems. The answer to the question of whether source iteration is more eﬀective than the

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full linear system solution via Krylov subspace algorithms is highly dependent on both the problem and the accelerators or preconditioners employed. It is diﬃcult to make general statements regarding the relative merits of one approach versus another. There are a number of directions for future research in this area. The possibility of hiding communication by utilizing asynchronous message passing to exchange ghost zones while computation is being carried out on the interior of each subdomain can increase scaling in many circumstances. Similarly, research into the subject of reducing the number of global reduction operations by restructuring Krylov subspace algorithms could yield scalability improvements. Finally, there is a continuing need for the development of more eﬀective parallel preconditioners for the SN equations.

8 Acknowledgments The author would like to acknowledge ﬁnancial support from Lawrence Livermore National Laboratory in the form of a ASCI contract and computing support from the National Energy Research Scientiﬁc Computing Center.

References [AL02]

Adams, M.L. and Larsen, E.W.: Fast Iterative Methods for DiscreteOrdinates Particle Transport Calculations. Prog. in Nucl. Eng. 40, 3–159 (2002) [AL77] Alcouﬀe, R.E.: Diﬀusion Synthetic Acceleration Methods for the Diamond-Diﬀerenced Discrete-Ordinates Equations. Nucl. Sci. Eng. 64, 344 (1977) [AX96] Axelsson, O.: Iterative Solution Methods. Cambridge University Press, Cambridge (1996) [BA94] Barrett, R. and et al.: Templates for the Solution of LInear Systems: Building Blocks for Iterative Methods SIAM, Philadelphia (1994) [BE02] Benzi, M.: Preconditioning Techniques for Large Linear Systems: A Survey. J. Comp. Phys. 182, 418–477 (2002) [DM05] D’azevedo, E.F. and Messer, B. and Mezzacappa, A. and Liebendorfer, M.: An ADI-Like Preconditioner for Boltzmann Transport. SIAM Sci. Comput., 26, 810–820 (2005) [GE01] Gentile, N.: Implicit Monte Carlo Diﬀusion–An Acceleration Method for Monte Carlo Time-Dependent Radiative Transfer Simulations. J. Comput. Physics 29, 543–571 (2001) [KBA92] Kock, K.R. and Baker, R.S. and Alcouﬀe, R.E.: Solution of the FirstOrder Form of the 3-D Discrete Oridnates Equation on a Massively Parallel Processor. Trans. Amer. Nucl. Soc., 65, 198 (1992) [LA82] Larsen, E.W.: Unconditionally Stable Diﬀusion Synthetic Acceleration Methods for the Slab Geometry Discrete Ordinates Eqautions. Part 1: Theory. Nucl. Sci. Eng. 82, 47 (1982)

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F.D. Swesty Lewis, E.E. and Miller, W.F.: Computational Methods of Neutron Transport. American Nuclear Society Inc, La Grange Park (1993) Mezzacappa, A. and Bruenn, S.W.: A numerical method for solving the neutrino Boltzmann equation coupled to spherically symmetric stellar core collapse. Astrophys. J., 405, 669–684 (1993) Mihalas, D, and Weibel-Mihalas, B.: Foundations of RadiationHydrodynamics. Dover, Mineola (1999) Morel, J.E.: A Synthetic Acceleration Method for Discrete Ordinates Calculations with Highly Anisotropic Scattering. Nucl. Sci. Eng. 82, 34– 46 (1982) Palmer, T.S.: Curvilinear Geometry Transport Discretizations in Thick Diﬀusive Regions. Ph.D. Thesis, University of Michigan, Ann Arbor (1992) Patton, B.W., and Holloway, J.P.: Application of Preconditioned GMRES to the Numerical Solution of the Neutron Transport Equation. Annals Nucl. Eng., 28, pp. 109–136 (28) Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd ed. SIAM, Philadelphia (2003) Saad, Y. and Schultz, M.: GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems. SIAM J, Sci. Statist. Comput. 7, 856–869 (1985) Swesty, F.D. and Smolarski, D. C. and Saylor, P.E.: A Comparison of Algorithms for the Eﬃcient Solution of the Linear Systems Arising from Multi-Group Flux-Limited Diﬀusion Problems. Astrophys. J. Suppl. 153, 369–387 (2004) Smolarski, D. C. Diagonally Striped Matrices and Approximate Inverse Preconditioners. J. Comput. and Appl. Math., in press, 2005 van der Vorst, H.: BiCGSTAB: A Fast and Smoothly Converging Variant of B-CG for the Solution of Nonsymmetric Linear Systems. SIAM J, Sci. Statist. Comput., 13, 631–644 (1992)

Diﬀerent Algorithms of 2D Transport Equation Parallelization on Random Non-Orthogonal Grids Shagaliev R.M., Alekseev A.V., Beliakov I.M., Gichuk A.V., Nuzhdin A.A., Rezchikov V.Yu.

Numerical solution of a non-stationary kinetic equation at a multi-group setting in the 2D spatial approximation demands much computer operating memory and calendar execution time. One of the ways of solving such problems is using algorithms of parallelization. Solving the problem is particularly diﬃcult in the general case, when the boundary problem for the transport equation is put in the space-domains with complex shapes, and this equation is approximated on non-orthogonal spatial grids. As a rule, implicit time-dependant schemes are used to numerically solve these tasks, while the system of grid equations on the steps by time is solved with the method of sweep computation. This means that at this time step grid transport equations are solved in strict succession, these successions are diﬀerent for diﬀerent directions of the particle motion and change at time steps. In other words, when the transport equation is being solved in the kinetic approximation at multi-processor systems, one has no possibility to predetermine the linkage layout between the processor elements (order of data exchange), which crucially complicates the problem of development of eﬃcient algorithms of transport problem parallelization. Several algorithms of parallelization a 2D non-stationary kinetic transport equation at a multi-group setting on non-orthogonal spatial grids are compared in the present Report. The ﬁrst algorithm is a large-block parallelization algorithm. When such an algorithm was used one of the space-domains of investigated system was calculated at one of the processors. (As a space-domain we take a family of system points, joined by similar physical properties. In each space-domain the kinetic transport equation was solved independently). To carry out the computation correctly iterations by inner boundary conditions were applied, each iteration was followed by inter-processor exchange of inner boundary conditions. The linkage between the multi-processor computer units was held by means of a special library of MPI data transport. One space-domain, as well as arbitrary number of space-domains, was successfully calculated on the processors at a time. All the applied transports are asynchronous, the points of transport asynchronization were at the end of the iteration cycle before the transport of inner boundary conditions.

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The eﬃciency of this algorithm during solving a test problem was 78%. The test parameters are presented in Table 1 below. It is evident that this parallelization algorithm has considerable restrictions concerned, ﬁrst of all, with the dimensions of the computed system and its fragmentation. The problem of unbalanced computation load on the processors for the reasons of diﬀerent number of points in the space-domains in the general case is acute for practically every computation. Table 1. Test problem parameters

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An other algorithm, which in its turn provides possibility for balanced load of the processors, is the algorithm of minor block parallelization, where the principle of space decomposition of the initial system into sub-domains was applied. System fragmentation into sub-domains, which were then disposed at separate processors (further on “para-domains”) was carried out here only in one direction, viz along the columns of the spatial grid. This parallelization algorithm were somehow similar to the event dynamic system. At each parallelization cycle the boundary conditions for the covered boundaries of para-domains, provided from adjacent processors, were analyzed, after which, results dependant, the further behavior scenario was speciﬁed. The main events included in this dynamically speciﬁed scenario were initialization of asynchronous exchanges, their completion, processing the provided boundary conditions, computation of a para-domain, computation of the resolution ratio for a family of cells. The basic principles of the algorithm of minor block parallelization with decomposition along the columns were as follows: • Spatial decomposition of the space-domain into para-domains; the spacedomain was split into columns of the spatial grid; • The system of grid equations was solved independently by the angle variable; • Complete integration of the inter-processor exchange with the computation, which was achieved by the application of asynchronous receive/send operations. Numerical investigations of this parallelization algorithm were carried out on a one-space-domain spherically symmetrical model problem. The thermody-

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namic and optical properties of the medium were analogous to those in the Flack’s task. The problem space-domain was a hemi-sphere with the radius 1 cm. This space-domain was covered with the spatial grid comprised of 3×Np columns (Np – number of processors) and 50 rows. The incoming energy ﬂow on the outer surface equaled zero, the initial radiation density equaled zero as well. Fore the ﬁrst computation series the angular grid with 126 directions by the space angle was selected. The computation was carried out with the seven-group approximation. The problem was solved in 10 steps by time. The right-hand-side iterations were carried out along with the solution of a correction equation with the KM-method and completed with the error 0.001. The numbers of processors during the computation were: 1, 10, 20, 30, 40, 50. The main parameters of the parallel algorithm eﬃciency, namely, the speedup factor and the parallelization eﬃciency, are presented in Table 2 and in Figs. 1, 2. Table 2. Speedup and parallelization eﬃciency in the ﬁrst computation series Number of Processors

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The second computation series was carried out on the angular grid with 160 directions by the space angle. The computation was carried out with the twenty-eight-group approximation. The problem was also solved in 10 steps by time. The right-hand-side iterations were carried out along with the solution of a correction equation ˆ`I-method and completed with the error 0.001. with the E The numbers of processors during the computation were: 1, 10, 20, 30, 40, 50. The main parameters of the parallel algorithm eﬃciency, namely, the speedup and the parallelization eﬃciency, are presented in Table 3 and Figs. 3, 4. The requirements of higher precision of the solution of multi-group transport problems at numerical simulation make important the use of big number of intervals by an energy variable (groups). The third parallelization

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60

50

Sp

40 Computation Ideal

30 C 20

10

0 0

10

20

30 N

40

50

60

Fig. 1. Speedup dependence on the number of processors

120

100

100

100.5

101 91.85 84.18

En

80

79.34

60

40

20

0 1

10

20

30

40

50

N Fig. 2. Dependence of the parallelization eﬃciency on the number of processors

Diﬀerent Algorithms of 2D Transport

491

60 50

Sp

40 Computation Ideal

30 20 10 0 0

10

20

30 N

40

50

60

Fig. 3. Speedup dependence on the number of processors

120 100

100 95.7

84.7 79.6

En

80

74.3

73.4

40

50

60 40 20

0

1

10

20

30 N

Fig. 4. Dependence of the parallelization eﬃciency on the number of processors

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Table 3. Speedup and parallelization eﬃciency in the second computation series Number of Processors

Speedup

Parallelization Eﬃciency

1 10 20 30 40 50

1 9.6 16.9 23.9 29.7 36.7

100 95.7 84.7 79.6 74.3 73.4

algorithm (combined) combines both the parallelization by space and energy variables. When a multi-group transport equation is being sold, the computations of the intervals or groups of intervals by the energy variable can be carried out independently, each on a separate processor or a group of processors. Further on, such groups of interval are called “group-domains”. Such an approach allows to use a considerable number of processors at solving a 2D kinetic equation, reasonable eﬃciency kept. Numerical investigations of the combined parallelization algorithm eﬃciency were carried out at three model problems. The method of problem splitting was applied for computing the speedup and the parallelization efﬁciency. For this method the main parameters of the parallelization were calculated using the following formulae: The speedup: t1 , Spn = tn where tn is the average time of the problem computing at one processor when N processors are run; t1 is the time of the problem computing at one processor; The parallelization eﬃciency: En =

t1 ∗ 100 . n∗ tn

For the ﬁrst computation series one hemi-spherical space-domain model problem had been selected as a test one: 80 rows were homogenously distributed at the radius, 200 columns were homogenously distributed at the angle. The order of the angle quadrature was 12, in total there were 96 directions of the particles motion. Number of energy groups was 28. The result speedup and the parallelization eﬃciency are presented in Table 4. The total number of processors, as well as the number of para-domains

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Table 4. Speedup (Sp) and the parallelization (E) eﬃciency in the ﬁrst test computation series Number of Processors 1 4 7 10 14 25 28 40 50

