Analysis and Numerical Computation of Conversation Laws

Gerald Warnecke (Ed.) Analysis and Numerics for Conservation Laws Gerald Warnecke Editor Analysis and Numerics for Co...

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Gerald Warnecke (Ed.) Analysis and Numerics for Conservation Laws

Gerald Warnecke Editor

Analysis and Numerics for Conservation Laws With 236 Figures and 18 Tables

123

Editor

Gerald Warnecke Institut für Analysis und Numerik Otto-von-Guericke-Universität Magdeburg Postfach 4120 39016 Magdeburg, Germany e-mail: [email protected]

Library of Congress Control Number: 2005922932

Mathematics Subject Classification (2000): 35L65, 65M99, 65Z05, 76N15

ISBN-10 3 -540-24834-X Springer Berlin Heidelberg New York ISBN-13 978-3-540-24834-7 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 © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany 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 author using a Springer TEX macro package Cover design: Erich Kircher, Heidelberg, Germany Printed on acid-free paper

46/3142/sz - 5 4 3 2 1 0

Preface

What do a supernova explosion in outer space, flow around an airfoil and knocking in combustion engines have in common? The physical and chemical mechanisms as well as the sizes of these processes are quite different. So are the motivations for studying them scientifically. The supernova is a thermo-nuclear explosion on a scale of 108 cm. Astrophysicists try to understand them in order to get insight into fundamental properties of the universe. In flows around airfoils of commercial airliners at the scale of 103 cm shock waves occur that influence the stability of the wings as well as fuel consumption in flight. This requires appropriate design of the shape and structure of airfoils by engineers. Knocking occurs in combustion, a chemical process, and must be avoided since it damages motors. The scale is 101 cm and these processes must be optimized for efficiency and environmental considerations. The common thread is that the underlying fluid flows may at a certain scale of observation be described by basically the same type of hyperbolic systems of partial differential equations in divergence form, called conservation laws. Astrophysicists, engineers and mathematicians share a common interest in scientific progress on theory for these equations and the development of computational methods for solutions of the equations. Due to their wide applicability in modeling of continua, partial differential equations are a major field of research in mathematics. A substantial portion of mathematical research is related to the analysis and numerical approximation of solutions to such equations. Hyperbolic conservation laws in two or more space dimensions still pose one of the main challenges to modern mathematics. This is due to the fact that the fundamental question of an existence theory for solutions to these equations has remained an open problem for many decades, despite intensive efforts of some of the leading mathematicians during the last fifty years. Due to their importance in applications the development of efficient numerical methods had to proceed despite the fact that mathematics could not prove that the solutions, to be approximated computationally, actually exist. The preferred practice is to seek numerically objects that are mathematically proven to exist and whose properties are well understood.

VI

Preface

In this situation it is important that analysis and numerical computations, including the work in fields of application, are studied in close cooperation. By numerical experiments we can explore the structure and properties of solutions in the hope of inspiring the search for the right analytical framework for these equations. Researchers developing computational methods or applying numerical method need as much analytical information on solutions as possible in order to gain confidence in their computations. Especially applied researchers have to be aware of this situation in order to make the right assessments in their comparisons with experimental or observational data. Hyperbolic conservation laws for fluid flows in conjunction with other mechanisms such as material elasticity, magneto-hydrodynamics or combustion were at the center of a major research effort in Germany in recent years. The priority research program Analysis and Numerics for Conservation Laws was funded by the German research foundation Deutsche Forschungsgemeinschaft (DFG) for a period of six years from 1997. The program consisted of three periods of two year grants for about 25 projects in each period including funds for visitor programs and workshops. The participating groups came from astrophysics, fluid mechanics, mathematical analysis and numerical mathematics. The span of research interests was from specific applications to fundamental mathematical questions. A number of workshops were organized in each year and in 2000 the 8th International Conference on Hyperbolic Problems: Theory, Numerics and Applications was held in Magdeburg, Germany. Another special event was a Symposium on Entropy jointly organized with two other DFG priority research programs Interacting Stochastic Systems of High Complexity as well as Ergodic Theory, Analysis and Efficient Simulation of Dynamical Systems (DANSE). Together with the two co-organizers of the symposium Andreas Greven and Gerhard Keller specially selected contributions to the topic were compiled in a book titled Entropy. The diversity of topics, represented in the volume you are looking at, was one of the strengths of the research program. It brought together research groups of very different background most of which were interacting with each other for the first time. This volume can be seen as a research report of the priority research program. It contains contributions from most participating projects and gives an overview of some results achieved in the program. The authors intend to convey their results to readers outside of their own particular field. They cover a wide range of conservation laws modeling, for instance, bubbly flows, retrograde and BZT fluids, detonation waves in combustion, magnetoplasmadynamic propulsion of space craft, solar physics, chemotaxis in mathematical biology, and type Ia supernovae. Kinetic models using the Boltzmann equation or the Boltzmann-Peierls equation are explored in relation to conservation laws. The numerical approaches range over finite volume methods, central schemes, the method of transport, discontinuous Galerkin methods, kinetic schemes to meshless particle methods. The efficient use of level-set methods for problems needing better interface resolution was a research topic for a number of groups. The majority of the contributions deal

Preface

VII

with the development and study of properties of numerical schemes. Additionally, mathematicians present results on error estimates for scalar conservations laws, existence of traveling wave solutions, structural stability, relaxation dynamics and scaling limits, as well as the value of dual a posteriori error estimates for adaptive algorithms. For some computational results a color scale is a convenient way to plot field variables. Therefore, a number of color plates have been included in the back. I would like to take this opportunity to thank those who helped in many ways. Robert Paul K¨ onigs and Bernhard Nunner were in turn the program officers of the DFG for mathematics during the conception and implementation of the program. They were exceptionally helpful with very sound advice and contributed substantially to the success of this program. A big thanks also to all participating colleagues for their input into devising the program and their continual support in running the program, this especially to the members of the coordinating board Dieter H¨ anel, Wolfgang Hillebrandt, Rupert Klein, Dietmar Kr¨ oner and Willi J¨ ager. Other colleagues supported this effort by taking on the difficult task of refereeing the project proposals that always outnumbered the budgetary possibilities of the DFG. They are Wolfgang Hackbusch, Rolf Jeltsch, Wilhelm Kegel, Wilhelm Kordulla, Stephan Luckhaus, Hans Ruder, Martin Sommerfeld, Peter Szmolyan and Harry Yserentant. Most important to me was the success of a quite substantial number of doctoral students funded within the program or taking part in activities generated by the program. I enjoyed their lively interaction very much and would like to thank them for their contributions to the activities and the scientific results of this program. The manuscripts of this book were handled by my secretary Stephanie Wernicke and the final draft for the publisher was expertly compiled by R¨ udiger M¨ uller. Magdeburg December 2004

Gerald Warnecke

Contents

Wave Processes at Interfaces S. Andreae, J. Ballmann, S. M¨ uller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Numerics for Magnetoplasmadynamic Propulsion J. Heiermann, M. Auweter-Kurtz, C. Sleziona . . . . . . . . . . . . . . . . . . . . . . . 27 Hexagonal Kinetic Models and the Numerical Simulation of Kinetic Boundary Layers H. Babovsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 High-resolution Simulation of Detonations with Detailed Chemistry R. Deiterding, G. Bader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Numerical Linear Stability Analysis for Compressible Fluids A.S. Bormann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Simulation of Solar Radiative Magneto-Convection M. Sch¨ ussler, J.H.M.J. Bruls, A. V¨ ogler, P. Vollm¨ oller . . . . . . . . . . . . . . . 107 Riemann Problem for the Euler Equation with Non-Convex Equation of State including Phase Transitions W. Dahmen, S. M¨ uller, A. Voß . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Radiation Magnetohydrodynamics: Analysis for Model Problems and Efficient 3d-Simulations for the Full System A. Dedner, D. Kr¨ oner, C. Rohde, M. Wesenberg . . . . . . . . . . . . . . . . . . . . . 163 Kinetic Schemes for Selected Initial and Boundary Value Problems W. Dreyer, M. Herrmann, M. Kunik, S. Qamar . . . . . . . . . . . . . . . . . . . . . 203

X

Contents

A Local Level-Set Method under Involvement of Topological Aspects F. V¨ olker, R. Vilsmeier, D. H¨ anel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen, K.P. Hadeler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Travelling Waves in Systems of Hyperbolic Balance Laws J. H¨ arterich, S. Liebscher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 The Role of the Jacobian in the Adaptive Discontinuous Galerkin Method for the Compressible Euler Equations R. Hartmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 The Multi-Scale Dust Formation in Substellar Atmospheres Ch. Helling, R. Klein, E. Sedlmayr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 Meshless Methods for Conservation Laws D. Hietel, M. Junk, J. Kuhnert, S. Tiwari . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Simulations of Turbulent Thermonuclear Burning in Type Ia Supernovae W. Hillebrandt, M. Reinecke, W. Schmidt, F.K. R¨ opke, C. Travaglio, J.C. Niemeyer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Hyperbolic GLM Scheme for Elliptic Constraints in Computational Electromagnetics and MHD Y.J. Lee, R. Schneider, C.-D. Munz, F. Kemm . . . . . . . . . . . . . . . . . . . . . . 385 Flexible Flame Structure Modelling in a Flame Front Tracking Scheme H. Schmidt, R. Klein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 Riemann-Solver Free Schemes T. Kr¨ oger, S. Noelle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 Relaxation Dynamics, Scaling Limits and Convergence of Relaxation Schemes H. Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 Multidimensional Adaptive Staggered Grids S. Noelle, W. Rosenbaum, M. Rumpf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 On Hyperbolic Relaxation Problems W.-A. Yong, W. J¨ ager . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 Appendix: Color Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

Wave Processes at Interfaces Sigrid Andreae1 , Josef Ballmann1 , and Siegfried M¨ uller2 1

2

Lehr- und Forschungsgebiet f¨ ur Mechanik der RWTH Aachen sigrid.andreae|[email protected] Institut f¨ ur Geometrie und Praktische Mathematik, RWTH Aachen [email protected]

Summary. We investigate the interaction of shock waves in a heavy gas with embedded light gas bubbles next to a rigid wall. This may give insight regarding cavitation processes in water. Due to the highly dynamical, unsteady processes under consideration we use an adaptive FV scheme for the computations to resolve accurately all physically relevant eﬀects. The results are validated by comparison with tube experiments.

1 Introduction The formation and collapse of vapor bubbles in a liquid is called cavitation. Lord Rayleigh discovered that pressure waves emitted during the process of cavitation [Ray17] may damage solids, e.g., marine screw propellers. Since then, the mechanism of cavitation damaging has been subject of experimental [Lau76, LH85] and analytical research. However, it is still unclear whether the shock and rarefaction waves or the liquid jet onto the solid is the main reason for the erosion of the material. The loading on an elastic-plastic solid exposed to shock and rarefaction waves in water was investigated by Specht in [ASB00]. Hanke and Ballmann showed one-dimensional results for a bubble collapse in water in [HB98]. Cavitation is induced by a pressure drop in the liquid below vapor pressure. Such a pressure decrease may occur due to local acceleration of the liquid flow caused by geometrical constraints, e.g., if the liquid flows through a narrow orifice or around an obstacle. In case the pressure drops below vapor pressure, the liquid bursts and creates a free surface filled with gas and vapor – the bubble. Due to changes in the flow field, the pressure in the liquid may increase afterwards causing the bubble to collapse. The collapse is accompanied by strong shock and rarefaction waves running into the bubble and the surrounding liquid. The shock wave focuses in the center of the bubble. This leads to extreme physical states in the interior. In addition, the shrinking of the bubble leads to a compression of the vapor. Both effects evoke an increase

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S. Andreae, J. Ballmann, S. M¨ uller

of pressure which bulges the bubble. Hereby, a dynamic oscillation process is initiated which finally leads to the collapse of the bubble. If the collapse takes place next to a solid, the pressure distribution becomes asymmetric and a liquid jet develops [PL98] which is either directed towards or away from the solid boundary. The direction of the jet depends on the elasticity of the solid and on a ratio determined by the initial distance between the boundary and the center of the bubble divided by the maximum extension of the bubble. This has been shown experimentally by Brujan et al. in [BN+ 01], [BN+ 01a]. In order to investigate experimentally the dynamics of a bubble collapse, the bubble is produced by a laser pulse. Thereby, the fluid is heated in the focus of the laser and forms a small, hot gas bubble at very high temperature. This experimental setup provides an exact positioning of the bubble. The processes taking place in the interior of the collapsing and oscillating bubble and the prediction of onset and extent of the cavitation damaging are still subject of theoretical and experimental research. However, small time and space scales as well as the complicated dynamics make an experimental approach difficult. Therefore numerical investigations are developed to reveal information about the wave dynamics in the fluid as well as the damaging of the solid. Of particular interest are pressure contours and velocity vectors in the liquid phase as requested in [BN+ 01a]. The primary objective of the present work is to provide an accurate prediction of all occurring wave phenomena. This concerns wave interactions among each other, with phase boundaries or neighboring solids. Of particular interest is the occurrence of instabilities as, e.g., the Richtmyer-Meshkov instability. Since all present methods for simulating two-phase flows suffer from pressure oscillations at the phase boundary, we use a very dense and heavy gas instead of water. The occurring wave phenomena are expected to be qualitatively comparable with those in water. For two-phase flow problems with different equations of state the phase boundary can be tracked or treated in a Lagrangian manner as a sharp, interior boundary using two meshes [Dick96]. The latter suffers from the drawback that the mesh has to be updated in every step which is expensive and time consuming and may result in a poor mesh quality for large displacements of the interface. Instead, we track the phase boundary using a level set method. Consequently, the phase boundary is represented as a mathematically sharp boundary as we will explain in Section 2. We do not implicate surface tension and mixing of the two fluids. For the sake of completeness, it is to be mentioned that for a two phase flow of one fluid a homogenized approach is possible, as will be presented by Voß in [Vos04]. There, one equation of state is used for the liquid as well as the gaseous phase and the so-called mixture region. Here, even states consisting of gas and vapor fractions can be modeled. The small time scales of the unsteady problem require the numerical scheme to be highly efficient regarding computational time and memory requirements. This is realized by a local grid refinement strategy. Furthermore,

Wave Processes at Interfaces

3

all physically relevant phenomena have to be reliably detected and adequately resolved. Moreover, the scheme has to be robust and must not exhibit numerical oscillations, e.g., pressure oscillations at the phase boundary. The details of the numerical scheme are presented in Section 3. In Section 4 numerical results for a bubble collapse near a rigid wall are presented and the arising dynamic wave pattern is discussed.

2 Level Set For modeling two-phase flows, there are mainly two different approaches to treat the two media, a fitting of the phase boundary with two separate grids connected by interface conditions, [Dick96], or one grid with a suitable algorithm to track the phase boundary. Since in the first case the grid has to be redesigned in every time step which is very time consuming, we use only one grid and the level set method to distinguish the two fluids. The level set method, proposed by Osher and Sethian in [OS88], is a tool to track propagating interfaces without an explicit description like a function of the interface under consideration. Instead a scalar field given in the domain is used to represent the motion of the interface. Consider the case of two domains Ω1 , Ω2 separated by a contact surface Γ . Now, a scalar field φ = φ(x, t) is introduced which is φ(x, t) < 0 for Ω1 x ∈ Ω1 and φ(x, t) > 0 for x ∈ Ω2 , see Fig. 1. The Γ Ω2 interface Γ is evolved in time by the fluid velocity v. Therefore, we may describe the time evolution of Fig. 1. Domains. the scalar field φ by ∂φ + v · ∇φ = 0 . (1) ∂t There are two different methods to exploit the evolution of the scalar field for tracking a moving interface. The most common approach suggested by Osher and Sethian in [Set96] is to define the scalar field as a smooth, signed distance function to the front under consideration, whereby the material interface corresponds to φ = 0. Sussman et al. used this approach in computing incompressible two phase flows in [SSO94, SA+ 99, SF99]. The smoothness of φ has to be sustained by a reinitialization after each time step. This way, it is guaranteed that the level set itself will not steepen and develop shocks. A disadvantage of this method is the loss of conservativity. Nguyen et al. suggested methods to recover the conservativity, see [NGF02]. Here, we follow an idea of Mulder et al. [MO92] where the level set function is not a smooth but a discontinuous scalar field. Initially, we assign φ(x, t) = −1 for x ∈ Ω1 and φ(x, t) = +1 for x ∈ Ω2 . This notation for φ is sometimes called “color”-function. The sign (color) of φ decides which fluid occupies which domain. Thereby, the choice of the equation of state is

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S. Andreae, J. Ballmann, S. M¨ uller

controlled. It has to be noticed that we track the jump in φ and not the zero level set. Multiplying equation (1) with the density ̺ and employing the continuity equation of fluid dynamics yields a conservative form of the transport equation for φ, i.e., ∂̺φ + ∇ · (̺φv) = 0 , (2) ∂t Written in this form equation (2) can be added to the system of conservation equations (1) as an additional equation. The main advantage of this approach is the preservation of conservativity.

3 Governing Equations and Method of Solution The fluid flow is modeled by the time-dependent 2D Euler equations for compressible fluids. Appending the evolution equation (2) this leads to the system of conservation equations ∂ U dV + F · n dS = 0 with (3) ∂t V ∂V ⎛

⎞ ̺ ⎜ ̺v ⎟ ⎟ U=⎜ ⎝̺E ⎠, ̺φ

⎛

⎞ ̺v ⎜ ̺ v ◦ v + p1 ⎟ ⎟ F=⎜ ⎝ v (̺ E + p) ⎠ . φv

Here, U is the array of the mean conserved quantities: density of mass, momentum, specific total energy and level set. p is the pressure and v the fluid velocity. The quantity V denotes a time-independent control volume with the boundary ∂V and the outer normal n. The flux F contains only the convective terms. Since the two fluids under consideration are gaseous both, there is no need to deal with the surface tension at their contact surface. The system of equations is closed by the perfect gas equations of state for the fluid i = 1, 2 present in the domain Ωi , i.e. the affiliated thermal equation, p = Ri ̺ T , and the caloric equation, e = cvi T . Herein, e is the internal energy and T the temperature. cvi and Ri are the heat capacity at constant volume and the special gas-constant for fluid i = 1, 2, respectively. The material properties, cvi and Ri , are listed in Table 1. The evaluation of the equations of state is governed by the scalar field φ, i.e.,

p1 = p (R1 , ̺, T ) : φ < 0 p= . (4) p2 = p (R2 , ̺, T ) : φ > 0 The conservation equations (3) are discretized by a finite volume method. The convective fluxes are determined by solving quasi–one dimensional Riemann problems at the cell interfaces. For this purpose we employ a two-phase

Wave Processes at Interfaces

5

Roe Riemann solver designed for the coupled system of the 2D Euler equations and the evolution equation (2) of the level set φ. For the construction of this solver we proceed similarly to [LV89] for real gases. In order to avoid non-physical expansion shocks we use Harten’s entropy fix. The spatial accuracy is improved by applying a quasi one-dimensional second order ENO reconstruction. Due to the strong dynamic behavior of the considered flow problems the time integration is performed explicitly. In order to properly resolve all physical relevant phenomena we need a very fine discretization of the computational domain. Due to the heterogeneity of the flow field, this high resolution is not needed throughout the entire computational domain but only locally near discontinuities. For this purpose we employ a dynamic local grid adaptation strategy to resolve the physically relevant phenomena at the expense of possibly few degrees of freedom and correspondingly reduced storage demands. The main distinction from previous work in this regard lies in the fact that we employ here recent multi-resolution techniques, see [M¨ ul02]. The starting point is to transform the arrays of cell averages associated with any given finite volume discretization into a different format that reveals insight into the local behavior of the solution. The cell averages on a given highest level of resolution are represented as cell averages on some coarse level where the fine-scale information is encoded in arrays of detail coefficients of ascending resolution. This requires a hierarchy of meshes. The multiscale representation is used to create locally refined meshes. For details we refer to [M¨ ul02]. Following Mulder [MO92] we chose φ as a color function in our computations. Mulder observed in [MO92] that using this formulation of φ the pressure shows spurious oscillations at the phase boundary. To reduce these oscillations we use averaged pressure and energy equations near the interface, i.e., ⎫ ⎧ : φ < −ǫ ⎬ p1 ⎨ , (5) p = (1 − αǫ (φ)) p1 + αǫ (φ) p2 : |φ| < ǫ ⎭ ⎩ p2 : φ > ǫ ⎧ ⎫ cv1 : φ < −ǫ ⎬ ⎨ (1 − αǫ (φ)) cv1 + αǫ (φ) cv2 : |φ| < ǫ e = T cv = T . (6) ⎩ ⎭ : φ > ǫ cv2 Here, the function αǫ (φ) is chosen as a linear interpolation between 0 and 1 in the interval [−ǫ, ǫ], i.e., αǫ (φ) = (φ/ǫ + 1) /2 for |φ| < ǫ. For our computations we chose ǫ = 0.5. Results of a comparison between the approximate Riemann solver using this averaging method and an exact Riemann solver are given in Figs. 2(a) and 2(b). Since φ is initialized by +1 for fluid 1 and by −1 for fluid 2, ǫ has to be chosen less than 1 to make sure that the modification of the pressure law is only applied in the vicinity of the interface. Note that the initial jump of φ is smeared by the FV scheme. The width of this numerical transition layer depends on the underlying grid resolution. It becomes smaller with finer

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S. Andreae, J. Ballmann, S. M¨ uller

grids. In particular, when the material boundary is a phase boundary and different equations of state have to be applied on both sides, a grid adaption strategy is strongly recommended to provide a high resolution of the interface. From this point of view, equation (5) describes not really a physically relevant phase transition, it serves more as a numerical stabilizer of the phase boundary.

4 Numerical Results The current work focuses on the ability of the scheme to accurately resolve the dynamics and wave pattern occurring in the presented test configurations for two-fluid flow. At first, we validate our scheme using experiments performed by Haas and Sturtevant, [HS87]. Herein, a shock runs across a bubble filled with helium in the one case and with R22 gas in the other. R22 is the heavy refrigerant chlorodifluromethane (CHClF2 ). In Table 1 the physical properties of the gases under consideration are given. The surrounding fluid is air in both cases. We compare our numerical results with the Schlieren photographs taken by Haas and Sturtevant. The third configuration is a helium bubble surrounded by R22 and placed next to a rigid wall, with initial conditions corresponding to an explosion problem. The main focus lies on the interaction of the emitted waves with the wall. Since we expect the occurring wave phenomena have something in common with the formation and collapse of a cavitation bubble, this configuration indicates possible causes for the damaging mechanisms accompanying cavitation in the related experiments. The Table 1. Molecular weight umol , special gas-constant R, ratio of speciﬁc heats γ and speed of sound c (at 293.15 K, 101, 35 kPa) for air, helium and R22. ﬂuid

umol [103 kg/mol]

R [J/kg/K]

γ

c [m/s]

air helium R22

28.964 4.003 864.687

287.0 2077.0 96.138

1.4 1.66 1.178

343.3 1007.4 184.0

characteristic physical quantity in dealing with wave interactions with boundaries is the acoustic impedance ̺c. Herein, c is the speed of sound. The ratio of the acoustic impedances of two fluids governs what happens to a shock wave traveling through fluid 1 and impinging on the phase boundary between fluid 1 and 2. According to the acoustic wave theory, the impinging shock wave is split up in a transmitted part traveling through Fluid 2 and a reflected part. The larger the jump of the acoustic impedance the more energy is reflected. In case, (̺c)1 >> (̺c)2 most of the energy is reflected with a phase change of 180◦ . If (̺c)1 << (̺c)2 most of the energy is reflected, too, but without change

Wave Processes at Interfaces

7

of phase angle. The results for shock impacts on the helium and the R22 bubble, presented in Section 4.2, show the influence of the acoustic impedance on the wave pattern. This will be discussed later in this chapter. 4.1 Validation To validate the solver the solution of a 1D Riemann problem is compared to the exact solution, see Figure 2(a), evaluated with an exact Riemann solver by Colella and Glaz [CG85] which is capable to deal with two phases. Apart from some slight pressure wiggles at the phase boundary, our approximate solver gives satisfying results. For time t = 0 the membrane at x = 0 cm bursts; on the left hand side in the high pressure region is helium, on the right hand side R22. A shock runs into the low pressure region. Behind the shock the density jumps from 10 kg/m3 to 67.2107 kg/m3 . The shock is followed by a very fast contact discontinuity. Over the contact discontinuity the density drops to 10.5365kg/m3 which is nearly its right initial value. Into the high pressure region runs a rarefaction wave. Table 2. Initial conditions for two phase Riemann problem with helium and R22.

3

̺ [kg/m ] p [N/m2 ] T [K] ̺ c [kg/s/m2 ] vx , vy [m/s]

helium bubble

R22 surrounding

20.0 1.217 × 107 293.0 20101.85 0.0 , 0.0

10.0 281684.3 293.00 1821.78 0.0 , 0.0

3

5

ρ [kg/m ]

2

p [10 N/m ]

60

100

40 50

20 0

-4

-2

0

(a)

2

4 x [m]

0

-4

-2

0

2

4 x [m]

(b)

Fig. 2. Exact and approximative solution for the two-phase Riemann problem helium (left) and R22 (right); density (a) and pressure (b).

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S. Andreae, J. Ballmann, S. M¨ uller

4.2 Shock Bubble Interaction Haas and Sturtevant performed experiments with shocks passing gas inhomogeneities of helium or R22 in an air surrounding to clarify the mechanisms of turbulence and mixing caused by shock waves, see [HS87]. They carried out two-dimensional and three-dimensional (for consistency) experiments. In the two-dimensional experiments, we compare with a cylindrical volume that is first enclosed by a 0.5 µm thick nitrocellulose membrane which bursts under the impact of the shock wave. With shadowgraph photography wavefronts and the topology of the bubble were made visible. The geometrical setup shown in Figure 3 is the same for both problems. The initial mesh has 125 × 10 cells and 5 levels of refinement are used. Due to the mirror symmetry of the problem, only the upper half was computed. A shock coming from left impinges on a gas bubble. Initially, the bubble and the surrounding pre-shocked air are at rest and in thermal and mechanical equilibrium. The initial conditions for the problems discussed here are given in Tables 3 and 4. pre−shocked

r = 0.025m at 0.4895 m

shocked

air

shock position at 0.02225 m

air

vshock

helium

r

0.0445 m

post−

axsis of symmetry 0.445 m

Fig. 3. Computational domain for subsection 4.2.

Helium Bubble in Air In the first computation the setup is the helium filled gas bubble surrounded by air. Figure 8 shows the time evolution of the density gradient in the domain given in Figure 3. In Figures 9(a) and 9(b) the pressure p is plotted in zdirection over the x-y-plane and the phase boundary is marked as a black line. In the x-y-plane the pressure gradients are shown (shock fronts visible as black lines). These figures show results for the same instant as Figures 8(b) and 8(d). Figure 4 shows a comparison between the experimental and our computational results. Two clippings of the adapted mesh are shown in Figures 5(a) and 5(b). In Figure 8(a) the incoming shock (marked as i) has already crossed the most left part of the bubble boundary. It is partly transmitted as a refracted shock (rr) and partly reflected as a rarefaction wave (rw). This behavior is governed by the ratio of the acoustic impedances, see values given in Table 3. Inside the bubble the transmitted shock runs ahead since the speed of sound

Wave Processes at Interfaces

9

Table 3. Initial conditions for shock interaction with helium bubble.

̺ p ̺c vx , vy

post-shocked air

pre-shocked helium

pre-shocked air

1.376 1.575 1.742 0.396 , 0.0

0.138 1.0 0.479 0.0 , 0.0

1.0 1.0 1.183 0.0 , 0.0

in helium is higher than in air at the same temperature. The shock-front is curved due to the spherical shape of the undisturbed phase boundary. The fore-running shock in helium arches as a thin, black line from x = 0.039 m to x = 0.0315 m at the phase boundary where the shock just hits the boundary. Outside the bubble the incident shock is visible as a straight, black, vertical line. The density jump at the phase boundary is a thin, opaque line marked as (pb). Behind the shock, the reflected rarefaction wave appears as a dark area. Since in the very beginning of the shock bubble interaction the shock front is parallel to the phase boundary, all the waves travel in x-direction. Later on, the shock impinges on the phase boundary under an increasing angle, see Fig. 8(b). Due to the laws of geometrical optic the rarefaction wave is reflected under the same angle as the shock impinges on the helium surface. Since the shock inside is faster than outside, a shock wave (marked s as “ side” shock) emanates where the refracted shock meets the phase boundary. A complicated four shock configuration develops which Henderson explained in [HCP91] and called twin regular reflection refraction. In Figure 8(c) the refracted shock is just passing the most right boundary of the bubble at x = 0.073 m, whereas the incident shock is at x = 0.043 m. The acoustic impedance in the post-shocked helium is only 0.542 kg/m2 /s but in the pre-shocked air 1.183 kg/m2 /s. Therefore, the air acts in the sense of a rigid boundary which makes that the reflected part of the shock wave hitting this boundary is a shock. This reflected shock (rl) focuses on the x-axis at 0.059 m which is visible as a small, light dot in the density gradients of Fig. 8(d) (marked by an arrow). The focus is more distinct in the corresponding pressure gradients, see Fig. 9(b). Here, the pressure is plotted in z-direction over the x-y-plane. Below, in the same figure the pressure gradient isolines are shown in gray-scale. As a result of the higher shock speed the helium near the x-axis is stronger accelerated than the air above it. Thereby, an anti-clockwise rotation of the bubble content is induced and at the symmetry axis the bubble constricts and develops a small throat. The helium volume remains rotating, splits up at the x-axis and travels circulating upstream, see Figs. 8(h)–8(j). We compared our results to photographs taken by Haas and Sturtevant, see [HS87], and found a good agreement see Figs. 4. In particular, the numerical results exhibit all waves visible in the Schlieren photographs. However, since the numerical results do neither include the complete experimental setup, e.g., the support for cylindrical membrane in Fig. 4(c), nor perturbations due

10

S. Andreae, J. Ballmann, S. M¨ uller

Fig. 4. Comparison between numerical (a,b) and experimental (c,d) results for helium bubble in air. Experimental pictures scanned from [HS87]. The ring is part of the experimental setup. For the indices at the waves, see Fig. 8.

0.04 y [m]

0

0.1

0.15

0.2

x [m]

0.25

0.3

(a) Resolution of bubble (left) and shock (right).

0.04

y [m]

0

0.1

0.15

x [m]

(b) Grid resolution in the bubble region. Fig. 5. Parts of adapted mesh corresponding to Fig. 8(i).

Wave Processes at Interfaces

11

to the rupture of the membrane, the comparison can only be qualitative. Nevertheless, as indicated by the same labels as in Fig. 8 all the waves from the experiment are resolved in the computation. In particular, the topology of the bubble is excellently reproduced. Notice that the ring which was necessary to fix the bubble in the experiment must not be confused with a wave surface. To show qualitatively the grid refinement, a part of the adapted mesh is presented in Fig. 5(a). The shock at x = 0.27 m as well as the bubble contour are well resolved. The re-coarsening of the grid inside the bubble is visible in Fig. 5(b). R22 Bubble in Air For the case of an R22 bubble in air, physical data are given in Table 4. The density gradients corresponding to the experimental Schlieren images are shown in Fig. 10. Figure 6 shows the absolute value of the velocity and integralcurves of the instantaneous velocity field. In Figure 11(a) the density is plotted in z-direction over the x-y-plane. In the x-y-plane the density gradient isolines are plotted in gray-scale; Figure 11(b) shows a similar plot for the pressure in z-direction, but here the phase boundary is marked with a black line, and in the x-y-plane the pressure gradient isolines are presented. Since the acoustic impedance of R22 is only slightly higher than the acoustic impedance for the post-shocked air, see Table 4, the incident shock (i) is mainly transmitted and only a small portion is reflected as a shock (rl), see Fig. 11(a). There, the transmitted and thereby refracted shock (rr) is visible as a concave curved black line extending from x = 0.037 m to x = 0.048 m. Compared to the incident shock (i) the refracted shock (rr) is slower, because the sound speed in R22 is lower than in air. This is also the reason for the higher acceleration of the air above the bubble compared to that of the gas R22. This fact leads to a clock-wise rotation of the material in the R22 bubble later on. The outwards running shock diffracts since it is decelerated at the phase boundary, see Fig. 10(c). Between the incident shock (i) and the inside running shock (rr) develops a compression wave (cw). By the compression wave the flow direction is turned by 90◦ towards the symmetry axis as indicated in Fig. 6. The front of the refracted shock bends more and more until it focuses on the x-axis at the phase boundary, see box on left side in Table 4. Initial conditions for shock interaction with R22 bubble.