Number of GroupDomains

Number of ParaDomains

Sp

E,%

1 4 7 1 14 1 28 4 1

1 1 1 10 1 25 1 10 50

1 3.5 5.5 11.6 8.8 25.4 12.3 39.5 43.9

100 89 78 116 63 102 44 99 88

at parallelization by space and the number of group-domains at parallelization by groups, are shown for each run. The two-space-domain model problem for the second computation series was obtained from the previous problem by means of splitting into rows. Two space-domains. In each space-domain the number of rows was 40, homogenous by the radius, the number of columns was 200 homogenous by the angle. The number of energy groups and other parameters of the problem coincide with those of the one-space-domain problem. The incoming ﬂow on the outer edge of the ﬁrst space-domain, corresponding to the temperature 1, was taken as the boundary condition. The speedup and the parallelization eﬃciency, obtained in the course of numerical research for diﬀerent numbers of para-domains and group-domains, are presented in Table 5. Table 5. The speedup (Sp) and the parallelization eﬃciency (E) in the second test computation series Number of Processors 1 10 25 40 50

Number of GroupDomains

Number of ParaDomains

Sp

E, %

1 1 1 4 1

1 10 25 10 50

1 9.4 20.7 31.1 36.6

100 94 83 78 73

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The three-space-domain model problem for the third computation series was obtained from the previous two-space-domain problem by means of splitting the ﬁrst space-domain into columns. Three space-domains. In the ﬁrst and the second ones the number of rows was 40, homogenous by the radius, the number of columns was 100, homogenous by the angle. In the third space-domain the number of rows was 40, homogenous by the radius, and the number of columns was 200, homogenous by the angle. The number of energy groups and other parameters coincide with those of the one-space-domain problem. The incoming ﬂow on the outer edges of the ﬁrst and the second space-domains, corresponding the temperature 1, was taken as the boundary condition. The speedup and the parallelization eﬃciency, obtained in the course of numerical research for diﬀerent numbers of para-domains and group-domains, are presented in Table 6. Table 6. The speedup (Sp) and the parallelization eﬃciency (E) in the third test computation series Number of Processors 1 10 25 40 50

Number of GroupDomains

Number of ParaDomains

Sp

E, %

1 1 1 4 1

1 10 25 10 50

1 7.5 20.3 24.9 34.9

100 75 81 62 70

Nowadays the most perfect parallelization algorithm is the one where the matrix decomposition of the initial system is used. Unlike splitting the system only into columns, here the space-domain was cut both into columns and rows. The main principles of the algorithm of minor block parallelization were as follows: • Application of spatial decomposition of the space-domain into para- domains, the space-domain was split into columns and rows; • The system of grid equations was solved independently by the angle variable; • Each para-domain of the current direction was solved with inner boundary conditions, computed at the current iteration, which allows to keep the solution precision and does not increase the total number of iterations comparing with the step-by-step method; • Complete integration of inter-processor data exchange with the computation, achieved by the application of asynchronous receive/send operations.

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A one-space-domain spherically symmetrical transport problem in the one-group iteration was used as a test problem for the investigations of the eﬃciency of the matrix decomposition method for solving 2D transport equations. The spatial grid comprised 1200 rows and 1200 columns. The problem was computed on the angular grid with 6, 8, 12 and 14 intervals by µ for the analysis of the inﬂuence of the interval numbers on the parallelization eﬃciency. There were 5 computation steps with 6 iterations in each one. The number of processors was selected so that the computation was balanced, that is the number of calculated points was equal on each processor at a time. The speedup and the parallelization eﬃciency are presented in Tables 7–10. Table 7. Parallelization eﬃciency at µ = 6 Np 1 50

Time 3381.12 118

Sp

En, %

1.00 29

100.00 57.01

Table 8. Parallelization eﬃciency at µ = 8 Np

Time

1 50

5402.21 160

Sp

En, %

1.00 34

100.00 67.05

Table 9. Parallelization eﬃciency at µ = 12 Np

Time

1 50

10717.53 274

Sp

En, %

1.00 39

100.00 78.23

Table 10. Parallelization eﬃciency at µ = 14 Np

Time

1 50

13882.33 350

Sp

En, %

1.00 40

100.00 79.28

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The following conclusion can be drawn out basing on the presented material: • Application of the combined parallel algorithm, that is the combination of matrix geometrical and energy decomposition, is the most eﬃcient one for solving the 2D multi-group transport equations, as extra possibilities for balancing the computation load on processors occur.

Part V

Neutron Transport

Parallel Deterministic Neutron Transport with AMR C.J. Clouse Lawrence Livermore National Laboratory, Livermore CA [email protected]

AMTRAN, a one, two and three dimensional Sn neutron transport code with adaptive mesh reﬁnement (AMR) has been parallelized with MPI over spatial domains and energy groups and with threads over angles. Block reﬁned AMR is used with linear ﬁnite element representations for the ﬂuxes, which are node centered. AMR requirements are determined by minimum mean free path calculations throughout the problem and can provide an order of magnitude or more reduction in zoning requirements for the same level of accuracy, compared to a uniformly zoned problem.

1 Introduction AMTRAN, a two and three dimensional Sn neutron transport code designed to run eﬀectively on large parallel machines that have both distributed memory and shared memory parallelism, ﬁrst began development in 1995 under an industrial partnership agreement with several oil well logging companies and LLNL. Due to shortened time lines and dwindling DOE funding in support of industrial partnership agreements, AMTRAN never became a major contributor in oil well logging calculations. However, it was recognized as an excellent R&D test bed for trying out new parallel algorithms and AMR techniques as applied to the Boltzmann equation. Development has continued over the years and, although the code and its algorithms have been presented at several conferences [1, 2] this is the ﬁrst publication of the basic code and it’s techniques. At the time development began in 1995, the notion of applying AMR techniques to neutron transport problems was unexplored. Since that time, a number of other AMR neutron transport codes have been developed [3,4]. AMTRAN, however, remains unique in its combination of degrees of parallelism (MPI in space and energy and threading in angle) and its use of ﬁnite element node centered ﬂuxes (the other referenced AMR codes use face or zone centered ﬂuxes). As in the case of hydrodynamics, where spatial AMR was ﬁrst developed, many transport applications have widely varying resolution needs within the same application. Stability and accuracy considerations require that spatial zoning be able to resolve length scales less than a neutron mean-free-path for most commonly used algorithms. This can be on the order of a millimeter in ﬁssionable materials of nominal density to a meter

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or more in air. An example of particular interest in our work is the interrogation of cargo containers for ﬁssionable material using a 14 MeV neutron source. The cargo container could be a semi-tractor trailer with the neutron source situated on one side of the trailer and a detector located on the opposite side. The goal is to be able to match the detector signal with known conﬁgurations of various types of ﬁssionable materials. In this example, the distance between source and detector could be many meters; the material throughout most of which is probably air or some other material that is relatively transparent to neutrons, but the ﬁssionable target would require good spatial resolution. Spatial AMR allows us to get the needed resolution in the target without making the overall calculation unwieldy.

2 Code Overview AMTRAN operates in 2D cylindrical and 3D cartesian geometries. Node centered ﬂuxes are represented with continuous linear ﬁnite elements, similar to the methods employed by Greenbaum and Ferguson [5]. Angular discretization in 2D and 3D is with standard discrete ordinates and, therefore, requires half angle approximations to maintain acceptable conditions on the ordinates when ﬁnite diﬀerencing the angular derivative in cylindrical coordinates (see [6] for a discussion of this topic). In 1D, a quadratic ﬁnite element approximation for the angular unknown has been implemented and provides signiﬁcant computational savings over standard diﬀerencing (see [7] for a detailed discussion). AMTRAN has several simple internal generators: nested spheres with point to point linearly interpolated densities, nested cylinders with constant densities and constant density cartesian blocks. It is also capable of reading COG [9] input and using the geometry generator routines in COG to construct a mesh. 2.1 Production of AMR Blocks The AMR algorithm in AMTRAN is block based and thus produces a hexahedral (quadrilaterals in 2D) block decomposition of the problem domain. Like previous AMR work in the ﬁeld of hydrodynamics, e.g. Berger and Colella [8], the zone size in each direction is halved for each increase in level of reﬁnement, unlike most hydrodynamic AMR techniques, though, no level nesting is required, i.e. it is possible to have adjacent blocks diﬀer by any power of 2 in zoning, as illustrated in Fig. 1. Uniform zoning within a block allows fast and eﬃcient computation of the transport equation. The reﬁnement criteria is based on neutron mean free path considerations, ag λg (1) h < min g

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501

Fig. 1. 2D example where dashed lines indicate block boundaries

where λg is the minimum neutron mean free path by energy group, h is the zonal width and ag is a user deﬁned multiplier, which can vary with energy group. Blocks are created by beginning at the ﬁnest level and tagging all zones that need to remain at that level. The tagged zones are boxed up using a “smart bisection” algorithm that can be brieﬂy described as follows for 3D with the obvious extension to 2D. Count up all the tagged zones in each 2D plane of the problem. Let yi represent the number of tagged zones in plane i, then deﬁne fi to be the discrete function fi = yi . The splitting plan, i = isplit, is chosen such that 2 d d2 (2) isplit = max 2 f (xi+1 ) − 2 f (xi ) dx dx where the second derivatives are deﬁned using a standard central diﬀerence formula, d2 f (xi ) ≡ f (xi+1 ) − 2f (xi ) + f (xi−1 ) (3) dx2 A block is subject to further splitting until it satisﬁes one of two conditions: 1) it no longer contains any tagged zones, in which case it is discarded, or 2) it satisﬁes the following condition, number of tagged zones ≥ irattag total number of zones where irattag is a user speciﬁed eﬃciency ratio. Generally, more and smaller blocks will be produced as the value of irattag is increased. 2.2 Sweeping the Mesh The time-dependent transport equation can be written as follows,

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→ 1 ∂ g ˆ • ∇ Ψ g + σtg Ψ g = Ψm + Ω (2l + 1)Pl σlg g φgl + υσfg φ + q (4) m m vg ∂t l

g

where g Ψm = angular ﬂux for energy group g and angle m, Pl = Legendre polynomial term l, σtg = total cross section for energy group g, σlg g = the lth component of a Legendre polynomial expansion of the differential scattering cross section from group g to group g, vg = the neutron velocity for group g. g φgl = 12 m Pl (µ)Ψm where m is the discrete angular index. g υσf ϕ = represents the production of the scalar ﬂux into group g, and q represents a possible, externally driven source term.