γ ̺ p ̺c vx , vy

post-shocked air

pre-shocked R22

pre-shocked air

1.4 1.376 1.575 1.742 0.396 , 0.0

1.178 2.985 1.0 1.875 0.0 , 0.0

1.4 1.0 1.0 1.183 0.0 , 0.0

12

S. Andreae, J. Ballmann, S. M¨ uller

Fig. 6. Absolute value of velocity with integral-curves for vx and vy (arrows) and phase boundary (φ = 0-level) as a black dashed line.

Fig. 10(e). The arising pressure peak is depicted in Fig. 11(b). The shock is reflected after focusing and runs outwards, see wave (rf) in Fig. 10(f). Again the shock (rf) is traveling slower in the R22 than in the helium. When the incident shock has passed the bubble it crosses its symmetric counterpart, see Fig. 10(f) at x = 0.06 m. Thereby, a reflected shock (s) running upstream is induced. These two shocks (s and rf) pass across the bubble in upstream direction and cause reflected and refracted waves inside the R22 bubble visible in the density gradients of Figs. 10(g)–10(i). Since they do not produce new physical effects they are not discussed further. The bubble migrates downstream and thereby it prolongates and rolls up its top (t). In Figure 10(j) the phase boundary represented by the zero level φ = 0 is indicated by a solid, black line. Obviously, there are growing instabilities on the top of the structure. It is assumed that these are Rayleigh-Taylor instabilities due to the shock passing across a curved phase boundary. The comparison with the experiment is given in Fig. 7. Again, the ring from the experimental setup is visible. The wave (w) at the bottom of Fig. 7(c)

Fig. 7. Comparison between numerical (a,b) and experimental (c,d) results for R22 bubble in air. Experimental pictures scanned from [HS87]. The ring is part of the experimental setup. For the indices at the waves, see Fig. 10.

Wave Processes at Interfaces

13

is a reflection from the shock tube wall. A good agreement between the phase boundaries (pb), the incident and refracted shocks (i and rr) is visible in in Figs. 7(a) and 7(c). Note that even the compression wave (cw) is resolved. In Figures 7(b) and 7(d) the phase boundary (pb) as well as the shock (rf) – reflected from the focus of the refracted shock – match perfectly their experimental counterparts. At the left border the reflected shock (s) from the crossing of the incident shocks at the symmetry axis is visible. 4.3 Explosion Problem Helium in R22 Next to a Rigid Wall This computation is motivated by the idea of a laser induced vapor bubble in a liquid next to a rigid surface. The resulting bubble expands very fast and causes a pressure wave in the liquid which interacts with the wall and after reflection again with the bubble. In the case discussed in the sequel, we study the sudden expansion of a hot, high pressure helium bubble with cylindrical shape surrounded by the heavy gas R22 at low temperature. The midpoint of the bubble of diameter 0.01 m is placed at x = 0.14 m , y = 0.08 m at the right of the computational domain which has the size 0.16 m × 0.32 m. Since the problem is mirror-symmetric with respect to the x-axis we computed only the upper half. Both fluids are treated as perfect gases. The chemical data is given in Table 1. The zero level set lies on the phase boundary between helium and R22; it is indicated as a white, dashed line in the Figs. 12–17. The initial conditions correspond to a the two-phase Riemann problem in Section 4.1. The computational results for the two-dimensional explosion problem are presented in Figures 12–21. The series of Figs. 12–17 show pressure (a), density (b) and the corresponding grid (c) at different times. The phase-boundary which is evaluated from the level set function as its zero level is indicated as a white dashed line in every picture. In the very beginning a circular shock wave (s) runs undisturbed outwards, see Figs. 12(a) and 12(b). It is closely followed by the contact discontinuity (white dashed line). Inwardly, a rarefaction wave runs concentrically into the center of the bubble. There, it is reflected as a now outwards running rarefaction wave which follows the shock and the contact discontinuity, see Figs. 13(a) and 13(b). In Figures 12(c) and 13(c) it is clearly visible that the mesh is refined close to the waves and is re-coarsened in the center after the reflection of the rarefaction wave. The first wave hitting the rigid wall is the shock. It is then reflected as a shock wave and the pressure level increases due to the superposition, see pressure and density legend in Fig. 13. The reflected shock, see rs in Figure 14, passes over the contact discontinuity which is decelerated thereby. Due to the gas R22 present in the high pressure region behind the shock, the contact discontinuity does not reach the wall, but is repelled before it, see Fig. 21(a) at 20 µs. The outwards running waves set the material inside the bubble in an outwards directed motion, see Fig. 18. In the center, the rarefaction wave is reflected as an outwards running rarefaction wave which decelerates the outwards running material producing an inverse

14

S. Andreae, J. Ballmann, S. M¨ uller

pressure gradient. This gradient steepens forming a shock inside the helium domain, see s1 in Fig. 14(a) and Fig. 21(a) at time t = 29 µs and x = 0.13 m x = 0.15 m, respectively. The shock that was reflected at the solid wall catches up the right branch of shock s1 in the helium region at x = 0.149 m and y = 32 µs and merges with it. Due to the interaction a weak contact discontinuity appears, see Fig. 21(b) at x = 0.147 m and t = 35 µs. In theses pictures, the wave dynamics on the x-axis in a range from x = 0.08 − 0.16 m is plotted as pressure (Fig. 21(a)) and density (Fig. 21(b)) isolines in an x-t plane. Later on, the shock reflected from the wall also interacts with left branch of the shock s1 in the interior of the helium bubble. They cross each other mainly undisturbed and a contact discontinuity emerges from the interaction, see x = 0.137 m and t = 45 µs in Fig. 21(a) (the phase boundary is marked as a black, dotted line). Since the initial shock front is cylindrical, it hits the wall under a continuously increasing angle. At first, the shock impinges onto the wall under an angle of 0◦ and is reflected regularly. With increasing angle, however, a regular reflection becomes impossible as explained by, e.g., BenDor in [BD91] and a Mach stem (ms) develops, see Fig. 17 on the x-axis at y = 0.072 m. The slip line (sl) coming from the Mach stem is visible in Fig. 19. In the case of a cavitation bubble collapsing next to a elastic or elastic-plastic wall in water the Mach stem of the reflected shock produces surface waves in the wall which could be one reason for cavitation damaging, see Specht in [Spe00]. The pressure distribution at the rigid wall in Fig. 20 displays the loading jump evoked by the Mach stem – visible at y = 0.0517 m – on the solid.

5 Conclusion and Future Work We presented results for highly dynamical two-fluid flow problems with wave interactions at material boundaries. The comparison with experiments in Section 4.2 verifies that our solver is adequate for computing two-fluid flow problems for different perfect gases. The advanced grid refinement strategy automatically detects all appearing waves and provides a perfect resolution of those waves. For the test case of a strongly expanding high pressure cylindrical gas bubble next to a rigid wall, the complicated wave interactions are well resolved and give hope for further computations with gas and liquid. Acknowledgement. The authors would like to thank Dr. Alexander Voß, Institut f¨ ur Geometrie und Praktische Mathematik, RWTH Aachen, for helpful discussions on Riemann problems and programming issues. In particular, the discussions on the physical modeling have been very helpful. This work was ﬁnancially supported by the DFG Priority Research Program “Analysis and Numerics for Conservation Laws”.

Wave Processes at Interfaces

i

(e)

rr

rw pb rl

Fig. 8. Shock-bubble (helium) interaction, (a)-(e): density gradients.

15

(j)

Fig. 8. Shock-bubble (helium) interaction, (f)-(j): density gradients.

Wave Processes at Interfaces

17

i rw Z

pb

fo c u

pb

s rl

rr

(b) Fig. 9. x-y-plane: pressure gradients, z-axis: Pressure over x-y-plane with phase boundary marked as a black, solid line. For annotations, see Fig. 8.

18

S. Andreae, J. Ballmann, S. M¨ uller

(e)

Fig. 10. Shock-bubble (R22) interaction, (a)-(e): density gradients.

Wave Processes at Interfaces

(j)

t

pb

pb

Fig. 10. Shock-bubble (R22) interaction, (f)-(j): density gradients.

19

20

S. Andreae, J. Ballmann, S. M¨ uller

pb

pb

cw Z

fo c u

s

s rr

i

(b) Fig. 11. x -y-plane: (a) density gradients, z -axis: density over x -y-plane. (b) pressure gradients, z -axis: pressure over x -y-plane. For annotations, see Fig. 10.

Wave Processes at Interfaces

s

s

0

0

(a) Pressure.

(b) Density. Fig. 12. t = 7.0µs, φ = 0 white dashed line.

s

s

0

0

(a) Pressure. (b) Density. Fig. 13. t = 18.53µs, φ = 0 white dashed line.

(c) Grid.

s

s

rs

rs s1

0

s1

0

(b) Density. (a) Pressure. Fig. 14. t = 29.00µs, φ = 0 white dashed line.

(c) Grid.

s

s

rs

rs s1 0

(c) Grid.

s1 0

(b) Density. (a) Pressure. Fig. 15. t = 40.00µs, φ = 0 white dashed line.

(c) Grid.

21

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S. Andreae, J. Ballmann, S. M¨ uller

s

s rs

s1

rs

s1

0

0

(b) Density. (a) Pressure. Fig. 16. t = 59.99µs, φ = 0 white dashed line.

ms

ms s

s

rs 0

(c) Grid.

s1

rs

s1

0

(a) Pressure. (b) Density. Fig. 17. t = 90.01µs, φ = 0 white dashed line.

(c) Grid.

rs

Fig. 18. Absolute value of velocity with integral-curves.

Fig. 19. Isolines of Mach number; mach stem (ms), slipline (sl).

Wave Processes at Interfaces

ms s

rs s1 0

Fig. 20. Pressure distribution at wall for t = 90.01µs.

t [µ s]

t [µ s]

x [m]

x [m]

(b) Density isolines. (a) Pressure isolines with φ = 0 as black dashed line. Fig. 21. Wave dynamics at x-axis for all computed time steps.

23

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S. Andreae, J. Ballmann, S. M¨ uller

References [ASB00]

[BD91] [BN+ 01] [BN+ 01a]

[CG85] [Dick96]

[HS87]

[HB98] [HCP91] [Lau76] [LH85]

[LV89]

[MO92] [M¨ ul02]

[NGF02]

[OS88]

[PL98] [Ray17] [Set96]

Andreae, A., Ballmann, J., Specht, U.:Wave phenomena at liquid-solid interfaces. In: Proceedings of Eighth International Conference on Hyperbolic Problems - Magdeburg, Birkh¨ auser (2000) Ben-Dor, G.: Shock Wave Reﬂection Phenomena. Springer (1991) Brujan, E.-A., Nahen, K., Schmidt, P., Vogel, A.: Dynamics of laserinduced cavitation bubbles near an elastic boundary. J. Fluid Mech., 433, 251–281 (2001) Brujan, E.-A., Nahen, K., Schmidt, P., Vogel, A.: Dynamics of laserinduced cavitation bubbles near elastic boundaries: inﬂuence of the elastic modulus. J. Fluid Mech., 433, 283–314 (2001) Colella, Ph., Glaz, H. M.: Eﬃcient solution algorithms for the Riemann problem for real gases. J. Comp. Phys., 59, 264–289 (1985) Dickopp, Ch.: Ein Navier Stokes L¨oser zur Simulation kollabierender Kavitationsblasen in der N¨ ahe elastischer Festk¨ orperoberﬂ¨ achen. Shaker Verlag, Aachen (1996) Haas, J. F., Sturtevant, B.: Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities. J. Fluid Mech., 181, 41–76 (1987) Hanke, M., Ballmann, J: Strong changes of state in collapsing bubbles. ZAMM, 78, 453–453. (1998) Henderson, L. F., Colella, P., Puckett, E. G.: On the refraction of shock waves at a slow-fast gas interface. J. Fluid Mech., 224, 1–27 (1991) Lauterborn, W.: Numerical Investigation of Nonlinear Oscillations of Gas Bubbles in Liquids. J. Acoust. Soc. Am., 59 2,283–293 (1976) Lauterborn, W., Hentschel, W.: Cavitation bubble dynamics studied by high speed photography and holography: Part One. Ultrasonics, 23, 260–268 (1985) Liu, Y., Vinokur, M.: Nonequilibrium ﬂow computations. I. An analysis of numerical formulations of conservation laws. J. Comp. Phys., 83, 373–397 (1989) Mulder, W., Osher, S.: Computing interface motion in compressible gas dynamics. J. Comp. Phys., 100, 209–228 (1992) M¨ uller, S.: Adaptive Multiscale Schemes for Conservation Laws, Lecture Notes on Computational Sciences and Engineering, Vol. 27, Springer, 2002 Nguyen, D., Gibou, F., Fedkiw, R.: A fully conservative ghost ﬂuid method & stiﬀ detonation waves. 12th Int. Detonation Symposium, San Diego, CA (2002) Osher, S., Sethian, J. A.: Fronts Propagating with CurvatureDependent Speed: Algorithms Based on Hamilton-Jacobi Formulations. J. Comp. Phys.,79, 12–49 (1988) Philipp, A., Lauterborn, W.: Cavitation erosion by single laser-produced bubbles. J. Fluid Mech., 361, 75–116 (1998) Lord Rayleigh: On the pressure developed in a liquid during the collapse of a spherical cavity. Phil. Mag., 34, 94–98 (1917) Sethian, J. A.: Level Set Methods, Cambridge Monographs on Applied and Computational Mathematics (1996)

Wave Processes at Interfaces [Spe00] [SSO94] [SA+ 99]

[SF99]

[Vos04]

25

Specht, U.:Numerische Simulation mechanischer Wellen an Fluid-Festk¨ orper-Mediengrenzen. Fortschritt-Berichte VDI, Reihe 7, 398 (2000) Sussman, M., Smereka, P., Osher, S.: A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow. J. Comp. Phys.,114, 146–159 (1994) Sussman, M., Almgren, A. S., Bell, J. B., Colella, P., Howell, L. H., Welcome, M. L.: An Adaptive Level Set Approach for Incompressible Two-Phase Flows. J. Comp. Phys., 148,81–124 (1999) Sussman, M., Fatemi, E.: An eﬃcient, interface-preserving Level Set Redistancing Algorithm and its Application to interfacial incompressible Fluid Flow. SIAM J. Sci. Comput.,20/4, 1165–1191 (1999) Voß, A.: Exact Riemann Solution for the Euler Equations with Nonconvex and Nonsmooth Equation of State, PhD thesis, RWTH Aachen, 2005

Numerics for Magnetoplasmadynamic Propulsion J¨ org Heiermann1 , Monika Auweter-Kurtz2 , and Christian Sleziona3 1

2

3

Institute of Space Systems, University of Stuttgart [email protected] Institute of Space Systems, University of Stuttgart [email protected] IHI, Heidelberg [email protected]

Summary. A ﬁnite volume method has been developed in this work for solving the conservation equations of argon plasma ﬂows in magnetoplasmadynamic self–ﬁeld accelerators. These accelerators can be used for interplanetary spaceﬂight missions because of their high speciﬁc impulse and high thrust density. Calculations show in agreement with the experiment that a primary reason for plasma instabilities at high current settings – which are limiting the operational envelope and the thruster lifetime – is the depletion of density and charge carriers in front of the anode because of the pinch eﬀect. The calculated thrust data agree well with experimental values, so that the newly developed method can be used for the design and optimization of new thrusters.

1 Introduction Manned interplanetary spaceflight is the next milestone after the Apollo missions on the moon, the introduction of the partly reusable space transportation system Space Shuttle, the commercialization of Earth–observing, navigation and telecommunications satellites and the building of the International Space Station. The choice of the propulsion system is most crucial to achieving reasonably short flight durations for interplanetary missions. Electric propulsion systems [Jah68, Auw92] are particularly suitable because they produce considerably higher specific impulses than chemical propulsion systems [BWB90]. In magnetoplasmadynamic (MPD) self–field thrusters the fuel is heated and ionized by an electric arc (Fig. 1). The plasma is accelerated by thermal expansion and magnetic forces caused by the self–induced magnetic field so that exhaust velocities above 20 km/s and thrust levels of 100 Newton can be achieved. Depending on the thruster design, the input power for MPD self–field engines is between 100 kW and 1 MW, which must be delivered by

28

J. Heiermann, M. Auweter-Kurtz, C. Sleziona

big solar generators or nuclear energy sources. Because nuclear reactors have a compact design and because of the great distance from the sun, nuclear power is the only reasonable energy source for interplanetary flights to the outer solar system so that MPD self–field engines – besides nuclear thermal rockets – represent an advantageous propulsion concept due to their high exhaust velocity and high thrust density. Since the 1980s MPD thrusters have been investigated experimentally [Weg94, ABH98], theoretically [Wag94, Kae91] and numerically [Sle92, Sle98, Boi99] at IRS. Technology aspects for the thruster design [Bue86, Auw94] and basic plasmaphysical processes [Sch82, Sch87] have been considered in order to achieve higher efficiencies and to avoid power–limiting instabilities. Reasons for thruster instabilities, which are characterized by voltage oscillations and increasing anode losses [MAH88], are the depletion of charge carriers at the anode [Hue80], micro turbulence [Cho91, Cho99] and space charge, drift and gradient driven plasma instabilities [Sch82, WKA98]. Taking into account the high cost of experiments, the application of numerical methods is necessary for detailed basic investigations of MPD flows and for thruster optimization.

2 Conservation Equations The configurations of the IRS–built nozzle–type self–field MPD accelerators DT2 and HAT, which are investigated in this work, are axisymmetric so that the flow is rotationally symmetric. Hence, the conservation equations can be written in a two–dimensional form using cylindrical coordinates. The assumption of axisymmetry allows only instabilities to occur in an axisymmetric manner as well. Argon is the propellant considered because this noble gas is used at IRS. Assuming continuum flow and quasi neutrality of the plasma [Cap94, Sch93], conservation equations for a two–fluid flow in thermal and reaction non–equilibrium are formulated without explicitly treating the electrode sheaths. Heavy particle species (neutral, singly and multiply ionized argon) and electrons are considered. The heavy particle gas and the electron gas are treated as ideal gases. Thermal non–equilibrium between electrons and heavy particles is assumed because the densities and thereby the collision frequencies and the energy exchange between the particles can become relatively low for expanding supersonic flows, and because heavy particles are cooled on solid walls while the electrons behave adiabatically. Reaction non–equilibrium is assumed because of the high flow speeds. Because of their low mass and their high mobility the electrons are assumed to be carrying the electric current in the plasma. Hence, the electron gas is being ohmically heated, and part of the energy is transferred to the heavy particles by collisions.

Numerics for Magnetoplasmadynamic Propulsion

29

In order to be able to simulate turbulence in the interior of the thruster and in the free plasma jet, a single–equation model is employed. The arc discharge is described by the Maxwell equations of classical electrodynamics and by Ohm’s law for plasmas. Including boundary conditions and constitutive equations, one obtains a closed system of thirteen conservation equations for the species densities ni , the axial momentum ρvz , the radial momentum ρvr , the heavy particle energy eh , the turbulence ρR, the electron energy ee , and the stream function Ψ of the self–induced magnetic field: ∂ni = − div (ni v) − div jD,i + ωi ∂t

for i = 0...6 ,

(1)

∂(ρvz ) = − div f z,invisc − div f z,visc , ∂t

(2)

∂(ρvr ) = − div f r,invisc − div f r,visc + qr , ∂t

(3)

∂eh = − div f h,invisc − div f h,visc + qh , ∂t

(4)

√ νT ∂(ρR) = − div(ρR v) + div ρ νh + ∇R + ρ(Cε2 f2 − Cε1 ) R P ∂t σε νT ρ − ∇ ρ νh + · ∇R − ∇νT · ∇R , σε σε (5) 5k 1 ∂ee = − div(ee v) − pe div v + j · ∇Te − j · ∇pe ∂t 2e ene 6 3 kTe − div Zi jD,i + div(λe ∇Te ) 2 i=1 +

6 i=0

(6)

5

ne ni αei (Th − Te ) +

|j|2 − ωi+1 χi→i+1 , σ i=0

Ψ Ψ vr 1 ∂Ψ = − div v + 2 r ∂t r r ⎤ ⎞ ⎡ ⎤ ⎡ ⎡ ⎤ ⎛ 0 0 0 β 1 rot⎣Ψ/r⎦ + rot⎣Ψ/r⎦ × ⎣Ψ/r⎦ − β∇pe ⎠ . (7) − rot⎝ µ0 σ µ0 0 0 0 ϕ

30

J. Heiermann, M. Auweter-Kurtz, C. Sleziona

The system is hyperbolic–parabolic and includes source terms. The equations are formulated in such a way that by using fluxes a maximum of conservativity is achieved in the finite volume discretization. The details concerning the physical modelling, the complete system of equations, the boundary conditions and the definitions of all variables can be found in Ref. [Hei02].

3 Numerical Methods The solving of the conservation equations is performed on unstructured, adaptive triangular meshes with dual cells. Unstructured meshes can be produced and modified significantly easier and faster for the complex geometries of self–field accelerators than structured meshes. By appropriate adaption, the computational time for approximating a solution can be reduced considerably. For the calculation of the inviscid fluxes the robust flux vector splitting scheme of Eberle is modified and extended [HAE99, HAK99, Ebe00]. With the heavy particle pressure ph , the electron pressure pe and the magnetic field B the magnetoacoustic speed is defined as γ(ph + pe ) B2 + . (8) c = ρ µ0 ρ One now defines a new reference speed q2 s = αc2 + q 2 1 − 2α + α 2 , c

(9)

with q 2 = min(c2 , qn 2 ) . α is chosen as α =

2 γπ

(10) (11)

with γ = 5/3 being the adiabatic coefficient. The velocity normal to the cell face of two neighbouring dual cells is qn = vz nz + vr nr .

(12)

In the case of qn vanishing, the choice of α leads to the mass flux from one side being equal to the effusion flux. The eigenvalues of the heavy particle flow are now defined as λ0 = qn , λ1 = λ0 + s , λ2 = λ0 − s .

(13)

Numerics for Magnetoplasmadynamic Propulsion

31

With the physical states left (l) and right (r) of a cell face the following equations are chosen for the splitting: 1 (λ1l + |λ1l |), 4 1 h1r = (λ1r − |λ1r |), 4 1 h2l = (λ2l + |λ2l |), 4 1 h2r = (λ2r − |λ2r |). 4 The upwind flux function for the species densities ni is now h1l =

(14) (15) (16) (17)

ni Finvisc = nil (h1l + h2l ) + nir (h1r + h2r ) .

(18)

With r1 = ρh1

and r2 = ρh2

(19)

the axial and the radial inviscid momentum fluxes are given by ρvz Finvisc ⎡⎛

⎢⎜ ⎜ =⎢ ⎣⎝vz + ⎡⎛

⎢⎜ ⎜ +⎢ ⎣⎝vz +

B2 ph + pe + 2µ0 ρs

ph + pe +

B2 2µ0

ρs

⎞

⎛

nz ⎟ ⎜ ⎟ r1 + ⎜vz − ⎝ ⎠

B2 ph + p e + 2µ0 ρs

⎞ ⎤ nz ⎟ ⎥ ⎟ r2 ⎥ ⎠ ⎦

⎞ ⎤l ⎛ ⎞ B2 p + p + nz ⎟ nz ⎟ ⎥ h e ⎜ 2µ0 ⎟ r1 + ⎜vz − ⎟ r2 ⎥ ⎝ ⎠ ⎠ ⎦ ρs

r

(20)

and ρvr Finvisc ⎡⎛

⎢⎜ ⎜ =⎢ ⎣⎝vr + ⎡⎛

⎢⎜ ⎜ +⎢ ⎣⎝vr +

B2 ph + p e + 2µ0 ρs

ph + pe + ρs

B2 2µ0

⎞

⎛

nr ⎟ ⎜ ⎟ r1 + ⎜vr − ⎠ ⎝

B2 ph + p e + 2µ0 ρs

⎞ ⎤ nr ⎟ ⎥ ⎟ r2 ⎥ ⎠ ⎦

⎞ ⎤l ⎛ ⎞ B2 p + p + nr ⎟ nr ⎟ ⎥ h e ⎜ 2µ0 ⎟ r1 + ⎜vr − ⎟ r2 ⎥ . ⎝ ⎠ ⎠ ⎦ ρs

The heavy particle energy flux is

r

(21)

32

J. Heiermann, M. Auweter-Kurtz, C. Sleziona eh Finvisc

= +

ph B2 ρ + (vz 2 + vr 2 ) + ph + pe + γ−1 2 2µ0 2

ph ρ B + (vz 2 + vr 2 ) + ph + pe + γ−1 2 2µ0

r1 + r2 ρ r1 + r2 ρ

l

(22) .

r

For the turbulence, the electron energy and the magnetic field, the upwind flux functions are constructed like equation (18): ρR Finvisc = (ρR)l (h1l + h2l ) + (ρR)r (h1r + h2r ) ,

(23)

ee Finvisc = eel (h1l + h2l ) + eer (h1r + h2r )

(24)

B = Bl (h1l + h2l ) + Br (h1r + h2r ) . Finvisc

(25)

and

This upwind formulation is also being used for the discretization of the term div v, because a central discretization would lead to numerical oscillations: div v Finvisc = h1l + h1r + h2l + h2r .

(26)

The upwind algorithm is very robust with respect to strong shocks, and it is simple and requires only a few calculations to obtain a good approximation of the solution of the Riemann problem. The variables which are needed at the cell faces for the approximate solution of the Riemann problem are linearly reconstructed by a Weighted Essentially Non–Oscillatory (WENO) scheme [Fri98] so that the discretization achieves a second–order accuracy in space. The WENO algorithm is explained in the following for the density ρ. A dual cell i is surrounded by J triangles. On each triangle Tj the gradient ⎤ ⎡ ρz,j ⎦ (∇ρ)Tj = ⎣ (27) ρr,j

can be computed with Cramer’s rule. The weight for the gradient is chosen as ωj =

(ε + ρ2z,j + ρ2r,j )−2 J (ε + ρ2z,k + ρ2r,k )−2

(28)

k=1

so that the contributions of the biggest gradients are dampened. ε is a small number (10−20 ) which helps to avoid divisions by zero. Furthermore, J j=1

ωj = 1 ,

(29)

Numerics for Magnetoplasmadynamic Propulsion

33

so that the WENO algorithm is linear preserving. The WENO gradient for a dual cell is now J (∇ρ)WENO (30) ωj (∇ρ)Tj . = i j=1

The non–linear weights ωj can be computed easily. They assure oscillation– free solutions in the vicinity of discontinuities and very smooth solutions for regions with smooth solutions.

The discretization of the viscous fluxes is done by a central scheme using average fluxes of neighbouring triangles. For all other derivatives which cannot be written in divergence or curl form a least–squares discretization is chosen. All conservation equations are discretized using the theorems of Gauß and Stokes. For the discretization of the right–hand–side of the species conservation equation (1), K 1 ni ni FWENO (xωk ) ∆Aωk − RHS = ∆Vω k=1 (31) K ! 1 (jD,i )Tωjk + (jD,i )Tωkl ∆Aωk nωk + ωi − 2 k=1

ni (xωk ) for a dual cell ω, which is surrounded by K neighboring cells, FWENO denotes the upwind flux through a cell surface ∆Aωk at the Gauss point xωk , where the left– and right–hand values at the cell face are computed by WENO reconstruction. The flux vector splitting already includes the normal surface vector nωk . It has to be written here, though, in the central discretization of the diffusion fluxes jD,i on the adjacent left and right triangles Tωjk and Tωkl . The spatial discretization of all other equations follows the same way.

Since a steady–state solution is wanted, an explicit first order time stepping scheme is employed for time–stabilization. For example, the calculation of the explicit local time step for equation (7) takes into account the magnetic diffusion because of the finite conductivity σ which is in some sense comparable with a classical heat conduction problem. With xm,k denoting the center of mass of cell k, all K neighbouring cells are included: 2 K K 1 µ0 1 σω + ∆tB = CFLB · σk |xm,k − xm,ω | . (32) 2 K +1 K k=1

k=1

The CFLB number dampens the other mechanisms in Ohm’s law which are not taken into consideration in equation (32). Before the actual time stepping is performed, all local time steps are multiplied with different random numbers RND between 0 and 1:

34

J. Heiermann, M. Auweter-Kurtz, C. Sleziona

∆tB,RND = ∆tB · RND B . The time integration for the stream function Ψ ⎡

⎢ ⎢ 1 1 ⎢ n+1 n Ψ −Ψ = ∆tB,RND ⎢ 1 − RHS B ω rm,ω + ⎢ K K ⎣

(33)

K

⎤

RHS B k rm,k ⎥ ⎥ ⎥ k=1 ⎥ . (34) ⎥ K ⎦

includes local residual smoothing. The time integration of all other equations follows the same way. The requirement of robustness is the reason for the choice of first–order time– stepping. Higher–order schemes like Runge–Kutta schemes generally accelerate convergence as long as the right hand sides are relatively simple. In the case of the complex non–linear right hand side presented here, they tend to produce numerical oscillations. The random time stepping also adds stability. It allows an increase of the CFL number by about an order of magnitude. Although its mathematical properties are not yet understood, it is clear that it introduces a non–linear, dampening dissipation which disappears once the steady state solution is achieved. It should be mentioned here that results obtained with and without random time steps are the same. For adapting the meshes, a combination of an analytically obtained error indicator [SW97, IWH00] and a gradient indicator is used. The local and weighted indicator νω for a dual cell ω with the volume ∆Vω is computed from a discrete solution of the equations (1) to (4). The basic concept is to calculate the jumps of the fluxes across the cell surfaces which is denoted by [ ]E : max {|[ρv · n]E | ∆AE } E⊂∂ω νω(1) = , (35) ρ ∆Vω # $ "# max #[(f z,invisc + f z,visc ) · n]E # ∆AE E⊂∂ω % νω(2) = , (36) (ρvz )2 + (ρvr )2 ∆Vω # # # # # # "# $ # max #[(f r,invisc + f r,visc ) · n]E # ∆AE + ## qr dV ## E⊂∂ω # # ω % νω(3) = , (37) 2 2 (ρvz ) + (ρvr ) ∆Vω # # # # # # $ # "# # # # max [(f h,invisc + f h,visc ) · n]E ∆AE + # qh dV ## E⊂∂ω # # ω , (38) νω(4) = eh ∆Vω

Numerics for Magnetoplasmadynamic Propulsion

1&

35

'

(39) ν (1) + νω(2) + νω(3) + νω(4) . 4 ω Choosing the conserved variables as weights proved to be very helpful in numerous numerical experiments. The new local mesh size g1 is now defined as ( Cε1 g1 = . (40) νω νω :=

The mathematically based indicator is combined with the gradient indicator ) * Cε2 *

. g2 = * (41) + |∇Th | |∇vz | |∇vr | |∇ρ| + + + |vz | |vr | ρ Th

The constants Cε1 and Cε2 are specified by the user so that the arithmetic average 1 g = (g1 + g2 ) (42) 2 is computed for the new local mesh size g which is required by the advancing front algorithm for remeshing the domain. Boundary layers, shocks and regions with strong source terms are refined reasonably from a physical point of view so that the computational time can be reduced.