AMTRAN can solve (4) as either a ﬁxed source calculation (where q is non-zero) or the usual k eigenvalue calculation, where k is a multiplier on the ﬁssion source term, or α eigenvalue calculation where the time dependent term is included in (4) and the time dependence is modeled as eαt . Equation (4) is solved through standard source iteration and angular sweeps in which the source terms on the right hand side of (4) are evaluated using the previous iterates values for the ﬂuxes. Then, with the value of S determined, inversion of the sweeping term on the left hand side of (4) is accomplished by sweeping through the mesh in the direction of neutron ﬂow; one sweep for each unique combination of direction and energy group. This downwind sweeping is complicated by block decomposition on a domain decomposed mesh in which diﬀerent domains reside on diﬀerent processors. In order to avoid idling processors, AMTRAN’s default domain decomposition is limited to 8 domains (4 domains in 2D). By ensuring that each domain includes one of the corners of the problem, all domains can immediately begin sweeping. Figure 2 illustrates a simple 2D example assuming four spatial domains and

Fig. 2. Dashed lines indicate domain (and block) boundaries. Solid lines indicate block boundaries

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12 angles. Each block designated “A” can be swept immediately by any one of the three angles that originate from its corner. After any “A” block is swept, its neighboring “B” blocks would have suﬃcient boundary information to begin their sweep, followed by the “C” blocks, etc. A domain continues to sweep blocks until no more blocks can be swept without receiving information from neighboring domains, at which point it sends out all of its downwind boundary information to the necessary neighboring domains and waits to receive upwind boundary information from any domains. AMTRAN assigns an estimated weight to each zone in the generator mesh based on the mean free path in that zone. This allows AMTRAN to estimate where to place domain boundaries such that each domain has roughly equal weight and, therefore, will have roughly equivalent zone counts after the AMR blocks are made. If each domain has roughly equivalent zone counts, then each will ﬁnish their sweeps at about the same time, pass information to and receive information from neighboring domains, and continue sweeping with little or no idle time. If there are reﬂecting boundaries in the problem, then the corners that lie on reﬂecting boundaries will begin their sweeps with “old” boundary information from the previous iteration. (Our deﬁnition of the term “iteration” is comparable to the standard textbook deﬁnition of an “outer iteration”, which implies all angles have been swept through the entire mesh.) This causes some degradation in the rate of convergence, but the fractional increase in the number of iterations it takes to converge is usually substantially smaller than the relative speedup achieved by simultaneously beginning sweeps from all corners. For example, in 3D critical sphere calculations with one reﬂection plane, the number of iterations required to achieve convergence is about 20% more than the number of iterations required by ordering the sweeps such that reﬂecting boundaries are not swept until incoming sweep information is received, but the calculation will run roughly twice as slow by ordering the sweeps since half the processors will be idle at any given time. Because eight spatial domains (four in 2D) can be very limiting when attempting to scale up to thousands of processors, recent work in AMTRAN has focused on eﬃcient use of processors with more than eight spatial domains. Basically, the idea for achieving high eﬃciency is through the use of domain overloading techniques. We have created the construct of a domain master which represents a unique collection of domains that may or may not be located contiguously in space. Domains are assigned to domain masters in such a way as to keep the domain master busy as much of the time as possible. Systematic algorithms have been worked out that asymptotically approach 100% theoretical eﬃciency as the number of domains per domain master is increased. The diﬀerence between theoretical and actual eﬃciency is dependent on how well the code is able to produce domains that are roughly equal in computational work, since the algorithm assumes equal weight domains. Details of the algorithm have been presented at an international conference [10], and will be outlined in a journal article in the near future.

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Fig. 3. A single zone from a coarse block that borders a ﬁner block along it’s x-face

2.3 Block Interfaces As the sweeps proceed from block to block, three scenarios can occur at block boundaries: (1) no change in zoning, (2) go from a coarser mesh to a ﬁner mesh, (3) go from a ﬁner mesh to a coarser mesh. The ﬁrst scenario obviously requires no special treatment. The second scenario can be dealt with in a straight forward fashion through bilinear interpolation of the coarse grid ﬂuxes onto the ﬁne mesh. This is consistent with the linear ﬁnite element representation of the ﬂuxes at the nodes. Unfortunately, the ﬁnite element representation of the ﬂuxes at the nodes does not provide an obvious unique solution to the third scenario, which is illustrated in Fig. 3. Over time, three diﬀerent methods evolved in the code for treating scenario 3. The original method, referred to as the pseudo-source method, is constrained by two criteria: ﬂuxes of nodes at the same physical location on two diﬀerent mesh should have the same value and ﬂux must be conserved across the interface. To satisfy the ﬁrst criterion, we require that ﬂuxes at nodes 1, 5, 21 and 25 in Fig. 4 have the same value on the coarse and ﬁne blocks. If we integrate (1) over a zone and focus on just the streaming term for the speciﬁc example

Fig. 4. Y-z interface of coarse to ﬁne zone. Fine zone nodes are numbered 1 to 25. Coarse zone nodes are nodes 1, 5, 21 and 25

Parallel Deterministic Neutron Transport with AMR

505

shown in Figs. 3 and 4 with neutrons traveling in the +x direction, then the ﬂux leaving the ﬁne zones overlapping the coarse zone can be written as, y0+∆y z0+∆z 25 wyi wzi Ψi dz dy i=1

y0

(5a)

z0

where, wyi and wzi are the y and z components of the linear ﬁnite element weight functions at node 1, deﬁned as wyi = 1 − wyi = 1 − wyi = 0

(yi −y) dyf (y−yi ) dyf

for yi−1 ≥ y ≥ yi ,

wzi = 1 −

for yi+1 ≥ y ≥ yi , and,

wzi = 1 − wyi = 0

(zi −z) dzf (z−zi ) dzf

for zi−1 ≥ z ≥ zi for zi+1 ≥ z ≥ zi elsewhere,

and dyf = dzf = 1/4∆y where ∆y = ∆z is the zone size on the coarse grid and the coordinates of node 1 are given by (y0 , z0 ). Likewise, the ﬂux entering the coarse zone can be expressed as y0+∆y z0+∆z

c c wyi wzi Ψi dz dy

i=1,5,21,25

y0

(5b)

z0

c c where, wyi and wzi are the y and z components of the linear ﬁnite element weight functions on the coarse zone at node 1, deﬁned as

(yi − y) ∆y (y − yi ) =1− ∆y

(zi − z) ∆z (z − zi ) =1− ∆z

c =1− wyi

for yi ≥ y,

c wzi =1−

for zi ≥ z ,

c wyi

for y ≥ yi ,

c wzi

for z ≥ zi .

At the time boundary information is received, the diﬀerence between (5a) and (5b) is calculated and stored as an additional zone centered source term that is included in the solution of (4) and, thus, from the perspective of nodes downwind from the boundary, the total ﬂux crossing from the ﬁne mesh to the coarse mesh has been accounted for through the inclusion of an additional source term, which we will refer to as the pseudo-source term. In a similar fashion, the diﬀerence between the coarse and ﬁne mesh for the second term on the left hand side of (4), the absorption term, is also accumulated into the pseudo-source term. The second method is referred to as the marching method and is illustrated in Fig. 5. In this method, the value of the most downwind node (node A in Fig. 5) is copied to the corresponding physical node on the neighboring coarse grid (node F in Fig. 5). The value of node G is then simply determined by ﬂux conservation across the interface between nodes F and G. The value of node H is then determined by ﬂux conservation across the interface between node G and H, etc.

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C.J. Clouse E

H

D G

C B A

F Direction of neutron flow

Fig. 5. Marching Method

Overlapping face of Block II

Node A

Node C

Node B

Block I

Fig. 6. Area Weighting Method

The third method is referred to as the area weighting method and is illustrated with a 3D example in Fig. 6. In our example, we have a coarse block (block I) partially overlapped by a ﬁne block (block II) on the top face. For neutrons incident on the upper right front corner, node C would not only receive a ﬂux contribution from block II (indicated by the shaded area) but also from two other blocks that overlap the two faces that are orthogonal to the shaded face. The total contribution would be the area weighted sum of the three incident faces. A corner node, like node C, can be shared by up to 8 blocks (in 3D), each of which may have a diﬀerent ﬂux value for that node. Only in the case where all blocks sharing a node are at the same AMR level are we guaranteed that all blocks see the same physical value. In the case of node A, the lightly shaded area would contribute to the area weighted nodal ﬂux, but if node A were located on the left edge of block I, then the lightly

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507

shaded area would either be located on a separate block or, in the case of block I lying on the left edge of the problem, there would be no neighboring block. In either case, the lightly shaded area would not contribute to node A’s value. Likewise, if the direction of neutron ﬂow were from back to front, only the shaded face, representing block II’s contribution, would contribute to node C’s value. Thus, one can see that the area and number of faces contributing to a nodal ﬂux value is dependent on the direction of neutron ﬂow. Method 1 is diﬃcult to implement in time-dependent problems and method 2 is prone to instabilities. Thus, the default method used in AMTRAN is method 3. 2.4 Energy Group Parallelism Distributed memory parallelism (i.e. the number of MPI processes spawned) is equal to 8 (or 4 in 2D) x the number of energy groups per process. If the number of processes is not evenly divisible by 8 (or 4 in 2D) then the number of energy groups on a processor will vary by processor. In the case of maximum parallelization, the user runs with one energy group per process. Typically, in serial Sn codes, energy groups are swept sequentially from highest to lowest where the source term is updated after each energy group sweep so that eﬀects from higher groups are immediately included in the lower group sweeps, thus providing Gauss-Seidel like convergence on the iteration. Therefore, in the case of no upscatter, the solution would converge after a single iteration of all groups. In the case of maximum parallelization, which is frequently the case in a typical AMTRAN calculation, all energy groups are being solved simultaneously, using source terms calculated from the previous iterate ﬂuxes and, therefore, the iterative technique is Jacobi-like rather than Gauss-Seidel. One might expect a Jacobi solution to take more iterations to converge than Gauss-Seidel however, in practice, what we observed for problems dominated by ﬁssion, and thus have large upscatter components, there appears to be a break-even point at about 16 energy groups where, for problems with fewer than 16 energy groups, a Jacobi solution converges in fewer iterations and with more than 16 energy groups, a Gauss-Seidel solution converges in fewer iterations. The diﬀerence was not large, however, varying by about +/−20% from 6 to 24 energy groups. In AMTRAN, each process calculates it’s energy group(s) contribution to the source term of each energy group in the problem. At this point, two diﬀerent methods can be employed for communicating the results to the other processes. The ﬁrst method is a tree-summing algorithm illustrated in Fig. 5 for a 4 group calculation with one energy group per process. Many vendor implementations of MPI− Allreduce implement essentially the same algorithm, however, we have seen MPI− Allreduce performance on some machines to be substantially worse than our implementation of the above algorithm and, therefore, we do not rely on the MPI− Allreduce call for the summing of the sources since it can be a signiﬁcant fraction of the run time of a calculation.

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Sum Sources Process 1

Process 1

Sum Sources Process 1

Energy group 1 Process 2 Energy group 2 Process 3

Process 3

Broadcast results

Energy group 3 Process 4 Energy group 4

Fig. 7. Tree-summing algorithm

A second method takes advantage of the fact that a process only needs to know what the source contributions are to it’s energy group(s). Thus, after a process computes the contribution of its energy group(s) to all the others, it sends individual messages to each process containing the contribution to that process’ energy group(s). This is illustrated in Fig. 8. The method illustrated in Fig. 7 requires 3(N-1) communications while that illustrated in Fig. 8 requires N(N-1) communications, where N is the number of MPI processes per domain. The messages in the second method, though, are much smaller and less synchronized than those in the ﬁrst method and, as a result, provide about a factor of 2 reduction in wall clock time for a 16 energy group calculation on the IBM ASCI Paciﬁc Blue SP-2 machine at LLNL.

3 Numerical Results As a simple numerical demonstration of the eﬀectiveness of spatial AMR, the two-dimensional rod test case, as deﬁned in [4], will be used. This problem consists of a cylindrical rod of 235 U surrounded by vacuum with a density that decreases linearly from 66.71 g/cm3 at the center plane to 20.09 g/cm3 at the ends. All problems were run on LLNL’s Thunder machine, which is a 1024 node (four processors per node), 1.4 GHz Itanium machine. Aussourd4 speciﬁes the ﬁnest level zoning to be 1 mm and gives results for up to 8 levels of AMR, but states that eﬃciency gains beyond 3 levels are negligible and tend to degrade accuracy. In fact, since the diﬀerence in density between the peak value and the ends is only a little more than a factor of three and the neutron mean free path varies linearly with the density, allowing more than three levels violates AMTRAN’s default zoning criteria, since three levels of reﬁnement already represents a factor of four diﬀerence in zone size for

Parallel Deterministic Neutron Transport with AMR Process 1

Process 1

Energy group 1

Energy group 1

Process 2

Process 2

Energy group 2

Energy group 2

Process 3

Process 3

Energy group 3

Energy group 3

Process 4

Process 4

Energy group 4

Energy group 4

509

Fig. 8. Arrows for process 1 are labeled. Arrows for other processes would be labeled in an analogous fashion

R (in cm)

each direction. Figure 9 shows the zoning used by AMTRAN with three levels of reﬁnement. Table 1 shows the results of several serial variations of the calculations relative to a serial baseline calculation consisting of a single block, uniformly zoned with 1 mm zoning. The keﬀ of the baseline calculation was 1.98480, which diﬀers slightly from [4]. This isn’t surprising since the nuclear database and energy group resolution were not speciﬁed, so a direct comparison could not be made. The relative error in Table 1 is deﬁned as: 1 − keff (6) Error = abs 1 − baseline 1 − keff

Z (in cm) Fig. 9.