4 Results The two IRS–developed MPD thrusters HAT and DT2 have been investigated with the above described finite volume method. Both thrusters have an exit nozzle diameter of 100 mm. The results presented here are for an argon mass flow rate of 0.8 g/s. Fig. 2 shows the thruster HAT being fired in the laboratory. A typical discretization of the computational domain is shown in Fig. 3. In the nozzle area, the mesh is highly refined for a good resolution of the boundary layers and for capturing the oblique shock (Fig. 4) which is also visible in the experiment (Fig. 2). The current distributions at high currents of 4000 A (Fig. 5) and 5000 A (Fig. 6) show that the electric arc starts to constrict downstream of the nozzle throat because of the pinch effect. This effect also becomes slightly visible in the density distributions shown in Figs. 7 and 8. The density is being lowered a little bit downstream of the nozzle throat at the anode wall. However, the

36

J. Heiermann, M. Auweter-Kurtz, C. Sleziona

density stays at a fairly high level at the inner anode surface so that there are enough charge carriers available for the electric arc. So far, no plasma instabilities were observed for the HAT at 5000 A in the laboratory experiment. However, the situation is completely different for the DT2. The arc also constricts at 4000 A (Fig. 9) and 5000 A (Fig. 10). Due to the slightly different nozzle shape of the DT2, the density depletion in front of the inner anode wall (Figs. 11 and 12) is significantly stronger than in the HAT. In the laboratory experiment, a current of 5000 A cannot be achieved for a mass flow rate of 0.8 g/s. Instead, plasma oscillations start at 4650 A. This can be linked to the depletion of density, and thus, charge carriers, in front of the anode surface because of the pinch effect. The experimental instability at 4650 A is not directly predicted by CFD because it is a 3–D effect. Experimental and numerical data for the thrust are given in the tables 1 and 2. The experimental data for the HAT show a slight drifting of the data which is caused by thermal heating of the pendulum thrust balance [ABH98, Weg94]. Nevertheless, the comparison of numerical and experimental data shows a good agreement.

Table 1. Thrust HAT (num.: numerical, exp.: experimental [ABH98, Weg94]). I [A]

m ˙ [g/s]

2000 2500 3000 3500 4000 4500 5000

0,8 0,8 0,8 0,8 0,8 0,8 0,8

F [N] num. 4,3 5,2 5,9 6,8 8,5 9,2 10,1

F [N] exp. 3,7-4,7 5,3 5,7-7,0 7,4 7,8-9,4 10,0 11,1

Table 2. Thrust DT2 (num.: numerical, exp.: experimental [ABH98, Weg94]). I [A]

m ˙ [g/s]

2000 2500 3000 3500 4000 4500 5000

0,8 0,8 0,8 0,8 0,8 0,8 0,8

F [N] num. 4,3 4,8 5,8 6,5 7,1 7,7 8,2

F [N] exp. 4,6 5,2 6,2 7,1 8,3 9,1 -

Numerics for Magnetoplasmadynamic Propulsion

37

5 Summary For the numerical solution of the conservation equations for argon plasma flows in magnetoplasmadynamic self–field thrusters a new finite volume method for unstructured, adaptive meshes has been developed. It has been shown for the first time by numerical simulations of the MPD thruster DT2 that a strong depletion of density in front of the anode is caused by the pinch effect at high current settings. The correlation of the numerical simulations with experimental data at instability–producing current settings clearly points to the fact that a primary reason for plasma instabilities in nozzle–type self–field MPD thrusters is the depletion of density and charge carriers in front of the anode caused by the pinch effect, which has been predicted theoretically for a long time. The prediction of thrust for the MPD thrusters DT2 and HAT agrees well with experimental data. Also, the simulations show that the depletion of charge carriers in front of the anode is delayed for HAT because its nozzle expansion angle is slightly less than that of DT2. This is probably the reason why no plasma instabilities can be observed for HAT at high current settings during experiments. The numerical method can therefore already be used for the design and development of new MPD thrusters. It particularly makes an optimization of the nozzle with respect to thrust possible, and it helps to preselect a nozzle geometry so that the depletion of density in front of the anode can be shifted to higher current settings. In addition, the numerical data can be used for further instability analysis. Acknowledgement. We gratefully acknowledge the decisive support by the German Research Foundation DFG (Deutsche Forschungsgemeinschaft) through the project “Conservation Equations and Numerical Solutions for Technically Relevant Magneto–Plasmas” (Au 85/9-1,2,3), which belongs to the German Priority Research Program (Schwerpunktprogramm) “Analysis and Numerics for Conservation Laws”. We thank L. Cassady (Princeton, USA), E. Choueiri (Princeton, USA), C. Coclici (Bosch, Stuttgart), A. Eberle (EADS, Munich), U. Iben (Bosch, Stuttgart), S. Jardin (Princeton, USA), H. Kaeppeler (Stuttgart), A. Kodys (Princeton, USA), A. Kolesnikov (Moskau, Russia), A. Meister (L¨ ubeck), G. Moro¸sanu (Ia¸si, Romania), C.–D. Munz (Stuttgart), P. Nikrityuk (Dresden), K. Sankaran (Princeton, USA), T. Sonar (Braunschweig), G. Warnecke (Magdeburg) and W. Wendland (Stuttgart) for their dedication, their patience and their support of our project.

38

J. Heiermann, M. Auweter-Kurtz, C. Sleziona

Anode C a th o d e

Insulator

Fig. 1. MPD thruster: Technical drawing (left), principle of its functionality (right).

Fig. 2. HAT ﬁring in the laboratory, 2000 A, 0.8 g/s. (See also color ﬁgure, Plate 1.)

Numerics for Magnetoplasmadynamic Propulsion

39

Anode

•

m

Fig. 3. Partial view of the adapted mesh for the MPD self–ﬁeld thruster HAT (28034 nodes, 2000 A, 0.8 g/s).

Fig. 4. HAT: Heavy particle temperature Th , 2000 A, 0.8 g/s. (See also color ﬁgure, Plate 2.)

40

J. Heiermann, M. Auweter-Kurtz, C. Sleziona

Anode Insulator •

m Cathode

Fig. 5. HAT: Current distribution Ψ , 4000 A, 0.8 g/s. (250 A between 2 isolines)

Anode Insulator •

m Cathode

Fig. 6. HAT: Current distribution Ψ , 5000 A, 0.8 g/s. (250 A between 2 isolines)

Numerics for Magnetoplasmadynamic Propulsion

41

Fig. 7. HAT: Density log10 (ρ), 4000 A, 0.8 g/s. (See also color ﬁgure, Plate 3.)

Fig. 8. HAT: Density log10 (ρ), 5000 A, 0.8 g/s. (See also color ﬁgure, Plate 4.)

J. Heiermann, M. Auweter-Kurtz, C. Sleziona

Insulator

42

Anode Insulator •

m Cathode

Insulator

Fig. 9. DT2: Current distribution Ψ , 4000 A, 0.8 g/s. (250 A between 2 isolines)

Anode Insulator •

m Cathode

Fig. 10. DT2: Current distribution Ψ , 5000 A, 0.8 g/s. (250 A between 2 isolines)

Numerics for Magnetoplasmadynamic Propulsion

43

Fig. 11. DT2: Density log10 (ρ), 4000 A, 0.8 g/s. (See also color ﬁgure, Plate 5.)

Fig. 12. DT2: Density log10 (ρ), 5000 A, 0.8 g/s. (See also color ﬁgure, Plate 6.)

44

J. Heiermann, M. Auweter-Kurtz, C. Sleziona

References [ABH98] Auweter-Kurtz, M., Boie, C., Habiger, H., Kaeppeler, H.J., Kurtz, H.L., Sleziona, P.C., Wegmann, T., Winter, M.W.: Numerische Simulation von MPD–Triebwerken und Vergleich mit durchzuf¨ uhrenden experimentellen Untersuchungen. Endbericht zum DFG–Forschungsvorhaben Au85/5-2, Institut f¨ ur Raumfahrtsysteme, Universit¨ at Stuttgart (1998) [Auw92] Auweter–Kurtz, M.: Lichtbogenantriebe f¨ ur Weltraumaufgaben, B.G. Teubner Stuttgart (1992) [Auw94] Auweter–Kurtz, M.: Plasma Thruster Development Program at the IRS. Acta Astronautica, 32, No. 5, 377–391 (1994) [BWB90] Bennett, G.L., Watkins, M.A., Byers, D.C., Barnett, J.W.: Enhancing Space Transportation: The NASA Program to Develop Electric Propulsion. IEPC–91–004, 21st AIAA/DGLR/JSASS International Electric Propulsion Conference, Orlando, Florida (1990) [Boi99] Boie, C.: Numerische Simulation magnetoplasmadynamischer Eigenfeldtriebwerke mit hochauﬂ¨ osenden adaptiven Verfahren. Dissertation, Institut f¨ ur Raumfahrtsysteme, Fakult¨ at Luft- und Raumfahrttechnik, Universit¨ at Stuttgart (1999) [Bue86] B¨ uhler, R.D.: Plasma thruster development: magnetoplasmadynamic propulsion, status and basic problems, AFRPL TR–86–013, Air Force Rocket Propulsion Laboratory, Edwards AFB, California (1986) [Cap94] Cap, F.: Lehrbuch der Plasmaphysik und Magnetohydrodynamik. Springer Verlag (1994) [Cho91] Choueiri, E.Y.: Electron–Ion Streaming Instabilities of an Electromagnetically Accelerated Plasma. Ph.D. thesis, Princeton University, New Jersey, USA (1991) [Cho99] Choueiri, E.Y.: Anomalous resistivity and heating in current–driven plasma thrusters. Physics of Plasmas, 6, No. 5, 2290–2306 (1999) [Ebe00] Eberle, A.: A Nonlinear Flux Vector Split Defect Correction Scheme for Fast Solutions of the Euler and Navier Stokes Equations. In: Freist¨ uhler, H., Warnecke, G. (eds.) Hyperbolic Problems: Theory, Numerics, Applications. Birkh¨ auser (2001) [Fri98] Friedrich, O.: Weighted Essentially Non–Oscillatory Schemes for the Interpolation of Mean Values on Unstructured Grids. Journal of Computational Physics 144, 194–212 (1998) [Hei02] Heiermann, J.: Ein Finite–Volumen–Verfahren zur L¨osung magnetoplasmadynamischer Erhaltungsgleichungen. Dissertation, Institut f¨ ur Raumfahrtsysteme, Fakult¨ at Luft– und Raumfahrttechnik und Geod¨ asie, Universit¨ at Stuttgart (2002) [HAE99] Heiermann, J., Auweter–Kurtz, M., Eberle, A., Iben, U., Sleziona, P.C.: Robuste hochauﬂ¨ osende Methoden zur Simulation magnetoplasmadynamischer Raketentriebwerke. DGLR-JT99-034, DGLR Jahrbuch, ISSN 0070–4083, 923–929 (1999) [HAK99] Heiermann, J., Auweter-Kurtz, M., Kaeppeler, H.J., Eberle, A., Iben, U., Sleziona, P.C.: Recent Improvements of Numerical Methods for the Simulation of MPD Thruster Flow on Adaptive Meshes. IEPC–99–169, 26th International Electric Propulsion Conference, Kitakyushu, Japan (1999)

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[Hue80] H¨ ugel, H.: Zur Funktionsweise der Anode im Eigenfeldbeschleuniger. Habilitation, Fakult¨ at f¨ ur Luft- und Raumfahrttechnik, Universit¨ at Stuttgart, DFVLR Forschungsbericht 80–20, (1980) [IWH00] Iben, U., Warnecke, G., Heiermann, J., Auweter-Kurtz, M.: Adaptive Numerics for the Simulation of Magneto–Plasmadynamic Rocket Thrusters. In: Neittaanm¨ aki, P., Tiihonen, T., Tarvainen, P. (eds.) Proceedings of The Third European Conference on Numerical Mathematics and Advanced Applications, Jyv¨ askyl¨ a, Finland. World Scientific, Singapore (2000) [Jah68] Jahn, R.G.: Physics of Electric Propulsion. McGraw–Hill Series in Missile and Space Technology (1968) [Kae91] Kaeppeler, H.J.: Basic equations and elements of the dispersion relation for a four–ﬂuid formalism of magneto–plasmadynamics with non–equilibrium ionization. IEPC–91–059, Proceedings of the 22nd International Electric Propulsion Conference, Viareggio, Italy (1991) [MAH88] Merke, W.D., Auweter–Kurtz, M., Habiger, H., Kurtz, H.L., Schrade, H.O.: Nozzle Type MPD Thruster Experimental Investigations. IEPC–88– 028, Proceedings of the 20th International Electric Propulsion Conference, Garmisch–Partenkirchen, Deutschland (1988) [Sch82] Schrade, H.O.: Basic Processes of Plasma Propulsion. Interim Scientific Report, Air Force Office of Scientific Research Grant 82–0298, Institut f¨ ur Raumfahrtsysteme, Universit¨ at Stuttgart (1982) [Sch87] Schrade, H.O.: Basic Processes of Plasma Propulsion. Interim Scientific Report, Air Force Office of Scientific Research Grant 86–0337, Institut f¨ ur Raumfahrtsysteme, Universit¨ at Stuttgart (1987) [Sch93] Schumacher, U.: Fusionsforschung. Hanser Verlag (1993) [Sle92] Sleziona, P.C.: Numerische Analyse der Str¨omungsvorg¨ ange in magnetoplasmadynamischen Raumfahrtantrieben. Dissertation, Institut f¨ ur Raumfahrtsysteme, Fakult¨ at Luft- und Raumfahrttechnik, Universit¨ at Stuttgart (1992) [Sle98] Sleziona, P.C.: Hochenthalpiestr¨omungen f¨ ur Raumfahrtanwendungen. Habilitationsschrift, Fakult¨ at Luft- und Raumfahrttechnik, Universit¨ at Stuttgart (1998) [SW97] Sonar, T., Warnecke, G.: On Finite Diﬀerence Error Indication for Adaptive Approximations of Conservation Laws (revised 2nd printing). Hamburger Beitr¨ age zur Angewandten Mathematik, Reihe A, Preprint 122, Hamburg (1997) [Wag94] Wagner, H.P.: Theoretische Untersuchung drift- und gradientengetriebener Instabilit¨ aten in MPD–Triebwerksstr¨ omungen. Dissertation, Institut f¨ ur Raumfahrtsysteme, Fakult¨ at Luft– und Raumfahrttechnik, Universit¨ at Stuttgart (1994) [WKA98] Wagner, H.P., Kaeppeler, H.J., Auweter–Kurtz, M.: Instabilities in MPD thruster ﬂows: 1. Space charge instabilities in unbounded and inhomogeneous plasmas. 2. Investigation of drift and gradient driven instabilities using multi–ﬂuid plasma models. J. Phys. D: Appl. Phys. 31, 519–541 (1998) [Weg94] Wegmann, T.: Experimentelle Untersuchung kontinuierlich betriebener magnetoplasmadymamischer Eigenfeldtriebwerke. Dissertation, Institut f¨ ur Raumfahrtsysteme, Fakult¨ at f¨ ur Luft- und Raumfahrttechnik, Universit¨ at Stuttgart (1994)

Hexagonal Kinetic Models and the Numerical Simulation of Kinetic Boundary Layers Hans Babovsky Institute for Mathematics, Ilmenau Technical University, P. O. Box 100565, D-98684 Ilmenau, Germany, [email protected] Summary. The paper deals with the transition regime of gas ﬂows between the mesoscopic and the macroscopic levels. We survey theoretical results and provide numerical tools. As the basic numerical scheme for the solution of the Boltzmann equation we use a hexagonal model proposed in [1]. Key words: Boltzmann equation, discrete velocity model, kinetic boundary layer. MSC: 76N20, 76P05.