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C.J. Clouse Table 1. Mesh

Uniform, single block (baseline calculation) Uniform, multi-block 3 level AMR Ref. 4 with 3 level AMR

Relative Error

Relative compute time

Number of zones

0 0 2.0e-5 4.1e-5

1 0.95 0.20 0.30

20800 20800 5200 5760

and the relative compute time is just the ratio of the time for the calculation relative to the baseline calculation: time Compute time = (7) timebaseline The major diﬀerence between Aussourd’s [4] AMR method and our application is our block based approach versus his tree based hierarchy. As he points out, the advantage to a block based approach is it is more amenable to spatial parallelism, but is less eﬃcient in reducing zone counts. A tree based algorithm is better able to capture irregularly shaped gradients. Our experience, however, has been that calculations generally run more eﬃciently with liberal settings for the boxing eﬃciency; i.e. it is better to minimize the number of blocks at the expense of running with more total zones. This assumes, of course, that the diﬀerence in zone count is not too large; generally no more than about 20%. If the diﬀerence is signiﬁcantly more that 20%, it is probably worthwhile to increase the boxing eﬃciency. Aussourd [4] reports roughly 40% overhead associated with the AMR logic for this test problem. As can be seen from Table 1, we observe little, if any, overhead. In fact, the multi-block logic, which is the major cost associated with the AMR overhead of our block based approach, actually experiences a 5% reduction in run time for a uniform calculation relative to a single block uniform calculation. This is most likely do to improved cache performance of the multi-block approach since, if a block is small enough for all the unknowns to ﬁt into cache, the sweeps can be performed without cache swapping. We have seen this eﬀect in the past and, in fact, added an input variable which allows users control over the maximum size of a block so calculations can be tuned for diﬀerent architectures. This super-linear speedup is also seen in the three level AMR calculation, which runs 5 times faster than the baseline calculation despite the fact that the zone count is only reduced by a factor of 4. It should be noted, though, that this particular test problem is ideally suited for a block based AMR approach, since the gradients are planar. Figure 10 shows the relative speed improvement for the rod test problem as a function of processor count. All points were run with the three level AMR

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Fig. 10. Relative speedup as a function of the number of processors for the rod test problem (ﬁxed problem size)

version of the problem, giving a total of 5200 zones with 16 energy groups and S10 quadrature. The 4, 8 and 16 processor runs used 4 spatial domains. The 8 processor run has two processors assigned per domain, and, therefore, would be running with 8 energy groups per processor. The 16 processor run has 4 processors assigned per domain, thus giving 4 energy groups per processor. Since there are two reﬂecting planes in this problem (the axis and z = 0), the domain decomposed problems take more iterations (∼20% increase) to converge than does the serial calculation, so the timings have been normalized to the iteration count of the serial calculation. As can be seen from the plot, we achieve about a 12.8X speedup with 16 processors, giving an overall parallel eﬃciency of 80%. The small problem size limits the degree of parallelism we can employ for this particular test case (the 16 processor run completed 116 iterations in about 7 seconds). Two dimensional calculations are commonly run that exceed 50,000 zones with 32 or more energy groups. A large three dimensional calculation can have several million zones. These kinds of calculations require hundreds to thousands of processors.

4 Future Work Much of our recent eﬀort has been focused on the ability to reﬁne in direction. Problems such as the neutron interrogation of a cargo container, mentioned in Sect. 1, not only require spatial AMR because of the large problem dimensions, but one is generally only interested in a narrow region of directional phase space; basically the cone of angles, originating from a 14 MeV source,

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that passes through the container and strikes a detector on the far side of the container. This is the classic source/detector problem of trying to get adequate resolution at a detector that is far from the source; ray eﬀects can be severe. We have been working on techniques that allow local adaptive reﬁnement of the directional set of angles and expect to publish our results in the near future. We also hope to extend the quadratic ﬁnite element in angle [7] work to multi-dimensions to see if the substantial improvements in convergence as a function of the number of angles holds for higher dimensions.

References 1. Clouse, C.: “Parallel 3D Neutronics on an AMR Grid”, Fifth Joint Russian American Conference on Computational Mathematics, Sept. 22, 1997. 2. Clouse, C.: “Parallel Deterministic Neutron Transport with Adaptive Mesh Reﬁnement”, 16th International Conference on Transport Theory, Georgia Tech University, Atlanta, GA, May 14, 1999. 3. Baker R.: A Block Adaptive Mesh Reﬁnement Algorithm for the Neutral Particle Transport Equation. Nuc. Sci. & Eng., 141, 1–12 (2002) 4. Aussourd, C.: Styx: A Multidimensional AMR SN Scheme. Nuc. Sci. & Eng., 143, 281–290 (2003) 5. Greenbaum, A. and J. Ferguson: A Petrov-Galerkin Finite Element Method for Solving the Neutron Transport Equation. Journal of Comp. Phys., 64, 97–111 (1986) 6. Lewis, E. and W. Miller: Computational Methods of Neutron Transport, Wiley and Sons, New York, 137–140 (1984) 7. Tolar, R. and J. Ferguson.: Quadratic Finite Element Method for 1D Deterministic Transport. Trans. Am. Nucl. Soc., 90 (2004) 8. Berger, M. and P. Collela: Local Adaptive Mesh Reﬁnement for Shock Hydrodynamics. Journal of Comp. Phys., 82, 64–84 (1989) 9. Buck, R., E. Lent, T. Wilcox and S. Hadjimarkos: COG User’s Manual, ﬁfth edition. Lawrence Livermore National Laboratory. (2002) 10. Compton, J. and C. Clouse: Domain Decomposition and Load Balancing in the AMTRAN Neutron Transport Code, 15th Annual Conference on Domain Decomposition Methods, Berlin, July (2003)

An Overview of Neutron Transport Problems and Simulation Techniques Edward W. Larsen Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, MI 48103-2109 [email protected]

1 Introduction We brieﬂy summarize (i) the general characteristics of neutron (and photon) transport processes relevant to nuclear reactor problems, (ii) the nature of calculation techniques for these problems, and (iii) current areas of research aimed at improving the accuracy and eﬃciency of these techniques.

2 Physical and Mathematical Basics The problem of simulating the interaction of neutrons and photons with matter has been an important calculational problem since the beginning of “the nuclear era” – circa 1950. In large part, this is because nuclear reactors now generate a substantial fraction of the electricity used by humankind. In this brief review, we discuss (i) the physical and mathematical aspects of neutron transport processes relevant to nuclear reactor problems, and (ii) the main computational techniques that have been developed to solve these problems. A very large literature on this subject has been published ([1–25] are a few main sources), and we can do little but scratch the surface here. Nonetheless, we hope to provide a useful overview of this interesting and challenging ﬁeld. In typical neutron transport problems, a physical system V and a source of neutrons are speciﬁed. The source can be internal (within the system), or external (e.g. a beam of neutrons incident on the outer surface ∂V of the system). Individual neutrons, after being born, stream through the system with ﬁxed directions of ﬂight and energies until they interact with nuclei or leak out of the system. An interaction with a nucleus results in one of three distinct types of events: (i) a capture event, in which the neutron is captured by the nucleus, (ii) a scattering event, in which the neutron exchanges some of its kinetic energy with the nucleus and departs with a diﬀerent energy and direction of ﬂight, and (iv) a ﬁssion event, in which the nucleus splits into two daughter nuclei, and one or more high-energy ﬁssion neutrons and

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other radiation are emitted. Each neutron that is emitted in a scattering or ﬁssion event is subject to the same possible physical interactions as the parent neutron. The general neutron transport problem is: given the physical system and the neutron sources, determine the neutron ﬂux at all points within the system. For shielding problems, the neutron and photon ﬂuxes must both be calculated. (Photons, which are a byproduct of inelastic scattering and ﬁssion events, undergo a similar transport process as neutrons, except that photons do not induce ﬁssion.) For steady-state problems, the mathematical equations that describe the neutron transport process consist of the linear Boltzmann equation for the angular ﬂux: Ω · ∇ψ(r, Ω, E) + Σt (r, E)ψ(r, Ω, E) ∞ = Σs (r, Ω · Ω, E → E)ψ(r, Ω , E )dΩ dE 0 4π χ(r, E) ∞ + νΣf (r, E )ψ(r, Ω , E )dΩ dE 4π 0 4π Q(r, E) , r ∈ V , Ω ∈ 4π , 0 < E < ∞ , + 4π

(1a)

and a boundary condition that prescribes the ﬂux incident on the outer boundary of the system: ψ(r, Ω, E) = ψ b (r, Ω, E) , r ∈ ∂V , Ω · n < 0 , 0 < E < ∞ .

(1b)

The independent variables in (1) are the 3-D spatial variable r = (x, y, z), the 2-D angular or direction-of-ﬂight variable Ω (a unit vector), and energy E. Thus, unless spatial symmetries exist that reduce the number of independent variables, phase space is 6-dimensional. The to-be-determined angular ﬂux ψ is deﬁned by: 1 ψ(r, Ω, E)dV dΩdE = the number of neutrons located in dV about r, v traveling in a direction in dΩ about Ω with an energy in dE about E. (2) " where v = 2E/m is the neutron speed. (Thus, ψ/v = N is the neutron density.) The remaining terms in (1) are speciﬁed cross sections or sources. Thus, Σt (r, E) is the total cross section, deﬁned by: Σt (r, E)ds = probability that a neutron at r with energy E will experience an interaction with a nucleus while traveling a distance ds;

(3)

Σs (r, Ω · Ω, E → E) is the diﬀerential scattering cross section, deﬁned by:

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Σs (r,Ω · Ω, E → E)dsdΩdE = probability that a neutron at r , traveling with direction Ω and energy E , will scatter into dΩ about Ω and dE about E while traveling a distance ds;

(4)

Σf (r, E) is the ﬁssion cross section, deﬁned by: Σt (r, E)ds = probability that a neutron at r with energy E will initiate a ﬁssion event while traveling a distance ds;

(5)

ν is the mean number of ﬁssion neutrons emitted in a ﬁssion event, χ(r, E) is the ﬁssion spectrum, deﬁned by: χ(r, E)dE = the probability that a ﬁssion neutron, emitted at r, will have energy in dE about E;

(6)

Q(r, E) is an isotropic interior source: Q(r, E)dV dE = the rate at which neutrons are isotropically emitted in dV about r and dE about E;

(7)

and ψ b (r, Ω, E) is the prescribed incident ﬂux, which for r ∈ ∂V , n = the unit outer normal vector to ∂V at S, Ω · n < 0, and dS an increment of area on ∂V at r, is deﬁned by: |Ω · n|ψ b (r, Ω, E)dS dΩdE = the rate at which neutrons, traveling in dΩ about Ω and dE about E, ﬂow into V through dS. (8) Equations (1) describe a steady-state neutron transport problem with no photons. If photons were included, a second transport equation for the photon angular ﬂux would be required; this equation would not contain the ﬁssion term on the right side of (1a), but it would contain extra source terms, depending linearly on the neutron ﬂux ψ, that describe the production of photons due to neutron ﬁssion and inelastic scattering. Equations (1) are linear because neutrons are assumed to interact only with nuclei, not with each other. In a cm3 of a typical reactor core, there are roughly 1023 nuclei, and perhaps 1010 (plus or minus a few orders of magnitude) neutrons. Therefore, neutron-neutron interactions, which would be nonlinear, are suﬃciently rare that they can be neglected. If time-dependent problems were to be considered, extra equations for the neutron precursors – ﬁssion neutrons that are emitted by radioactive decay several seconds after the initiating ﬁssion event took place – would have to be formulated and coupled to the time-dependent version of (1a). The study of time-dependent reactor problems forms the specialized ﬁeld of reactor kinetics.