1 Introduction The evolution of a gas flow may be formulated on different levels of description. The most intuitive is the microscopic level, where a gas is considered as a huge N -point system of particles which move and interact as a Hamiltonian system (with short range interaction potentials). Of course from a numerical point of view this is certainly not the level of choice, since the number of particles is extremely too large for a simulation. However, the main method in practice for the simulation of rarefied gas flows is that of Monte Carlo systems, where features of the microscopic view point are taken over. In the mesoscopic level, particles as individuals are given up and replaced by density functions in the physical phase space. The corresponding equation is the Boltzmann equation. The influence of particle interactions (collisions) is balanced in the Boltzmann collision integral. Finally, on the macroscopic level, the densities in velocity space are given up and replaced with a system of moments. The resulting equations are the Euler resp. the Navier Stokes systems. Passing over from the microscopic to the macroscopic level results in a system reduction and thus in diminished complexity. For numerical purposes, reduced systems are to be preferred. However, at times the more complex description is necessary for an appropriate simulation. For example, calculating the Boltzmann collision

48

H. Babovsky

operator (in a fully six-dimensional phase space) requires in each time step the evaluation of a five-dimensional in each point of the discretized calculational domain. (See section 2 for the two-dimensional version.) This is a highly time and memory consuming task. To cope with this complexity, one may try to get rid of the collision operator and pass to simplified systems like relaxation schemes (like BGK; see [2]) or truncated moment hierarchies [16]. When there is a need for the collision operator, one may try to reduce the Boltzmann regime to a domain as small as possible or to find kinetic model systems which are more efficiently solvable than the original (discretized) operator. Progress in both directions is the subject of this paper. More specifically, • The evolution of the collision integral requires for any velocity pair (v, w) the integration over the surface of the ball (in 3D) resp. the circle (in 2D) between v and w (see Fig. 1). This lower dimensional manifold is not ”compatible” with a regular rectangular grid discretization which leads to an extremely low consistency order for classical integration schemes (see [10]). We propose the replacement of the circular structure with a hexagonal system as described in section 3. In [1], it was shown that kinetic equations based on hexagonal models possess all properties of classical kinetic systems (H-theorem, conservation laws, linearized collision operators etc.). From a numerical point of view, their efficiency has been demonstrated in [6]. • In flows around bodies, the only domain requiring the kinetic description by the Boltzmann equation is in many cases a very thin zone along the boundary of the body, called the kinetic boundary layer. The coupling of these layers to the macroscopic fluid flow has been e.g. investigated in [15]. The most promising way of coupling (for small Knudsen numbers) is the calculation of jump conditions which provide a kinetic correction to the macroscopic formulation. We develop an algorithm for this. The scope of the paper is as follows. In section 2, we survey some basic facts and some most recent results concerning the relationship between mesoscopic and macroscopic equations as far as they concern the topic of this paper. Section 3 shortly introduces the hexagonal system as proposed in [1]. This part also includes some demonstrations of the flexibility of the system when applied to different areas of fluid flows, and a continuum limit for the discretization parameter going to 0. Section 4 investigates boundary layers and proposes an algorithm for the calculation of jump conditions. From now on we switch exclusively (except section 2.2) to two-dimensional velocity models. Three-dimensional models exhibit similar properties. Their investigation is on progress; a publication is in preparation.

Hexagonal kinetic models and simulation of kinetic boundary layers

49

2 Structure of fluid flows 2.1 Boltzmann equation versus Navier Stokes system The Boltzmann equation is an equation for the density function f = f (t, x, v) for particles in the four-dimensional phase space. (In the following we use the conventions x = (x, y)T and v = (vx , vy )T .) Its stationary, spatially onedimensional version reads vx ∂x f (x, v) = J(f , f )(x, v)

(1)

where J(f , f ) is the Boltzmann collision integral J(f , f )(v) = k(|v − w|, η)[f (v′ )f (w′ ) − f (v)f (w)] dη d2 w lR2

(2)

S1

The velocity pairs (v′ , w′ ) and (v, w) are related via momentum and energy conservation and are parametrized by a unit vector η ∈ S 1 , v′ = v − η, v − w · η,

(3)

′

w = w + η, v − w · η

(4)

For given (v, w), the set of all possible ”pre-collision” velocities is located on the circle between v and w, as illustrated in Fig. 1.

V’ W

V W’ Fig. 1. Collision pairs.

For details concerning elementary properties of the Boltzmann equation we refer the reader to standard literature, e.g. [4, 11]. The only collision invariants of the collision operator are the four functions φ(v) = 1, vx , vy , |v|2 φ(v)J(f , f )(v) d2 v = 0 (5) lR2

50

H. Babovsky

Given a density function f (v), define mass ρ and the bulk velocities u = (ux , uy )T by ρ := f (v) d2 v (6) lR2 ρu := vf (v) d2 v (7) lR2

internal energy e and heat flux vector q = (qx , qy )T by 1 ρe = |v − u|2 f (v) d2 v 2 lR2 1 (v − u)|v − u|2 f (v) d2 v q= 2 lR2 and the components of the stress tensor Π = (pij )i,j∈{x,y} by pij := (vi − ui )(vj − uj )f (v) d2 v .

(8) (9)

(10)

lR3

Multiplying the Boltzmann equation (1) with the four abovementioned functions φ and integrating yields the four moment equations

∂x

&

∂x (ρux ) = 0 ' ρux u + p(1) = 0

(11) (12)

1 2 (1) ∂x ρux |u| + e + p , u + q1 = 0 2

(13)

where p(1) = (pxx , pxy )T denotes the first column of Π. Finally, the pressure p is defined by pii . (14) 2p = ρe = i∈{x,y}

Equations (11) – (13) represent a non-closed system of four equations for the nine unknowns ρ, ux , uy , pxx , pxy , pyy , qx , qy , e. This system may be closed by introducing some phenomenological relations expressing pij and qx in terms of the other quantities. For a Navier-Stokes-Fourier fluid these relations are (see [11, Sections II.8, IV.7]): pxx = p + µ(−∂x ux + ∂y uy ) pyy = p + µ(+∂x ux − ∂y uy ) pxy = −µ(∂x uy + ∂y ux ) q = −κ∇x e .

,

, ,

(15) (16) (17) (18)

Hexagonal kinetic models and simulation of kinetic boundary layers

51

with viscosity and heat conduction coefficients µ and κ. Inserting these yields the following Navier-Stokes equations. ∂x (ρux ) = 0 1 2 ∂x ρ u x + e = µ ∂x2 ux 2 ∂x

∂x (ρux uy ) = µ ∂x2 uy , , 2 1 µ ρux |u| + 3e = κ ∂x2 e + ∂x2 |u|2 2 2

(19) (20) (21) .

(22)

Assuming a non-permeable wall at x = 0 and with this ux (0) = 0, equation (19) yields ux ≡ 0

,

(23)

and a linear profile for uy , uy (x) = uy (0) + ∂uy · x .

(24)

In case of uy = 0, we also find a linear profile for the internal energy e. The problem of determining slip boundary conditions is to find relations between the boundary values at x = 0 and the constant gradients. Furthermore, equations (10) and (11) turn into ∂x (p11 ) = ∂x (p12 ) = 0 , ∂x (p12 uy + q1 ) = 0 .

(25) (26)

In numerical simulations, on may use these equations as a test which parts of the half space system belongs to the kinetic boundary layer and which to the fluid system. 2.2 Flows along walls Here. we switch to three-dimensional flows. Sone et al. [15] have studied the structure of a fluid flow governed by the Boltzmann equation in the presence of a physical wall. Consider a compressible steady fluid flow adjacent to some smooth wall ∂Ω, which is given by the Boltzmann equation v · ∇x f (x, v) = 1/ǫ2 · J[f (x, ·), f (x, ·)](v)

(27)

supplemented by the diffuse boundary condition at a ∈ Ω, f (a, v) = c(a) · mT (a) (v) for

n(a), v > 0

.

(28)

Here, J[f , f ] is the Boltzmann collision integral as described in the preceding section, and ǫ2 is proportional to the Knudsen number, i.e. to the mean free

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H. Babovsky

path; n(a) is the normal vector on ∂Ω at a pointing into the fluid domain; mT is a Maxwellian with zero mean velocity and temperature T , mT (v) = (2πT )−3/2 exp[−|v|2 /(2T )]

(29)

(where the Boltzmann constant has been for simplicity set equal to 1); c(a) is the normalizing constant guaranteeing that there is no flux through the wall, i.e. n(a), vf (a, v)d3 v = 0 . (30) lR3

In this paper we are interested in the case of small Knudsen number ǫ2 , with the curvature of ∂Ω being large compared to ǫ. As was analyzed in [15], the flow may be decomposed into a thin kinetic boundary layer (with a thickness ∼ ǫ2 ) which is appropriately described by a low order Hilbert expansion (and thus by the linearized Boltzmann equation), a viscous boundary layer (with thickness ∼ ǫ), and an outer domain well described by the Euler equations. One of the objectives of the paper is the coupling of the kinetic boundary layer to the viscous layer. We are going to investigate numerical tools for the calculation of slip boundary conditions as kinetic corrections to the viscous part. E.g., if the wall temperature is T (a), then the fluid temperature at a is in general different from T (a), with a correction term of the order of ǫ and dependent on the temperature gradient. Similar statements hold for the fluid velocity components tangential to ∂Ω. Sone et al. [15] pointed out that an appropriate scaling for the boundary region around a ∈ ∂Ω is given by a space-coordinate transformation taking into account the strong anisotropy of the flow in the boundary layer. For this we assume a bijection ξ(x) = (ξ1 , ξ2 , ξ3 )T such that x(0, ξ2 , ξ3 ) ∈ ∂Ω

,

(31)

and for x ∈ ∂Ω, the system {n, ω1 , ω2 } defined by ǫ·n=

∂x (0, ξ2 , ξ3 ), ∂ξ1

ω1 =

∂x (0, ξ2 , ξ3 ), ∂ξ2

ω2 =

∂x (0, ξ2 , ξ3 ) ∂ξ3

(32)

forms an orthonormal system. The stationary system then easily transforms into 1 ∂ ∂ 1 ∂ · vn · f + vω1 · f + vω2 · f = 2 · J[f , f ] . (33) ǫ ∂n ∂ω1 ∂ω2 ǫ In the kinetic boundary layer, the partial derivatives concerning the tangential vectors ω1 and ω2 may be neglected, and we end up with the half space problem 1 ∂ f = · J[f , f ] . (34) ∂n ǫ The analysis of this problem yields the details of the jump conditions mentioned above. vn ·

Hexagonal kinetic models and simulation of kinetic boundary layers

53

3 Hexagonal kinetic models 3.1 Hexagonal discretization of lR2 For h > 0, consider the partition Ph of lR2 which is given by regular hexagons with side length h as demonstrated in Fig. 2. In the sequel we will denote these as basic hexagons. We assume that the partition is numerated by some index set I.

Fig. 2. Hexagonal discretization of lR2 .

The above constellation gives rise to two regular grids on lR2 . The grid Ch is given by the set {cz |z ∈ I}

(35)

of all centers of basic hexagons. As one can check quickly, Ch is spanned by two vectors: Ch = c0 + {k · (c(1) − c0 ) + l · (c(2) − c0 )|k, l ∈ Z} .

(36)

Furthermore, Ch exhibits hexagonal symmetry. Denote by R the (e.g. clockwise) rotation around the angle π/3. Then c0 , c ∈ Ch

=⇒

c0 + Rk (c − c0 ) ∈ Ch

for k = 0, . . . , 5

.

(37)

All six-tupels of the form (37) are the nodes of a regular hexagon in Ch . A second grid Gh is given by the nodes gz of the basic hexagons. Obviously all nodes can be represented in the form 1 g = c + h · Rk (38) 0

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H. Babovsky

for some c ∈ Ch and for k ∈ {0, . . . , 5}. Gh has triangular symmetry: g0 , g ∈ Gh

=⇒

g0 + R2k (g − g0 ) ∈ Gh

for k = 0, 1, 2

.

(39)

We introduce a sign-function sign : Gh → {−1, 1}

(40)

such that two neighboring points have different sign: sign(g1 ) · sign(g2 ) = −1 if |g1 − g2 | = h .

(41)

Then it turns out that the two subsets of Gh of elements with definite sign have hexagonal symmetry. Following the arguments above, there are a lot of regular hexagons contained in the grid Gh . Those with the center belonging to Ch are denoted as ”class A”, and those with center in Gh as ”class B”. See Fig. 3 for the illustration of some examples (Class A on the left and Class B on the right). Let (g0 , . . . , g5 ) describe any regular hexagon in G, numbered in clockwise order. Then it turns out that for class A hexagons sign(gi ) · sign(gi+1 ) = −1 ,

(42)

while for class B hexagons sign(gi ) · sign(gi+1 ) = 1 .

(43)

In the following we parametrize the hexagons described above by tuples (η, ξ), where η ∈ C ∪ G denotes the center of the hexagon, and η + Rk ξ, k = 0, . . . , 5 its nodes. For short, we write H(η, ξ). Notice that H(η, ξ) = H(η, Rn ξ) for n ∈ Z. The set of hexagons H(η, ξ) is denoted as H.

( a) Fig. 3. Some regular hexagons in Gh .

Hexagonal kinetic models and simulation of kinetic boundary layers

55

3.2 A hexagonal collision model Assume H ∈ H to be given by the six-tupel H = H(η, ξ)=(g ˆ 0 , . . . , g5 )

(44)

with gi = η + Ri ξ

.

(45)

A binary hexagonal collision model is given by the collision relation (gi , gi+3 ) → (gj , gj+3 )

(46)

which maps opposite pairs (gi , gi+3 ) of velocities with equal probability into one of the other opposite pairs. The corresponding collision operator reads ⎞ ⎛ 2 1 (47) JH,bin [fH , fH ]i = γH,bin ⎝ · fj fj+3 − fi f(i+3) mod 6 ⎠ , 3 j=0 where the indices 0, . . . , 5 have been identified with the positions g0 , . . . g5 . Its weak version reads for any test function πH on H . 2 2 1 fi fi+3 πH , JH,bin [fH , fH ] = γH,bin · (πH,j + πH,j+3 ) 3 j=0 i=0 (48) / −(πH,i + πH,i+3 )

According to the rules of classical kinetic theory we assume shift invariance of the collision model and thus that γH,bin depends only on the diameter diam(H) = 2|ξ| of the hexagon. The binary collision operator Jbin on the whole grid Gh is given by summing up over all H ∈ H. Given v, w ∈ Gh , denote η := 0.5 · (v + w) as well as

,

0 0 kh (v, w) := 2γH,bin

ξ := v − η if H(η, ξ) ∈ / H, else .

(49)

(50)

Furthermore, the point measure with support on the nodes of the hexagon between two vectors is denoted as 5

µh (v, w) :=

1 i δ 6 i=0 η+R ξ

.

(51)

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H. Babovsky

Then π, Jbin [f , f ] =

v,w∈Gh

kh (v, w) (π, µ(v, w) − π(v)) f (v)f (w)

(52)

Unfortunately, Jbin contains – besides the physically correct invariants mass, momenta and kinetic energy – a further invariant sign, f . To suppress this, we are forced to introduce an additional ternary collision operator. Its restriction onto one single hexagon H ∈ H reads JH,ter [fH , fH , fH ]i = γter · (−1)i (f1 f3 f5 − f0 f2 f4 ) ,

(53)

and its weak version is πH , JH,ter [fH , fH , fH ] = γH,ter ·sign(g0 )·πH , sign·(f1 f3 f5 −f0 f2 f4 ) .

(54)

Since in general this term does not vanish for πH = sign, sign is no longer an artificial invariant. In [1] a complete kinetic theory on hexagonal grids has been developed proving that all of the main features of the classical Boltzmann collision operator are still valid. This includes the correct number of collision invariants as well as the set of equilibria (which stay close to the equilibria given by Maxwellians), the H-theorem and the linearization around an equilibrium. This theory is based on the interplay between class A and class B hexagons. The ternary part of the collision operator is cubic in f and thus dominates the binary operator, if the density ρ is large. Thus in numerical simulations we choose γter ∼ 1/ρ if necessary. 3.3 Grid refinement and continuum limit The refinement of the grid Gh with discretization parameter h to the grid Gh/2 may be achieved in two ways. On one hand, we may shrink the grid by a factor 0.5; i.e. we define – for some fixed g0 ∈ Gh – Gh/2 := g0 + {0.5 · (g − g0 )|g ∈ Gh }

.