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In many reactor design and safety problems, the space- and time-dependent temperature of the reactor, T (r, t) must be considered as an extra unknown. This is because cross sections are temperature-dependent; when temperature rises, Doppler-broadening causes narrow absorption resonances to widen, leading to a small but signiﬁcant increase in the ability of the reactor to absorb neutrons. When T is included, an extra equation for T must be formulated, and the cross sections in (1a) depend nonlinearly on T . Σ γ (E,T) T1 T1 < T2 T2 E

Fig. 1. Doppler-Broadening of Capture Cross Section

In typical neutron transport problems, the total cross section and mean free path satisfy 0.1 cm−1 < Σt < 10.0 cm−1 , 1 = mean free path < 10.0 cm , 0.1 cm < λ = Σt

(9a) (9b)

with nominal values of Σt ≈ 1 cm−1 and λ ≈ 1 cm. Thus, neutron mean free paths are generally orders of magnitude greater than electron or ion mean free paths; this is because neutrons, which have no electrical charge, only interact with atomic nuclei. The mean scattering cosine µ0 for neutron low-energy elastic scattering is: 2 , (10) µ0 = 3A where A is the mass number (number of neutrons plus protons) of the scattering nucleus. Thus, neutron scattering is generally much more isotropic than typical electron scattering, which tends to be highly forward-peaked, with µ0 ≈ 1. When fast neutrons scatter oﬀ nuclei, the neutrons typically exchange a signiﬁcant fraction of their kinetic energy with the nuclei. For example, a 1.0 Mev neutron scattering in hydrogen (uranium) requires about 30 (2500) scattering events to slow down to thermal (10−2 ev) energies. (The lighter the nucleus, the greater the energy exchange.) Electrons generally require orders of magnitude more scattering events to undergo comparable energy losses.

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Assembly

Overall, the relatively large mean free path, large scattering angle, and large energy loss of neutrons (compared with electrons) make (1) for neutrons considerably more amenable to direct numerical simulation than the corresponding equations for electrons. (Numerical methods for electron transport problems nearly always use approximations necessitated by the extremely small mean free path, scattering-angle, and energy-loss per collision.)

Shield

ln ψ

Core 4m

1 m2 m

1m

Fig. 2. Nuclear Reactor Schematic (Many Details Not Shown)

Figure 2 – a cartoon of a slice through a nuclear reactor – indicates a central cylindrical core, consisting of a large array of long fuel assemblies that contain the reactor fuel. (One fuel assembly is depicted.) Surrounding the core is a reﬂector, and then a shield. (Only the shield is depicted.) Not shown in the ﬁgure is the shielding on the top and bottom of the core. Also not shown are the multitude of coolant channels passing through the core; cold ﬂuid entering the core becomes hot as it passes through the core; when it exits, the hot ﬂuid powers steam turbines, which then generate electricity. In general, the reactor core contains a great deal of small-scale mechanical structure; the reﬂector and shield, however, are basically homogeneous. The fuel assemblies in reactor cores are generally arranged to maximize eﬃciency and to ensure that the neutron ﬂux is relatively “ﬂat” across the core. (However, due to the changes in cross sections from one material to another, important space-dependent eﬀects deﬁnitely occur.) In the neutron shield, surrounding the core, the neutron and photon density fall oﬀ – roughly exponentially – by about 10 orders of magnitude from the interior wall of the shield to the exterior wall. This fall-oﬀ (log scale) is depicted in Fig. 2.

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Equations (1) describe a ﬁxed-source problem for a physical system V . Practical nuclear reactor neutron transport problems generally occur in three categories: 1. Eigenvalue Calculations. Here the physical system V is the entire reactor, the internal and boundary sources are set to zero, and the eigenvalue k (the reactor criticality) is introduced in the denominator of the ﬁssion term. Equations (1) become: Ω · ∇ψ(r, Ω, E) + Σt (r, E)ψ(r, Ω, E) ∞ = Σs (r, Ω · Ω, E → E)ψ(r, Ω , E )dΩ dE 4π 0 1 χ(r, E) ∞ + νΣf (r, E )ψ(r, Ω , E )dΩ dE , k 4π 0 4π r ∈ V , Ω ∈ 4π , 0 < E < ∞ ,

(11a)

ψ(r, Ω, E) = 0 , r ∈ ∂V , Ω · n < 0 , 0 < E < ∞ .

(11b)

The purpose of k is to adjust the amplitude of the ﬁssion process (which adds new neutrons to the system), so that this process and the capture and leakage processes (which delete neutrons from the system) exactly balance, permitting a steady-state solution ψ to exist. The problem is to determine a positive value of the eigenvalue k such that (11) have a positive eigenfunction solution ψ. If k > 1, the ﬁssion process is suppressed to obtain a steady-state solution, and the reactor is said to be supercritical. If k < 1, the ﬁssion process is enhanced to obtain a steady-state solution, and the reactor is said to be subcritical. If k = 1, the the reactor is critical. A power reactor, operating at steady state, is critical (k = 1). To increase the power output of a critical reactor, control rods (which absorb neutrons) are slightly withdrawn, causing k to become slightly greater than 1, thus causing the neutron population within the reactor to slowly grow. Conversely, to decrease the power output of a critical reactor, control rods are slightly inserted, causing k to become slightly less than 1, thus causing the neutron population within the reactor to slowly decline. Determining the criticality k of a reactor is one of the fundamental problems in reactor design and operation. 2. Assembly Calculations. Here the physical system V is a single reactor assembly, all internal sources are set to zero, the eigenvalue k is introduced, and reﬂecting (symmetry) boundary conditions are imposed on the outer edges of the assembly to simulate an inﬁnite periodic system. If Ω r is the specular reﬂection of the incident direction vector Ω across the surface ∂V of the assembly (see Fig. 3), then (1) become:

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Ω · ∇ψ(r, Ω, E) + Σt (r, E)ψ(r, Ω, E) ∞ = Σs (r, Ω · Ω, E → E)ψ(r, Ω , E )dΩ dE 0 4π 1 χ(r, E) ∞ + νΣf (r, E )ψ(r, Ω , E )dΩ dE , k 4π 4π 0 r ∈ V , Ω ∈ 4π , 0 < E < ∞ , ψ(r, Ω, E) = ψ(r, Ω r , E) , r ∈ ∂V , Ω · n < 0 , 0 < E < ∞ .

(12a) (12b)

The problem is to determine a positive value of k such that (12) have a positive (eigenfunction) solution ψ. [Note that this assembly-speciﬁc value of k is generally not the criticality k of the reactor in (11).]

n

∂V V Ω

Ω r = Ω − 2(Ω ⋅ n)n

Fig. 3. Reﬂecting (Symmetry) Boundary Condition)

The purpose of this calculation is to model the neutron ﬂux within a single assembly. If all assemblies were identical, and if the the reactor core were inﬁnite in extent, then (12) would correctly predict the criticality k of the entire (inﬁnite) reactor, and the eigenfunction ψ, extended periodically outside of the single assembly V , would correctly predict the (periodic) eigenfunction of the inﬁnite reactor. However, reactor cores are not inﬁnite, and fuel assemblies are not identical. Therefore, the neutron ﬂux is not a periodic function of space. Nonetheless, for each assembly, (12) are solved, and the results are processed to yield homogenized diﬀusion coeﬃcients that, by means of a homogenized diﬀusion approximation, predict (approximately) the variation of the amplitude of the neutron ﬂux across the core. When this result is combined with the detailed eigenfunctions [obtained from (12) for each assembly], estimates of the neutron ﬂux are obtained that are suﬃciently accurate for many applications. This process, of solving many eigenvalue problems (one for each assembly) and then “stitching” together a global solution, is often much more eﬃcient than solving (11) for the entire reactor.

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3. Shielding Calculations. Here the physical system V is the reactor shield, all internal sources (within the shield) are set to zero, the ﬁssion term is set to zero (Σf = 0 in a shield), and a prescribed incident neutron ﬂux ψ b , representing the neutron ﬂux ﬂowing out of the core and into the shield, is prescribed. Equations (1) become: Ω · ∇ψ(r, Ω, E) + Σt (r, E)ψ(r, Ω, E) ∞ = Σs (r, Ω · Ω, E → E)ψ(r, Ω , E )dΩ dE , 0

4π

r ∈ V , Ω ∈ 4π , 0 < E < ∞ , ψ(r, Ω, E) = ψ (r, Ω, E) , r ∈ ∂V , Ω · n < 0 , 0 < E < ∞ . b

(13a) (13b)

The purpose of this calculation is to determine the extent to which the shield absorbs the neutron and photon radiation from the reactor core, and in particular, to calculate the radiation that successfully transmits through the shield. In these problems, neutrons are usually suppressed quickly; it is the high-energy photons, which are more deeply penetrating, that are problematic. To be complete, a second transport equation and boundary conditions, closely resembling (13), should be added to describe the photon ﬂux. This second transport equation contains a source term, depending linearly on the solution ψ of (13), which describes the production of photons due to inelastic neutron scattering. Unlike eigenvalue and assembly calculations, whose purpose is to determine the detailed neutron population in the reactor core, shielding calculations are primarily performed to ensure that the reactors are safe – that people can safely work outside the reactor, without an undue risk of exposure to the radiation generated within the reactor. For large commercial power reactors, practical neutron transport problems are rarely solved for the entire reactor system (full core plus shield); to attempt this would generate a numerical problem that is much too large and complex. Instead, much smaller “pieces” of the reactor are treated (e.g. a single assembly, or the reactor shield) and the results combined in ways that adequately approximate the neutron ﬂux across the entire system. This completes our discussion of the basic mathematical and physical description of neutron transport problems associated with the design and operation of nuclear reactors. More detailed information on these topics can be found in standard nuclear engineering texts: [1, 6–8], and [21]. Next, we discuss the two main computational approaches for simulating practical neutron transport problems: stochastic (i.e. Monte Carlo) and deterministic techniques. Although these two approaches simulate the same physical problem, they are (i) fundamentally diﬀerent in terms of their approach to the problem and their solution techniques, (ii) have been developed by distinct computer code groups, (iii) exist only in separate computer codes,

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and (iv) have complementary advantages and disadvantages. Because of this complementarity, Monte Carlo and deterministic methods have each found favor for certain types of problems. First, we discuss the basic concepts that form the foundation of these two approaches.

3 Basics of Stochastic and Deterministic Methods The physical process of neutron transport has at least one superﬁcially contradictory aspect: the history of a single neutron is eﬀectively a sequence of random events, while the solution of (1) contain no randomness. For example (see Fig. 4), a single neutron is born at a random point (r0 , Ω 0 , E0 ) in phase space, it streams to a random point r1 where (let us say) it scatters into direction and energy Ω 1 and E1 ; then it streams to a random point r2 where (let us again say) it scatters into direction and energy Ω 2 and E2 ; then it streams to a random point r3 where (let us again say) it scatters into direction and energy Ω 3 and E3 ; ﬁnally, it streams to a random point r4 where it might be captured – or, instead – it could stream to the point r5 where it leaks out of the system. Each particle history requires several random “decisions:” what is the distance between collisions? When a collision occurs, is the collision a scattering event, a ﬁssion event, or a capture event? If the particle scatters, what are its outgoing energy and direction? For each neutron, these questions are answered by the rules of probability dictated by nature (through the numerical values of the cross sections), and except for exceedingly rare cases, the answers are diﬀerent.

Source

scatter (r 2 ,Ω 2 , E2 )

birth (r 0 ,Ω 0 , E0 ) scatter (r1 ,Ω1 , E 1)

scatter (r 3 ,Ω 3 , E3 )

(r 4 ) capture

V

(r 5 ) leaks out Fig. 4. A Neutron History

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In spite of these random aspects of individual particle histories, (1) contain no randomness. That is, if these equations are solved, one obtains a result that seems to carry no information about what a single neutron might do. How can such a result be meaningful? The answer is that (1) describe the mean neutron ﬂux, averaged over an inﬁnite number of neutron histories. Typical neutron transport problems often involve well over 1015 neutrons, so the random ﬂuctuations from the mean – which do exist – are usually small. If one were to simulate a very large number of individual neutron histories and average the results, an estimate of the neutron ﬂux would be obtained that agrees with the solution of (1). The principle underlying Monte Carlo simulations is to ignore the mathematical description of the problem (1), and to directly simulate a large number of particle histories, just as discussed above. The goal is to simulate and average a suﬃciently large number of particle histories that a useful estimate of the ﬂux is obtained. The principle underlying deterministic simulations is to ignore the random aspects of individual particle histories and solve (1), by (i) discretizing these equations with respect to each of its variables – thereby converting the equations into a (typically) large system of algebraic equations – and (ii) solving this algebraic system of equations. Thus, although Monte Carlo and deterministic methods solve the same physical problem, they are based on fundamentally diﬀerent principles of numerical computation. Next, we discuss in somewhat greater detail (i) the ways in which Monte Carlo and deterministic methods are implemented, and (ii) the comparative advantages and disadvantages of these two appoaches. We begin with a discussion of Monte Carlo methods.

4 Stochastic (Monte Carlo) Methods Neutron and photon Monte Carlo methods are based directly on the physics of neutron and photon transport, as experienced by individual neutrons and photons. In the Monte Carlo process, individual particle histories are simulated (and averaged together); one does not need to work with the linear Boltzmann (1a). The details of Monte Carlo particle transport simulations are well-described in documents that are devoted to the subject ([4, 5, 9, 12], and [15]; see also [18] for shielding applications), and we will only touch on a few main points here. The ﬁrst important issue to discuss is the fact that, in principle, Monte Carlo simulations include no errors in the modeling of the physical system or the treatment of the transport physics. If one perfectly knows the geometry of the system and the the cross sections at each point within the system, then the analog Monte Carlo process (in which particle histories are simulated in a manner that is completely faithful to the actual physics) will, in the limit

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of an inﬁnite number of Monte Carlo particles, produce an exact estimate of the ﬂux. This fact is one of the most compelling reasons for employing Monte Carlo methods. However, actual Monte Carlo simulations do not produce exact solutions because no present or future computer will ever be able to process an inﬁnite number of particle histories. Realistic Monte Carlo calculations require that the simulation terminate after a ﬁnite number of histories have been processed. Thus, all Monte Carlo simulations will inherently contain statistical errors, due to the fact that the simulation ends after N histories have been processed. Can one say anything about the magnitude of these statistical errors? Fortunately, the answer to this question is “Yes.” To explain, let us consider a typical problem in which a system V , a neutron source within V , and a detector region R within V are all prescribed (see Fig. 5).

Source V

R Detector

Fig. 5. Source-Detector Problem

We wish to calculate: P = the probability that a neutron, born in the source region, will be captured within R.

(14)

Analog Monte Carlo simulations of this problem will generate and process the histories of a speciﬁed number N of Monte Carlo particles. These particles are all born within the source region, in a manner that is consistent with the speciﬁed source. Each particle will stream through V , undergoing collisions, etc., just as described above. Each history will end with the particle either being captured at some point within V , or leaking out of V . When the nth particle history ends, we assign a tally: % 1 if the particle is captured in R, τn = (15) 0 if the particle is not captured in R.

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After the N histories are completed, the probability P is estimated by the average: N 1 PN = τn . (16) N n=1 The Central Limit Theorem provides a relationship between the exact probability P and the estimate PN . This relationship holds for any Monte Carlo simulation in which tallys are averaged, as they are in (16). The Central Limit Theorem states that for each problem, there exists a constant σ (called the standard deviation) such that, for each a > 0, lim prob

N →∞

! 1/2 a 2 2 |P − PN | √
(17)

This says that the statistical errors in Monte Carlo estimates will on the average decrease as the number√of particle histories increase, and that the errors will be on the order of σ/ N : σ √ |P − PN | = O . (18) N This result is both good and bad. It is good because it shows that, indeed, as N → ∞, Monte Carlo estimate PN of P will converge to the correct result. It is bad because it shows that the convergence of PN to P is (i) slow, and (ii) not guaranteed to be (and almost never is) monotonic. For instance, if one has an estimate PN and one wants to generate a more accurate estimate with an error which is smaller by a factor of 10, one should rerun the Monte Carlo simulation with 100 times the number of particles as the ﬁrst calculation. Unfortunately, the Central Limit Theorem does not guarantee that for a single simulation the error will reduce by precisely a factor of 10; it only says that if one were to run a very large number of these more expensive simulations, the average error will reduce by a factor of 10 over the average error in the less √ expensive simulation. The O(1/ N ) behavior of the error is a fundamental feature of Monte Carlo simulations. Undeniably, this is a drawback to Monte Carlo simulations; they converge slowly. Another important aspect of Monte Carlo simulations is that the standard deviation σ in (18) can be large. This is the case if the detector region in Fig. 5 is distant from the source region, in which case P is very small (a source neutron ending its history by being captured in the detector region is a rare event). In this case, if the physical source emits 1015 neutrons and only 108 of these end their history by being absorbed in the detector, then P = 10−7 , and on the average, only one out of a million Monte Carlo particles will score in R. For simulations of rare events, it is almost never advisable to use analog Monte Carlo, in which the physics is reproduces faithfully; the resulting simulation is much too ineﬃcient.

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For simulations of rare events – which occur inevitably in deep shielding calculations – it is necessary to employ nonanalog techniques. In doing this, the “rules of the game” are modiﬁed in such a way that the same value of P is estimated, but the value of σ is reduced. The most basic technique for doing this is absorption weighting. Here particles are assigned an extra variable: w =particle weight. Particles are generally born with w = 1. The same rules concerning the calculation of distance-to-collision apply to these particles. However, when a particle undergoes a collision, the rules change. If pγ is the probability that the neutron will be captured: Σγ Σγ = , (19) pγ = Σt Σs + Σγ + Σf then when a collision occurs, (i) we do not allow the particle to be captured (we do permit all other scattering and ﬁssion processes to occur with their relative probabilities), and (ii) we reduce the particle’s weight by the multiplicative factor pγ . (By doing this, we greatly extend the lifetime of Monte Carlo particles, ensuring that many more of them will eventually travel into the detector.) If a Monte Carlo particle with weight w enters the detector region R, we ﬁx w and use analog Monte Carlo until the particle is captured or leaks out of R. If the particle does leak out of R, we turn the nonanalog process back on. Thus, Monte Carlo particles will end their histories in only two ways: they can leak out of the system, or they can be captured within R. In this procedure, when a particle ends its history, we assign: % ﬁnal value of w if the particle is captured in R, (20) τn = 0 if the particle leaks out of V . After N particles are generated and their histories are processed, PN is again calculated by (16). It is clear that the calculational cost per particle is greater with absorption weighting than with analog Monte Carlo, because the absorption-weighted Monte Carlo particles will have longer histories. However, many more absorption-weighted particles will score in the detector region, so the absorption-weighted statistical errors should be smaller. Given the same amount of computing time, which method (analog or absorptionweighting) will generally produce the more accurate answer? For rare-event problems, absorption-weighted Monte Carlo simulations are usually more eﬃcient than analog simulations. Also, in the limit N → ∞, the absorption-weighted estimate of PN will, like the analog estimate, converge to the exact value of P . (Although absorption-weighted Monte Carlo is not a faithful representation of the physics, it is a fair game.) Many more sophisticated nonanalog Monte Carlo techniques have been developed, all of which make use of the notion of particle weight, and all of which are designed to make the Monte Carlo simulation run more eﬃciently. (In diﬃcult problems, Monte Carlo with absorption weighting is still

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unacceptably ineﬃcient.) One of the most widely-used of these nonanalog techniques employs the use of weight windows. This procedure is based on the observation that it can be helpful to control the range of particle weights at each point in V . The weight window technique operates as follows: 1. The system V is divided into a subregions, each no more than a few mean free paths across. 2. In each (j th ) subregion, a lower weight wj− and an upper weight wj+ are assigned, with wj− < wj+ . (Usually, wj+ ≈ 10wj− .) 3. The Monte Carlo process with absorption weighting is started. In addition to the rules of absorption weighting described above, the following rules are also imposed: 4. If a particle has weight w which is greater than the upper weight window at that point, the particle is split into n particles, each with weight w/n lying as close as possible to the center of the weight window. Each of these n particles must then be tracked, and their histories run to completion. 5. If a particle has weight w which is lower than the lower weight window at that point, the particle is subjected to Russian roulette: if wjc is the “center” of the weight window, then with probability p = w/wc the particle survives with increased weight wc , and with probability 1 − p the particle is killed. The eﬀect of the weight window is to place rough bounds on the numbers of Monte Carlo particles at diﬀerent locations across V . It has been shown that if the upper and lower weight window weights are chosen optimally, the resulting Monte Carlo simulation becomes much more eﬃcient than absorption weighting. Other nonanalog strategies, such as geometric splitting, are based on related ideas. The use of nonanalog Monte Carlo techniques, such as weight windows, for practical deep penetration problems in nuclear engineering, places a signiﬁcant burden on the code user. The user must specify all the geometric details of the physical system V and the necessary neutron sources. In addition, the user must supply all the biasing parameters – for example, the numerical values of each of the weight windows (assuming that the weight window technique is used). If V is large, with many subregions, and if the weight windows also depend on energy, then there can be many hundreds of these parameters that must be chosen. Determining the biasing parameters in a way that leads to near-optimal eﬃciency of the Monte Carlo simulation can be a diﬃcult and time-consuming task, relying on the experience, skill, and luck of the code user. Furthermore, the biasing parameters depend strongly on the location and nature of the detector; if the detector changes, the biasing parameters also change. This reveals another fact about Monte Carlo simulations for diﬃcult nuclear engineering problems requiring nonanalog biasing: they are not userfriendly.

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To conclude this section, we brieﬂy summarize the advantages and disadvantages of Monte Carlo methods for neutron transport problems. First, Monte Carlo methods for neutron and photon transport are, at their root, quite simple – they directly apply the physics of particle transport to the simulation of individual particle histories. This can be done in a way that requires no approximations – each particle can be simulated in a manner that perfectly adheres to the underlying physics. Therefore, Monte Carlo simulations are capable of producing solutions which are free of any type of truncation error. However, Monte Carlo simulations of diﬃcult problems are often very costly to set up and run. To make the Monte Carlo code run with acceptable eﬃciency, the code user must specify a large number of biasing parameters, which are specialized to each diﬀerent problem. Determining these parameters can be diﬃcult and time-consuming. Also, even when the biasing parameters are well-chosen, Monte Carlo simulations converge slowly and non-monotonically with increasing run time. Thus, while Monte Carlo solutions are free of truncation errors, they are certainly not free of statistical errors, and it can be challenging to obtain Monte Carlo solutions with suﬃciently small statistical errors, and with acceptable cost. Finally, the nonanalog techniques that have been developed for making Monte Carlo simulations acceptably eﬃcient are useful for source-detector problems – in which a detector response in a small portion of phase space is desired – but are not useful for obtaining eﬃcient global solutions, over all of phase space. Generally, Monte Carlo solutions work best when very limited information about the ﬂux (e.g. a single detector response) is desired in a given simulation.