(55)

Alternatively, we may replace each element of Gh with some 4-stencil as is indicated in Fig. 4. In what follows we make use of this second interpretation. Our aim is to construct the limiting collision operator after successive refinement of the grid. The arguments are presented on a formal level but can be made rigorous using the classical theory of convergence of measures as described e.g. in [9]. Passing from Gh ⊂ Gh/2 to Gh/2 we can easily extend the function sign =: signh such that the restriction signh/2 |Gh = signh . For given h > 0, each g ∈ Gh is a node of three adjacent basic hexagons. Connect the centers of these to form a regular triangle ∆h = ∆h (g). Its area is given by

Hexagonal kinetic models and simulation of kinetic boundary layers

57

Fig. 4. Grid reﬁnement.

λ(∆h ) =

√ 3 · h2

.

(56)

A function µ : Gh → lR+ may be considered as a discrete measure on lR2 with support in Gh . However, we may reinterprete it as an absolutely continuous measure on lR2 with density which is piecewise constant on the triangles ∆h . In order to compare signed measures on Gh with (e.g. absolutely continuous) measures on lR2 we define for functions πh on Gh the discrete signed measures πh (g) · δg (57) πh , µh := g∈Gh

With the special function 1(g) l := 1 we define the measure µh := l 1, µh . Given a function f ∈ L1loc , we define the mapping Ph f : Gh → lR by Ph f (g) := f (v) dv .

(58)

(59)

∆h (g)

The following statements are well-known results in measure theory. 3.1 √Lemma: Denote by →w the weak convergence of measures on lR2 . Then (a) 3h2 µh →w dλ (Lebesgue measure) (b) Ph f , µh →w f (v)dλ(v) (c) signh , µh →w 0 . From Property (c) follows that the artificial invariant of the binary collision operator vanishes in the continuum limit. Thus there is no need to consider the ternary collision part. Now assume that kh (., .) is bounded (an assumption which may be weakened). Then by the usual techniques of measure theory it is straightforward to derive in the weak limit an integral expression for the right hand side of (52). From this we can deduce a ”classical” formulation for

58

H. Babovsky

the continuous hexagonal model. (See, e.g. [3, 4] for the weak formulation of collision integrals.) We summarize the results. 3.2 Theorem: Under the above assumptions, there exists a collision kernel k0 (|v − w|) such that the binary collision operator Jbin [f , f ] converges weakly to the kinetic collision operator k0 (|v − w|) · [f , µ(v, w) − f (v)] f (w)dvdw . (60) J[f , f ](v) = lR2

lR2

This operator has beyond the physical invariants mass, momenta and kinetic energy no further (in particular no artificial) invariants. 3.3 Remark: The main difference to classical two-dimensional collision operators is the summation f , µ(v, w) over the regular hexagon between v and w (i.e. over six angles) rather than the integration over the whole circle between v and w shown in Fig. 1. 3.4 Numerical examples: Channel flows We study several details of the collision model in the most simple twodimensional rectangular geometry. For numerical simulations we have to restrict to finite (i.e. truncated) grids. In all the calculations presented here we use a socalled ”two-layer” model. This is a 54-velocity model generated by attaching to a single central basic hexagon two ”layers” of further basic hexagons (see Fig. 5). It turns out that this model is rich enough to resolve a couple of interesting detailed structures of kinetic flows. The range of applications includes interior as well as exterior flows, extends from slow to fast (supersonic) flows and covers a wide range of Knudsen numbers. Numerical simulations of time dependent problems are easy to implement if they are based on an operator splitting scheme, where the free-flow part is evaluated by a standard upwind difference scheme. We have used such a scheme to obtain the results demonstrated below. (However, we should mention that for steady boundary value problems there are more efficient ways; in [7], a combination of FEM techniques with Newton methods has been proposed.) As an illustration, we present two numerical examples. Fig. 9 shows the propagation of a supersonic jet into a channel. We recognize a shock front in the neighborhood of the lower plate with a dense regime beyond and a rarefaction zone below the plate. Furthermore, the flow field between the shock front and the stagnation point (lower plate) is demonstrated. A second example is given by the convection problem demonstrated in Fig. 10. It exhibits a flow in a gravitational field generated by heating of the center of the lower plate. The flow field shown in the second part of Fig. 10 demonstrates the existence of convection rolls. In detail, we investigate features of channel flows concerning the boundary slip conditions and the flow parameters.

Hexagonal kinetic models and simulation of kinetic boundary layers

59

Fig. 5. The 2-layer model.

Boundary conditions Kinetic theory allows a variety of models for the design of gas-surface interactions which is much larger than for macroscopic models. ”Typical” models (linear, local in space and time) are described by so called reflection laws prescribing how a particle is reflected from a wall if it hits the wall at some point a with some velocity v. There are two main classes (and linear combinations of these). Deterministic reflection laws are given by mappings v → Rv telling that the velocity after the wall contact is Rv. If R is one-to-one, then the corresponding boundary condition for f is n(a), Rv · f (a, Rv) = |n(a), v| · f (a, v),

(61)

where n(a) denotes the inner normal on the wall surface at a. The main representatives are specular reflection: Rv = v − 2n(a), v · n reverse reflection: Rv = −v Stochastic reflection laws describe the result of the contact with the wall by some probability law P (.|v). The most prominent one is diffuse reflection, where particles leave the wall thermally distributed. The corresponding boundary condition is given by (2.28). Reflection laws highly influence the fluid flow along a boundary. In the following we restrict to the three laws described above. In terms of macroscopic flow parameters, one can find immediately some fundamental differences. Common to all of these is that there is no flux through the boundary, i.e. v⊥ = 0.

(62)

The only law guaranteeing the no-slip condition, i.e. v = 0

(63)

60

H. Babovsky

is reverse reflection. (This is true in a more general setting. As was shown in [13], reverse reflection is the only ”no-slip” law in a large class of physically reasonable laws.) Both deterministic laws above are adiabatic (i.e. there is no energy exchange between fluid and wall), while the diffuse law is not. The only law with vanishing shear stress is specular reflection. We have calculated a plane channel flow under each of these conditions. The results presented in Fig. 8 illustrate the differences. We find e.g. a non-vanishing velocity at the boundary in the case of diffuse reflection (in contrast to reverse reflection) which has to be described by a jump condition. For specular reflection, shear stress is constant zero. Flow parameters In the fluid dynamic limit, the viscosity coefficient and the heat conduction are two mayor parameters to characterize a fluid flow. We investigate these for the hexagonal model. Since both are coupled via the Prandtl number (which is fixed in our monatomic model) we restrict to the calculation of the viscosity coefficient. The derivation of hexagonal models with various Prandtl numbers (via internal energies) is under investigation. In the classical kinetic theory, flow parameters depend on the collision model. E.g. Maxwellian molecules (with bounded collision kernel) behave different than molecules with kernels given by some other law. Here we test how the flow is influenced by the choice of collision frequencies γH for different hexagons H. As expected, it turns out that the viscosity coefficient does not depend on the flow density. However, it strongly depends on the distribution of collision frequencies γH for the hexagons. As is usual in the classical theory, γH should depend only on the diameter of H. We assume a parameter dependent law γH = λ + (1 − λ) · 0.32 · diam(H)

.

(64)

For λ ∈ [0, 1.5], the corresponding collision kernels are strictly positive. For λ = 1, all collision frequencies are equal. For λ = 0, they vary from 0.640 (basic hexagons) to 2.790 (largest hexagons). For λ = 1.5, they change from 1.180 to 0.105, i.e. interactions within the largest hexagons are almost suppressed. The corresponding viscosity coefficients are linearly dependent on λ as is shown in Fig. 6. Here, we do not go into details concerning ”artificial” viscosity, i.e. the influence of the discretization parameters on the viscosity coefficient (cf. e.g. [14] in the context of lattice Boltzmann models). This will be due to further work. (For our calculation we used a 128 × 16- grid in position space, however doubling the grid points in each direction does not lead to significant changes.)

Hexagonal kinetic models and simulation of kinetic boundary layers

61

7

fi

0 0

ti

1.51

Fig. 6. Viscosity coeﬃcient versus collision model.

4 Kinetic boundary layers 4.1 Structure of linearized boundary layers We consider kinetic flows with finite (i.e. truncated) velocity space which we again denote by G. As pointed out in section 2.2, the coupling of kinetic boundary layers to fluid flows can well be modeled by linearized half space problems. For discrete velocity systems, these have been analyzed in [5]. (The continuous case has been treated in [12].) For a small perturbation φ(x, v) around an equilibrium, the linearized system of equations can be formulated in the form ∂x φ = Lφ

,

(65)

The operator L acts only in the velocity space and is (in 2D) similar to the matrix in Jordan normal form, J = diag(0, 0, α1 · J1 , α2 · J1 , Λ, −Λ) where J1 is the Jordan block given by 01 J1 = 00

(66)

,

(67)

and Λ is a positive diagonal matrix. In terms of the corresponding basis {b1 , . . . , bn } of the n-point discretization of the velocity space, the general solution of (3.1) is of the form φ(x) = β1 b1 + β2 b2 + (β3 + x · α1 β4 )b3 + β4 b4 + (β5 + x · α2 β6 )b5 + β6 b6 + exp(xΛ)b+ + exp(−xΛ)b−

.

(68)

Here, the basis vectors b1 , b2 , b3 and b5 are related to the invariants mass, momenta and energy. b4 and b6 are macroscopically not observable fluctuations which are responsible for the relation between the gradients of b3 and

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H. Babovsky

b5 and their values at the boundary x = 0 (jump conditions). The term exp(xΛ)b+ is a short notion for k i=0

exp(λi x) · b7+i .

A similar abbreviation is exp(−xΛ)b− . To formulate the boundary conditions, we denote + φ P+ φ φ= =: P− φ φ−

(69)

(70)

where P+ (resp. P− ) denotes the restriction of φ to the velocity points pointing to the right (resp. to the left). We call a solution (3.4) a kinetic boundary layer in [0, ∞), if P+ φ(0) = RP− φ(0),

(71)

for some linear (totally absorbing) reflection law R, and in addition for large x φ(x)/x is bounded .

(72)

(70) is satisfied if the exponentially increasing term vanishes, i.e. if b+ = 0

(73)

In order to obtain a unique boundary layer, we have to introduce two further conditions; e.g. we may fix the constant gradients α1 β4 and α2 β6 . Under these restrictions, the construction of a kinetic boundary layer is a well-posed problem [5]. Analyzing the algebraic structure of the problem it is possible to find analytic expressions for the jump conditions – at least for discrete models of no more than moderate size. Examples are given in [8]. For larger systems it is more convenient to replace the half space problem with an appropriate two-point boundary value problem. For its preparation, we introduce some properties of the eigenspace of L which have been derived in a quite general setting in [5] and which are valid also in the case of hexagonal models. We call a vector φ on G even, if P+ φ = P− φ, and odd, if P+ φ = −P− φ. 4.1 Properties of L: (a) The eigenvectors b1 , b3 and b5 are even, and b2 , b4 and b6 odd. (b) If for some λ > 0 and (b+ , b− )T = 0 L(b+ , b− )T = λ(b+ , b− )T

(74)

L(b− , b+ )T = −λ(b− , b+ )T

(75)

then

Hexagonal kinetic models and simulation of kinetic boundary layers

63

A straightforward consequence of this is − T 4.2 Lemma: Suppose φ(1) = (φ+ (1) , φ(1) ) is a solution of (3.1), (3.6) and + T (3.7) in [0, ∞). Define φ(2) (−x) := (φ− (1) (x), φ(1) (x)) . Then φ(2) is a kinetic boundary layer in (−∞, 0] satisfying the boundary condition P+ φ(2) (0) = RP+ φ(2) (0)

(76)

4.2 The two-point BVP We want to numerically construct the linearized kinetic boundary layer on [0, ∞) + b− φ(x) = φ(1) (x) = (β3 + x · α1 β4 )b3 + β4 b4 + exp(−xΛ) · (77) b− − In particular, we want to find out the relation between the gradient g1 := α1 β4 and the jump condition β3 for b3 . From theoretical arguments we easily find a linear relationship s1 g1 + s2 β3 = 0.

(78)

For the calculation we define for some H > 0 the shift operators SH φ(x) := φ(x − H)

,

S−H φ(x + H)

(79)

and the function ψ(x) :=

1, S−H φ(1) (x) − SH φ(2) (x) 2

(80)

on [−H, H], where φ(2) is defined as in Lemma 5. ψ has the representation ψ(x) = x · α1 β4 b3 + β4 b4 − + 1 b− b− + exp(−(x + H)Λ) · − − exp(−(H − x)Λ) · b− b+ 2 −

(81)

We assume H large enough and thus neglect terms of the form exp(−H · Λ). Then ψ satisfies ψ(0) = β4 b4 , ′

ψ (0) = α1 β4 b3 , ψ ′′ (0) = 0.

(82) (83) (84)

The boundary values are φ− (2H) φ+ (2H) + φ (2H) ψ(H) = −φ(2) (0) + φ− (2H)

ψ(−H) = φ(1) (0) −

(85) (86)

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H. Babovsky

where φ(2H) = β3 b3 + 2Hψ ′ (0) + ψ(0) = (β3 + α1 β4 )b3 + β4 b4 =: πb3 + β4 b4 .

(87)

The boundary conditions for ψ read ψ + (−H) + φ− (2H) = R[ψ − (−H) + φ+ (2H)] ψ − (H) − φ− (2H) = R[ψ + (H) − φ+ (2H)]

(88) (89)

Suppose the gradient ψ ′ (0) for b3 is prescribed. Because of the linear dependence (77) we may solve the linearized differential system with the boundary conditions (84), (85) and with φ(2H) given by (86) with some fixed π. The only unknown is β4 . Once this is known, all other quantities of interest can be reconstructed. Given a fixed β4 , a solver of the corresponding BVP is readily written applying an upwind technique as in section 3.4. The correct value of β4 is found using an iteration scheme to construct the correct solution ψ. This is found by exploiting the symmetry property of ψ with respect to the origin. A typical result of such calculations is shown in Fig. 7. which demonstrates the temperature distribution of a heat layer problem. Finally we want to mention that two-point BVPs are also successfully applied to calculate boundary layers on the basis of the nonlinear collision operator. The only problem is to formulate boundary conditions at the artificial boundary x = H. This can be managed with an iterative scheme which minimizes the curvature close to the right boundary.

1.2

1.1

T 1 i

1

0.9

0.8 0.8

0 0

20

40

60

80

100

i

Fig. 7. Linearized heat layer problem.

120 128

Hexagonal kinetic models and simulation of kinetic boundary layers

(A,1)

(A,2)

(B,1)

(B,2)

(C,1)

(C,2)

65

Fig. 8. Channel ﬂow with diﬀerent boundary conditions. A: diﬀuse, B: reverse, C: specular reﬂection; 1: velocity, 2: shear stress.

References 1. L. S. Andallah, H. Babovsky. A discrete Boltzmann equation based on hexagons. Math. Models Methods Appl. Sci, 13:1537–1563, 2003. 2. P. Andries, J.-F. Bourgat, P. Le Tallec, B. Perthame. Numerical comparison between the Boltzmann and ES-BGK models for rarefied gases. Preprint no. 3872, INRIA Rocquencourt, 2000. 3. H. Babovsky. Convergence proof for Nanbu’s Boltzmann simulation scheme. Europ. J. of Mech. B/Fluids, 1:41–55, 1989. 4. H. Babovsky. Die Boltzmann-Gleichung. Teubner, Stuttgart, 1998. 5. H. Babovsky. Kinetic boundary layers: on the adequate discretization of the Boltzmann collision operator. J. Comp. Appl. Math., 110:225–239, 1999. 6. H. Babovsky Design of numerically efficient kinetic models. Preprint 05/02, Inst. f. Math., TU Ilmenau. 2002 (zur Ver¨ oﬀentlichung eingereicht).