5 Deterministic Methods As we discussed earlier, deterministic methods for solving neutron transport problems are based on solving (1). The ﬁrst step is to discretize these equations with respect to all of the independent variables. We now brieﬂy discuss each of these discretizations. First, we discretize the energy variable E. To do this, we conﬁne the energies of the particles to a ﬁnite interval Emin < E < Emax and divide this interval into G energy groups as depicted in Fig. 6. gth energy group

Emin=EG

EG-1

Eg

Eg-1

E2

Fig. 6. Multigroup Energy Grid

E1

E0=Emax

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Next, we approximate all the cross sections in (1a) by histograms in energy, each histogram having one value within each energy group. (There are many prescriptions for doing this, but however it is done, it is done carefully, in order to minimize the total number of groups G.) Finally, the resulting equation is integrated over each group. This yields the multigroup transport equations: Ω · ∇ψg (r, Ω) + Σt,g (r)ψg (r, Ω) G Σs,g →g (r, Ω · Ω)ψg (r, Ω )dΩ = g =1

4π

G χg (r, E) + νΣf,g (r)ψg (r, Ω )dΩ 4π 4π g =1

Qg (r) , r ∈ V , Ω ∈ 4π , 1 < g < G , + 4π where

(21)

Eg−1

ψg (r, Ω) =

ψ(r, Ω, E)dE

(22)

Eg

are the group ﬂuxes. This multigroup approximation is used almost universally to treat the energy variable. The angular variable is usually discretized in one of two ways. The more common SN (discrete ordinates) approximation assumes that, instead of being able to travel in an arbitrary direction of ﬂight Ω on the unit sphere, neutrons are only able to travel in a ﬁnite number MN of speciﬁed directions, Ω n , for 1 ≤ n ≤ MN . Furthermore, each discrete ordinate Ω n is associated with a cone on the unit sphere of area wn . The set {(Ω n , wn )|1 ≤ n ≤ MN } is called an angular quadrature set of order N . (In 1-D, MN = N , but for historical reasons, in 2-D and 3-D, MN > N .) With this approximation, (21) becomes: Ω n · ∇ψg,n (r) + Σt,g (r)ψg,n (r) =

MN G

Σs,g →g (r, Ω n · Ω n )ψg ,n (r)wn

g =1 n =1

+

G MN χg (r, E) νΣf,g (r)ψg ,n (r)wn 4π g =1 n =1

+

Qg (r) , r ∈ V , 1 ≤ n ≤ MN , 1 < g < G . 4π

(23)

An alternate PN (spherical harmonics) method is available, in which the group angular ﬂuxes ψg (r, Ω) are expanded in a ﬁnite number of spherical harmonic functions ψg (r, Ω) ≈

+n N n=0 m=−n

ψg,n,m (r)Yn,m (Ω) ,

(24)

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and a coupled system of equations for the expansion coeﬃcients ψg,n,m (r) are obtained by multiplying (21) by the spherical harmonic functions Yn,m (Ω) and integrating Ω over the unit sphere. The SN approach is more widely-used because the system of SN equations closely resembles the original Boltzmann equation; the comparable system of PN equations has a very diﬀerent and more complicated mathematical structure. Finally, to discretize the spatial variable, one ﬁrst takes the physical system and imposes on it a spatial grid. Then, one discretizes each of (23) on this grid. Many diﬀerent spatial discretization techniques have been developed, among them ﬁnite diﬀerence, ﬁnite element, and nodal methods. It is not possible to discuss these methods here in any detail. Generally, all the methods work well for problems on which the spatial cells are optically thin. However, as the spatial cells become more coarse, the simpler ﬁnite-diﬀerence methods will generally exhibit undesirable behavior – they can become negative, or oscillatory. While the coarsening of the spatial mesh will degrade any discretization method, some methods are much more sensitive than others. Certain ﬁnite element methods are quite insensitive to the use of coarse spatial grids for optically thick problems dominated by scattering. Much eﬀort has gone into the development of spatial discretization methods that are as accurate as possible on coarse grids [16]. This concludes the ﬁrst part of a deterministic method – the discretization of the linear Boltzmann equation. One now usually has a very large system of algebraic equations to be solved, of the form: AΨ = B ,

(25)

where A is a matrix representing the linear Boltzmann operator, Ψ is the vector of unknowns, and B is a vector of interior and boundary sources. How large is this system? In a typical 3-D shielding calculation, G = 150 energy groups is often needed to achieve suﬃcient resolution of the energy variable, MN = 24 discrete ordinates (3 ordinates per octant) is the minimum number needed to resolve the angular variable, and a spatial grid of 100 × 100 × 100 cells is not unreasonable (and is often a low estimate). This implies 3.6 × 109 unknowns – a huge number, which is often much smaller than is seen in some problems. (The large number of unknowns is a direct and inevitable consequence of the high-dimensionality of phase space.) Generally, there is little hope to solve (25) by directly inverting the matrix A. Thus, the second part of a deterministic method – developing an eﬃcient solution strategy for solving (25) – comes to the forefront. Since it is generally impractical to directly invert A, solution strategies for solving (25) have focused on iterative methods. A great deal of work has been done on this topic and is described in a recent comprehensive review article [22]. It is impossible to discuss the ﬁne points of this work here. However, the general situation is as follows:

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1. A straightforward iteration method, called Source Iteration (SI), converges in one iteration if the problem has no scattering or ﬁssion. For problems with scattering and ﬁssion, SI converges rapidly if the processes of capture and leakage are not weak. In this case, neutron histories are generally short; it is unlikely that a neutron will scatter a very large number of times before being captured or leaking out of V . However, if a problem has a diﬀusive subregion, which is optically thick and is dominated by scattering, then SI will generally converge very slowly – because the particles in this subregion will have long histories. 2. Numerous acceleration methods have been developed to increase the rate of convergence of the SI technique. Most notably, the diﬀusion synthetic acceleration (DSA) method can, under ideal conditions, produce spectacular speedups over SI. However, DSA has stability issues associated with the use of advanced discretization schemes and unstructured grids [22]. 3. The use of Krylov subspace methods, with DSA as a preconditioner, has been a major topic of research in recent years. It now appears that for most problems of practical interest, the proper combination of DSA + Krylov will yield an eﬃcient solution technique [25]. We conclude this section with a brief summary of the advantages and disadvantages of deterministic methods. First, by their nature, deterministic methods automatically produce a global solution on the chosen grid in phase space. However, these solutions will contain truncation errors associated with the discretizations of the energy, direction, and spatial variables. While truncation errors generally decrease as the phase space grid is reﬁned, some errors can be notoriously diﬃcult to suppress, and because of the high dimensionality of phase space, it may be diﬃcult (due to computer memory limitations) to employ a suﬃciently ﬁne grid. Also, deterministic methods are generally formulated on spatial grids that have a limited capacity to model curved surfaces. Thus, the geometric modeling of the physical system may itself contain “truncation” errors. Finally, iterative techniques for solving the ﬁnal discrete set of equations are not always as eﬃcient as desired. As one might expect, all of these deﬁciencies are topics of current research. Detailed information on deterministic neutron transport methods can be found in [2, 3, 10, 11, 13, 14, 16, 22], and [25]. The dates of publication of these references indicate how up-to-date they are; this is relevant because the ﬁeld of deterministic transport simulations has changed immensely in the past 20 years. Two recent detailed reviews are [22] and [25].

6 Automatic Variance Reduction (Hybrid) Methods We have already discussed a serious deﬁciency of classic Monte Carlo codes: often, they are not user-friendly. (In addition to specifying the geometry and

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material structure of the system, the Monte Carlo code user must also specify biasing parameters. No analogous task is required of an SN code user.) In the past 10 years, it has been recognized that the biasing parameters – which (to repeat) have traditionally been chosen by the code user – can in fact be calculated by a relatively inexpensive deterministic adjoint calculation. In principle, and in fact, it is possible to: (i) ﬁrst, run a suitable (inexpensive) deterministic adjoint problem, (ii) use the results of that calculation to determine biasing parameters, (iii) implement these biasing parameters in a Monte Carlo code, and (iv) run the Monte Carlo code with the machine-generated biasing parameters. This general process has been described as an Automatic Variance Reduction method, or a Hybrid Monte Carlo-Deterministic method. The commercial hybrid neutron transport code MCBEND empoys deterministic diﬀusion with Monte Carlo transport [17, 20]. A similar eﬀort combining deterministic SN and Monte Carlo codes is described in [19]. This and other related work is discussed in a recent review [23], and case studies have been reported [24], showing the eﬃcacy of the approach. The computergenerated biasing parameters are usually obtained by the computer much more quickly than by human (trial and error) eﬀort, and are usually more efﬁcient than the ones obtained by the code user (the subsequent Monte Carlo simulation runs much more eﬃciently). The principle drawback to the user is that presently, two codes must be used: a deterministic code for the preliminary adjoint calculation, and a subsequent Monte Carlo code. The problem must be set up on both codes, and the results of the deterministic code must be shipped to the Monte Carlo code. Thus, in current implementations, hybrid methods are not as convenient to use as they could be, if they were implemented in one relatively easy-to-use code. Due to their relative unfamiliarity and the awkwardness of using two codes to implement them, the use of hybrid methods is currently very limited. However, hybrid methods oﬀer computational advantages that, sooner or later, will motivate code developers to implement them in user-friendly ways. When this happens, hybrid methods should become much more widely-adopted.

7 Discussion The simulation of neutron and photon transport processes for nuclear reactors is generally diﬃcult and costly. During the past 50-odd years, two distinct computational techniques have been developed to implement these simulations. These techniques (Monte Carlo and deterministic) are complementary, in ways that have been discussed above, and that have led to their wide use in diﬀerent types of problems. Current research in Monte Carlo and deterministic methods is aimed at making these methods run more eﬃciently and accurately. Perhaps the best sources of literature on these topics are (i) journal articles published in Nuclear Science and Engineering (the premier research journal of the American Nuclear Society), (ii) proceedings of

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the biannual topical conferences organized by the Mathematics and Computation Division of the American Nuclear Society (where most deterministic work and some Monte Carlo work is reported before journal publication), and (iii) proceedings of the biannual topical conferences organized by the Radiation Protection & Shielding Division of the American Nuclear Society (where most Monte Carlo work is reported before journal publication). This review should not end without a brief discussion of the technical relationship between Monte Carlo and deterministic methods. A basic question is: have advances in one of these methods led to advances in the other? The perhaps amazing answer to this question is: until the recent advent of hybrid methods, “No.” The fact is that, due to the fundamental diﬀerences in how Monte Carlo and deterministic methods are conceived, implemented, and used, distinct groups of code developers (and users) have grown up around these two classes of methods. Computer codes have been built around Monte Carlo methods, or deterministic methods, but not both; to this author’s knowledge, no “production” particle transport code exists in which Monte Carlo and deterministic transport methods can both be employed. In short, Monte Carlo and deterministic methodologies and codes have developed and built by diﬀerent communities of code developers; also, the Monte Carlo and deterministic user communities are largely distinct. In eﬀect, the numerical transport community consists of two separate cultures. To some extent, this technical and cultural dichotomy has persisted because it has not been obvious that implementing a Monte Carlo and a deterministic method in the same code is beneﬁcial. If one were to build a neutron transport code in which a physical system were speciﬁed, and one could with “the ﬂip of a switch” either solve the problem with Monte Carlo, or with deterministic methods, then constraints would be imposed that would limit the modeling ability of Monte Carlo. This is because deterministic methods today are not generally capable of modeling curved surfaces, while Monte Carlo methods routinely treat curved surfaces. Thus, to implement both methods in the same code, the physical description of the system would (from the Monte Carlo viewpoint) have to be compromised, to accommodate the deterministic technique. Thus, if the only consequence of implementing Monte Carlo and deterministic methods in the same code is that the code user can solve the same physical problem using both techniques, it is not obvious that this ability would be suﬃciently useful to warrant the eﬀort of developing the code. For this reason, and for the extreme diﬀerences in philosophy and methodology that go into the construction of Monte Carlo and deterministic codes, there have been few reasons for Monte Carlo and deterministic code developers to try to merge their methodologies. Because of this long historical perspective – that Monte Carlo and deterministic methods are too diﬀerent to be combined into something new and useful – it has taken many years to even begin to recognize that it is possi-

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ble to combine Monte Carlo and deterministic methods advantageously. The recent advent of hybrid methods is only beginning to dispel this old myth. Although hybrid methodology has currently had little impact, future work on hybrid methods should lead to signiﬁcant beneﬁts in simulating neutron transport problems. Computer codes will likely always exist that contain a single methodology (Monte Carlo or deterministic). However, because of the demonstrated advantages of hybrid methods, it seems inevitable that user-friendly computer codes containing both Monte Carlo and deterministic methodologies will become available that enable hybrid techniques to be used more easily and eﬃciently. When this happens, and when it is perceived that these codes are advantageous for certain types of problems, the codes should quickly become accepted and adopted, just as the current “singlemede” Monte Carlo and deterministic codes have been accepted and adopted for the kinds of problems to which they are best suited. Furthermore, any future advances in Monte Carlo or deterministic methodologies would automatically beneﬁt impact a hybrid code – because hybrid methods make direct use of both techniques. Thus, the current directions of research in computational neutron/photon transport methods involve (i) the suppression of statistical errors (variance) in Monte Carlo, (ii) the suppression of truncation errors in deterministic methods, (iii) the enhancement of eﬃciency in iterative algorithms for deterministic methods, and ﬁnally, (iv) the development of new hybrid methods that combine features of Monte Carlo and deterministic methods to solve transport problems in ways that are advantageous compared to a conventional Monte Carlo or deterministic method.

References 1. A.L. Weinberg and E.P. Wigner, The Physical Theory of Neutron Chain Reactors, University of Chicago Press, Chicago (1958). 2. B.G. Carlson, “The Numerical Theory of Neutron Transport,” in Methods in Computational Physics, Vol. 1, Academic Press, New York (1963). 3. B.G. Carlson and K.D. Lathrop, “Transport Theory – The Method of Discrete Ordinates,” in Computing Methods in Reactor Physics, edited by H. Greenspan, C.N. Kelber, and D. Okrent, Gordon and Breach, New York (1968). 4. M.H. Kalos, F.R. Nakache, and J. Celnik, “Monte Carlo Methods in Reactor Computations,” in Computing Methods in Reactor Physics, edited by H. Greenspan, C.N. Kelber, and D. Okrent, Gordon and Breach, New York (1968). 5. J. Spanier and E.M. Gelbard, Monte Carlo Principles and Neutron Transport, Addison-Wesley, Reading, Massachusetts (1969). 6. G.I. Bell and S. Glasstone, Nuclear Reactor Theory, Van Nostrand Reinhold, New York (1970). 7. A.F. Henry, Nuclear-Reactor Analysis, MIT Press, Cambridge, Massachusetts (1975). 8. J.J. Duderstadt and L.J. Hamilton, Nuclear Reactor Analysis, Wiley, New York (1976).

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9. L.L. Carter and E.D. Cashwell, Particle-Transport Simulation with the Monte Carlo Method, Technical Development Center, ERDA. Available as TID-26607 from the National Technical Information Service, U.S. Department of Commerce, Springﬁeld, VA 22161 (1977). 10. R. Sanchez and N.J. McCormick, “A Review of Neutron Transport Approximations,” Nucl. Sci. Eng. 80, 481 (1982). 11. E.E. Lewis and W.F. Miller, Jr., Computational Methods of Neutron Transport, Wiley-Interscience, New York (1984). [Reprinted: American Nuclear Society, La Grange Park, Il. (1993).] 12. M.H. Kalos and P.A. Whitlock, Monte Carlo Methods. Volume I: Basics, Wiley, New York (1986). 13. R.D. Lawrence, “Progress in Nodal Methods for the Solution of the Neutron Diﬀusion and Transport Equations,” Prog. Nucl. Energy 17, 271 (1986). 14. G.I. Marchuk and V.I. Lebedev, Numerical Methods in the Theory of Neutron Transport, Harwood, London (1986). (Originally published in Russian in 1981). 15. I. Lux and L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations, CRC Press, Boca Raton, Florida (1991). 16. E.W. Larsen, “The Asymptotic Diﬀusion Limit of Discretized Transport Problems,” Nucl. Sci. Eng. 112, 336 (1992). 17. Chucas S. and Grimstone M., “The Acceleration Techniques used in the Monte Carlo Code MCBEND,” Proc. 8th Int. Conf. Radiation Shielding, Arlington, Texas, April 24–28, 1994, 2, 1126, American Nuclear Society (1994). 18. J.K. Shultis and R.E. Faw, Radiation Shielding, Prentice-Hall PTR, Upper Saddle River, New Jersey (1996). 19. K.A. Van Riber, T.J. Urbatsch, P.D. Soran, D.K. Parsons, J.E. Morel, G.W. McKinney, S.R. Lee, L.A. Crotzer, F.W. Brinkley, J.W. Anderson, and R.W. Alcouﬀe, “AVATAR – Automatic Variance Reduction in Monte Carlo Calculations,” Proc. ANS M&C Topical Meeting: Mathematical Methods and Supercomputing in Nuclear Appliations, Vol. 1, p. 661, October 6–10, 1996, Saratoga Springs, New York (1996). 20. T. Shuttleworth, M. Grimstone, and S. Chucas, “Application of Acceleration Techniques in MCBEND,” Journal of Nuclear Science and Technology, Supplement 1 (March, 2000),Proceedings of Ninth International Conference on Radiation Shielding, p. 406, Tsukuba Ibaraki, Japan, Oct. 17–22, 1999. 21. W.M. Stacey, Nuclear Reactor Physics, Wiley, New York (2001) 22. M.L. Adams and E.W. Larsen, “Fast Iterative Methods for Discrete-Ordinates Particle Transport Calculations,” Prog. Nucl. Energy 40, 3 (2002). 23. A. Haghighat and J.C. Wagner, “Monte Carlo Variance Reduction with Deterministic Importance Functions,” Prog. Nucl. Energy 42, 25 (2003). 24. H.P. Smith and J.C. Wagner, “A Case Study in Manual and Automated Monte Carlo Variance Reduction with a Deep Penetration Reactor Shielding Problem,” Proc. ANS M&C Topical Meeting: Nuclear, Mathematical, and Computational Sciences: A Century in Review – A Century Anew, April 6–10, 2003, Gatlinburg, Tennessee, American Nuclear Society, 555 N. Kensington Avenue, La Grange Park, IL 60525 (2003). 25. E.W. Larsen and J.E. Morel, “Advances in Discrete-Ordinates Methodology,” to appear in Nuclear Computational Science: A Century in Review, Edited by Y. Azmy and E. Sartori, Springer, New York (2006).

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Lecture Notes in Computational Science and Engineering Vol. 1 D. Funaro, Spectral Elements for Transport-Dominated Equations. 1997. X, 211 pp. Softcover. ISBN 3-540-62649-2 Vol. 2 H. P. Langtangen, Computational Partial Differential Equations. Numerical Methods and Diffpack Programming. 1999. XXIII, 682 pp. Hardcover. ISBN 3-540-65274-4 Vol. 3 W. Hackbusch, G. Wittum (eds.), Multigrid Methods V. Proceedings of the Fifth European Multigrid Conference held in Stuttgart, Germany, October 1-4, 1996. 1998. VIII, 334 pp. Softcover. ISBN 3-540-63133-X Vol. 4 P. Deuﬂhard, J. Hermans, B. Leimkuhler, A. E. Mark, S. Reich, R. D. Skeel (eds.), Computational Molecular Dynamics: Challenges, Methods, Ideas. Proceedings of the 2nd International Symposium on Algorithms for Macromolecular Modelling, Berlin, May 21-24, 1997. 1998. XI, 489 pp. Softcover. ISBN 3-540-63242-5 Vol. 5 D. Kröner, M. Ohlberger, C. Rohde (eds.), An Introduction to Recent Developments in Theory and Numerics for Conservation Laws. Proceedings of the International School on Theory and Numerics for Conservation Laws, Freiburg / Littenweiler, October 20-24, 1997. 1998. VII, 285 pp. Softcover. ISBN 3-540-65081-4 Vol. 6 S. Turek, Efﬁcient Solvers for Incompressible Flow Problems. An Algorithmic and Computational Approach. 1999. XVII, 352 pp, with CD-ROM. Hardcover. ISBN 3-540-65433-X Vol. 7 R. von Schwerin, Multi Body System SIMulation. Numerical Methods, Algorithms, and Software. 1999. XX, 338 pp. Softcover. ISBN 3-540-65662-6 Vol. 8 H.-J. Bungartz, F. Durst, C. Zenger (eds.), High Performance Scientiﬁc and Engineering Computing. Proceedings of the International FORTWIHR Conference on HPSEC, Munich, March 16-18, 1998. 1999. X, 471 pp. Softcover. 3-540-65730-4 Vol. 9 T. J. Barth, H. Deconinck (eds.), High-Order Methods for Computational Physics. 1999. VII, 582 pp. Hardcover. 3-540-65893-9 Vol. 10 H. P. Langtangen, A. M. Bruaset, E. Quak (eds.), Advances in Software Tools for Scientiﬁc Computing. 2000. X, 357 pp. Softcover. 3-540-66557-9 Vol. 11 B. Cockburn, G. E. Karniadakis, C.-W. Shu (eds.), Discontinuous Galerkin Methods. Theory, Computation and Applications. 2000. XI, 470 pp. Hardcover. 3-540-66787-3 Vol. 12 U. van Rienen, Numerical Methods in Computational Electrodynamics. Linear Systems in Practical Applications. 2000. XIII, 375 pp. Softcover. 3-540-67629-5 Vol. 13 B. Engquist, L. Johnsson, M. Hammill, F. Short (eds.), Simulation and Visualization on the Grid. Parallelldatorcentrum Seventh Annual Conference, Stockholm, December 1999, Proceedings. 2000. XIII, 301 pp. Softcover. 3-540-67264-8 Vol. 14 E. Dick, K. Riemslagh, J. Vierendeels (eds.), Multigrid Methods VI. Proceedings of the Sixth European Multigrid Conference Held in Gent, Belgium, September 27-30, 1999. 2000. IX, 293 pp. Softcover. 3-540-67157-9 Vol. 15 A. Frommer, T. Lippert, B. Medeke, K. Schilling (eds.), Numerical Challenges in Lattice Quantum Chromodynamics. Joint Interdisciplinary Workshop of John von Neumann Institute for Computing, Jülich and Institute of Applied Computer Science, Wuppertal University, August 1999. 2000. VIII, 184 pp. Softcover. 3-540-67732-1 Vol. 16 J. Lang, Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems. Theory, Algorithm, and Applications. 2001. XII, 157 pp. Softcover. 3-540-67900-6 Vol. 17 B. I. Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition. 2001. X, 197 pp. Softcover. 3-540-41083-X

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Vol. 39 S. Attinger, P. Koumoutsakos (eds.), Multiscale Modelling and Simulation. 2004. VIII, 277 pp. Softcover. 3-540-21180-2 Vol. 40 R. Kornhuber, R. Hoppe, J. Périaux, O. Pironneau, O. Wildlund, J. Xu (eds.), Domain Decomposition Methods in Science and Engineering. 2005. XVIII, 690 pp. Softcover. 3-540-22523-4 Vol. 41 T. Plewa, T. Linde, V.G. Weirs (eds.), Adaptive Mesh Reﬁnement – Theory and Applications. 2005. XIV, 552 pp. Softcover. 3-540-21147-0 Vol. 42 A. Schmidt, K.G. Siebert, Design of Adaptive Finite Element Software. The Finite Element Toolbox ALBERTA. 2005. XII, 322 pp. Hardcover. 3-540-22842-X Vol. 43 M. Griebel, M.A. Schweitzer (eds.), Meshfree Methods for Partial Differential Equations II. 2005. XIII, 303 pp. Softcover. 3-540-23026-2 Vol. 44 B. Engquist, P. Lötstedt, O. Runborg (eds.), Multiscale Methods in Science and Engineering. 2005. XII, 291 pp. Softcover. 3-540-25335-1 Vol. 45 P. Benner, V. Mehrmann, D.C. Sorensen (eds.), Dimension Reduction of Large-Scale Systems. 2005. XII, 402 pp. Softcover. 3-540-24545-6 Vol. 46 D. Kressner (ed.), Numerical Methods for General and Structured Eigenvalue Problems. 2005. XIV, 258 pp. Softcover. 3-540-24546-4 Vol. 47 A. Boriçi, A. Frommer, B. Joó, A. Kennedy, B. Pendleton (eds.), QCD and Numerical Analysis III. 2005. XIII, 201 pp. Softcover. 3-540-21257-4 Vol. 48 F. Graziani (ed.), Computational Methods in Transport. 2006. VIII, 524 pp. Softcover. 3-54028122-3

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