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Fig. 9. Supersonic ﬂow propagating into a channel. (a) density; (b) ﬂow ﬁeld around the stagnation point. (See also color ﬁgure, Plate 7.)

7. H. Babovsky, D. G¨ orsch und F. Schilder. Steady kinetic boundary value problems in: Lecture Notes on the Discretization of the Boltzmann Equation, N. Bellomo and R. Gatignol Eds., pp. 131–156, World Scientiﬁc 2002. 8. H. Babovsky and T. Platkowski. Kinetic boundary layers for the Boltzmann equation on discrete velocity lattices. Preprint 07/03, Inst. f. Math., TU Ilmenau. 2003 (zur Ver¨ oﬀentlichung eingereicht). 9. P. Billingsley. Convergence of probability measures. John Wiley & Sons, New York, 1968. 10. A.V. Bobylev, A. Palczewski and J. Schneider. On the approximation of the Boltzmann equation by discrete velocity models. C. R. Acad. Sci. Paris I 320(5):639–644, 1995.

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Fig. 10. Thermally driven ﬂow. (a) density; (b) convection cells (ﬂow ﬁeld). (See also color ﬁgure, Plate 7.)

11. C. Cercignani. The Boltzmann equation and its applications. Springer, New York, 1988. 12. F. Golse, B. Perthame and C. Sulem. On a boundary layer problem for the nonlinear Boltzmann equation. Arch. Rational Mech. Anal., 103:81–96, 1988. 13. J. Schnute, M. Shinbrot. Kinetic theory and boundary conditions for ﬂuids. Can. J. Math., 25:1183–1215, 1973. 14. V. Sofonea and R. F. Sekerka. Viscosity of ﬁnite diﬀerence lattice Boltzmann models. J. Comput. Phys., 184:422–434, 2003. 15. Y. Sone, C. Bardos, F. Golse and H. Sugimoto. Asymptotic theory of the Boltzmann system, for a steady ﬂow of a slightly rareﬁed gas with a ﬁnite Mach number: General theory. Eur. J. Mech. B-Fluids, 19:325–360, 2000. 16. H. Struchtrup and M. Torrilhon. Regularization of Grad’s 13 moment equations: Derivation and linear analysis. Phys. Fluids, 15:2668–2680, 2003.

High-resolution Simulation of Detonations with Detailed Chemistry Ralf Deiterding1 and Georg Bader2 1

2

California Institute of Technology, 1200 East California Blvd., Mail-Code 158-79, Pasadena, CA 91125, [email protected] Institut f¨ ur Mathematik, Technische Universit¨ at Cottbus, Universit¨ atsplatz 3-4, 03044 Cottbus, [email protected]

Summary. Numerical simulations can be the key to the thorough understanding of the multi-dimensional nature of transient detonation waves. But the accurate approximation of realistic detonations is extremely demanding, because a wide range of diﬀerent scales need to be resolved. This paper describes an entire solution strategy for the Euler equations of thermally perfect gas-mixtures with detailed chemical kinetics that is based on a highly adaptive ﬁnite volume method for blockstructured Cartesian meshes. Large-scale simulations of unstable detonation structures of hydrogen-oxygen detonations demonstrate the eﬃciency of the approach in practice.

1 Introduction Reacting flows have been a topic of on-going research since more than hundred years. The interaction between hydrodynamic flow and chemical kinetics can be extremely complex and even today many phenomena are not very well understood. One of these phenomena is the propagation of detonation waves in gaseous media. A detonation is a shock-induced combustion wave, which internally consists of a discontinuous hydrodynamic shock wave followed by a smooth region of decaying combustion. The adiabatic compression due to the passage of the shock rises the temperature of the combustible mixture above the ignition limit. The reaction results in an energy release that drives the shock wave forward. In a self-sustaining detonation, shock and reaction zone propagate essentially with an identical speed dCJ that is approximated to good accuracy by the classical Chapman-Jouguet (CJ) theory, cf. [37]. But up to now, no theory exists that describes the internal flow structure satisfactory. The Zel’dovich-von Neumann-D¨ oring (ZND) theory is widely believed to reproduce the one-dimensional detonation structure correctly, but already early experiments [7, 30] uncovered that the reduction to one space dimension is not

70

R. Deiterding, G. Bader B

Trajectory D

A

G

Mach stem Transverse wave

C

Head of reaction zone

F E

Triple point

Incident shock

L

Fig. 1. Left: regular detonation structure at three diﬀerent time steps on triple point trajectories, right: enlargement of a periodical triple point conﬁguration. E: reﬂected shock, F: slip line, G: diﬀusive extension of slip line with ﬂow vertex.

even justified in long combustion devices. It was found that detonation waves usually exhibit non-neglectable instationary multi-dimensional sub-structures and do not remain planar. The multi-dimensional instability manifests itself in instationary shock waves propagating perpendicular to the detonation front. A complex flow pattern is formed around each triple point, where the detonation front is intersected by a transverse shock. Pressure and temperature are increased remarkable in a triple point and the chemical reaction is enhanced drastically giving rise to an enormous local energy release. Hence, the accurate representation of triple points is essential for safety analysis, but also in technical applications, e.g. in the pulse detonation engine. Some particular mixtures, e.g. low-pressure hydrogen-oxygen with high argon diluent, are known to produce very regular triple point movements. The triple point trajectories form regular “fish-scale” patterns, so called detonation cells, with a characteristic length L and width λ (compare left sketch of Fig. 1). Fig. 1 displays the hydrodynamic flow pattern of a detonation with regular cellular structure, how it is known since the early 1970s, cf. [29, 23]. The right sketch shows the periodic wave configuration around a triple point in detail. It consists of a Mach reflection, a flow pattern well-known from non-reactive supersonic hydrodynamics [4]. The undisturbed detonation front is called the incident shock, while the transverse wave takes the role of the reflected shock. The triple point is driven forward by a strong shock wave, called Mach stem. Mach stem and reflected shock enclose the slip line, the contact discontinuity. The Mach stem is always much stronger than the incident shock, what results in a considerable reduction of the induction length lig , the distance between leading shock and measurable reaction. The shock front inside the detonation cell travels as two Mach stems from point A to the line BC. In the points B and C the triple point configuration is inverted nearly instantaneously and the front in the cell becomes the incident shock. Along the symmetry line AD the change is smooth and the shock strength decreases

High-resolution Simulation of Detonations with Detailed Chemistry

71

continuously. In D the two triple points merge exactly in a single point. The incident shock vanishes completely and the slip line, which was necessary for a stable triple point configuration between Mach stem and incident shock, is torn off and remains behind. Two new triple points with two new slip lines develop immediately after D. But the experimental analysis of the described transient sub-structures is difficult. Direct numerical simulation of the governing equations can be an alternative here [24, 8, 11, 16, 36, 33]. In the following, we will demonstrate that recent parallel computers of moderate size, e.g. clusters of standard personal computers, allow detonation structure simulations in two and three space dimensions even for detailed non-equilibrium chemistry that provide deep insight into the internal flow structure far beyond previous experimental results [29, 23].

2 Detonation Modeling The governing equations for detonation propagation in premixed gases with realistic chemistry are the Euler equations for multiple thermally perfect species with chemically reactive source terms [10, 37]. In d-dimensional Cartesian coordinates these equations can be written as an inhomogeneous conservation law of the structure d ∂ ∂ q(x, t)+ fn (q(x, t)) = s(q(x, t)) , x = (x1 , . . . , xd )T ∈ Rd , t ∈ R+ 0 , ∂t ∂x n n=1 (1) where q = q(x, t) ∈ S ⊂ RM denotes the vector of conserved quantities. The functions fn (q) ∈ C1 (S, RM ), n = 1, . . . , d are the hydrodynamic fluxes, s(q) ∈ C1 (S, RM ) is the source term.

2.1 Euler Equations for Gas-Mixtures For the multi-component Euler equations with K species the vector of conserved quantities has M = K + d + 1 components. We choose the form q(x, t) = (ρ1 , . . . , ρK , ρu1 , . . . , ρud , ρE)T .

(2)

The partial density of the ith species denoted by ρi , where i = 1, . . . , K. is K The total density of the mixture ρ = i=1 ρi is a conserved quantity, too. The K ratios Yi = ρi /ρ are called mass fractions. They satisfy the relation i=1 Yi = 1. We denote the nth component of the velocity vector u = (u1 , . . . , ud )T by un and E is the total energy per unit mass. The flux functions are fn (q) = (ρ1 un , . . . , ρK un , ρu1 un + δ1n p, . . . , ρud un + δdn p, un (ρE + p))T (3)

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for n = 1, . . . , d. Herein, p is the hydrostatic pressure and δjn denotes the Kronecker-Symbol. We assume that all species are ideal gases in thermal equilibrium. Under this assumption the same temperature T can be used to evaluate the partial pressure of all species as pi = RT ρi /Wi with R denoting the universal constant and Wi the molecular weight, respectively. According to Dalton’s law the total pressure is given by p=

K i=1

pi = RT

K ρi . Wi i=1

(4)

Each species is assumed to be thermally perfect and has a temperaturedependent specific heat cpi (T ). The functions cpi (T ) are usually approximated by polynomials of degree 4 which are valid within a restricted temperature range, e.g. from 300 K to 5000 K [31, 20]. The enthalpies per unit mass are written as T

hi (T ) = h0i +

cpi (σ)dσ

T0

with h0i called the heat of formation at the reference temperature T 0 . For the K enthalpy of the mixture h = i=1 Yi hi (T ) holds true. Inserting this into the thermodynamic relation ρh−ρE +ρ u2 /2−p = 0 and inserting (4) for p yields ϕ(q, T ) :=

K i=1

ρi hi (T ) − ρE + ρ

K ρi u2 − RT =0. 2 Wi i=1

(5)

It can be proven rigorously [5], that for each q in the space of admissible states S a unique temperature T exists that satisfies (5). Unfortunately, a closed form of the inverse can only be derived under simplifying assumptions and the computation of T from (5) is in general unavoidable, whenever the pressure p has to be evaluated. The appropriate speed of sound for the described model is the frozen speed of sound, which is given by 2

c =

∂p ∂ρ

s,Y1 ,...,YK

=

K i=1

, Yi φi − (γ − 1) u2 − H

(6)

& u2 ' ∂p R u2 and φi := − hi (T ) + γ = (γ − 1) T . The 2 ∂ρi 2 Wi coefficient γ = γ(Y1 , . . . , YK , T ) can be calculated from the mixture quantities '−1 & K K Y c (T ) and W = by employing the relation Y /W cp = i pi i i i=1 i=1

with H = h +

γ = cp (cp − R/W )−1 . By inserting the previous expression for φi into (6) and by applying the ideal-gas law (4) it can be shown, that the frozen speed of sound of a thermally perfect gas-mixture satisfies the relation c2 = γp/ρ.

High-resolution Simulation of Detonations with Detailed Chemistry

73

Mathematical Properties The Euler equations for thermally perfect gas-mixtures inherit most mathematical properties of the standard Euler equations , - for a single polytropic gas with equation of state p = (γ − 1) ρE − ρ u2 /2 , cf. [28]. Utilizing expression (6) for the speed of sound the hyperbolicity of (1) with vector of state (2) and flux functions (3) can easily be proven [5]. For d = 2, 3 the proof uses the validity of the rotational invariance property that carries over from the standard Euler equations almost directly [32] and requires just the diagonalization of the Jacobian A1 (q) = ∂f1 (q)/∂q. For instance, in two space dimensions the matrix of right eigenvectors R1 (q) = (r1 | . . . |rK+d+1 ) that diagonalizes A1 (q) with R−1 1 (q) A1 (q) R1 (q) = Λ1 (q) for all q ∈ S with Λ1 (q) = diag(u1 − c, u1 , . . . , u1 , u1 + c) takes the form ⎡ ⎤ Y1 1 0 ... 0 0 Y1 ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ . . . . . . .. .. .. .. .. .. ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ R1 (q) = ⎢ Y ⎥ . (7) 0 ... 0 1 0 YK K ⎢ ⎥ ⎢ u1 − c u1 ... u1 0 u1 + c ⎥ ⎢ ⎥ ⎢ u2 ⎥ u2 ... u2 1 u2 ⎢ ⎥ ⎣ ⎦ φK φ1 ... u2 − u2 H + u1 c H − u1 c u2 − γ−1 γ−1 Furthermore, it can be shown that the flux functions fn (q) and their Jacobians An (q) satisfy the homogeneity property fn (q) = An (q)q for all q ∈ S. The Homogeneous Riemann Problem The profound understanding of the Riemann Problem (RP) in the nonreactive case provides the theoretical basis for the construction of a reliable Godunov-type method in Sec. 3.2 as a key ingredient for detonation simulation. For s ≡ 0 the solution structure of a quasi-one-dimensional RP with discontinuous initial data ql , xn < 0 , (8) q(x, 0) = qr , xn > 0 can be shown to be in principle identical to the standard case of a single polytropic gas that is discussed in detail for example in [28, 12, 32]. The first and last characteristic field with the eigenvalues un −c and un +c are genuinely nonlinear, provided that the condition ∂γ γ(γ + 1) = (1 − γ)T ∂T

(9)

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is satisfied for all q ∈ S [5]. All other characteristic fields are associated to the eigenvalue un and are linearly degenerate. If we assume that condition (9) is satisfied, it can be concluded that the solution of the RP consists of admissible shocks or smooth rarefaction waves in the first and last characteristic field, while the contributions in all other characteristic fields sum up to a single contact discontinuity. Unlike standard Euler equations, no complete set of Riemann invariants can be found for the Euler equations for thermally perfect gases [22]. The only Riemann invariants that can be derived are the mass fractions Yi , which are constant across the first and last characteristic field [5], and the velocity un and the pressure p, which are invariant across the contact discontinuity. 2.2 Reactive Source Terms We write the source term of detailed chemical reaction in the form s(q) = (W1 ω˙ 1 , . . . , WK ω˙ K , 0, . . . , 0, 0)

T

.

The chemical production of each species is expressed as product of its production rate in molar concentration per unit volume ω˙ i = ω˙ i (q) ∈ C 1 (S, R) and Wi . The rates ω˙ i (ρ1 , . . . , ρK , T ) are derived from a reaction mechanism of J chemical reactions K

f νji Si ⇋

K

r νji Si ,

j = 1, . . . , J ,

i=1

i=1

f r are the stoichiometric coefficients of species Si appearing as where νji and νji a reactant and as a product. The entire molar production rate of species Si per unit volume is then given by

ω˙ i =

f r 1 νjl νjl K K 1 ρl ρl f r (νji −νji ) kjf −kjr , Wl Wl j=1

J

l=1

i = 1, . . . , K , (10)

l=1

with kjf (T ) and kjr (T ) denoting the forward and backward reaction rate of each chemical reaction [37]. The reaction rates are calculated by the Arrhenius law f /r

kj

f /r

f /r

(T ) = Aj T βj

f /r

exp(−Ej

/RT ) .

(11)

Some backward reaction rates might be derived by assuming the corresponding chemical reaction to be in chemical equilibrium, but especially detonations usually require mechanisms that utilize non-equilibrium backward reaction rates at least for some of the reactions. A chemical kinetics package, e.g. Chemkin [19], can be utilized to evaluate (10), (11) according to the reaction mechanism and given thermodynamic data.

High-resolution Simulation of Detonations with Detailed Chemistry

75

3 Numerical Methods We use the time-operator splitting approach or method of fractional steps [17] to decouple hydrodynamic transport and chemical reaction numerically. This technique is most frequently used for time-dependent reactive flow computations. The homogeneous partial differential equation d

H(∆t) :

∂q ∂ + fn (q) = 0 , ∂t n=1 ∂xn

∆t ˜ κ+1 IC: Qκ =⇒ Q

(12)

and the usually stiff ordinary differential equation S (∆t) :

∂q = s(q) , ∂t

∆t ˜ κ+1 =⇒ IC: Q Qκ+1

(13)

are integrated successively with the data Q from the preceding step as initial condition (IC). 3.1 Finite Volume Schemes The appropriate discretization technique for conservation laws with discontinuous solution is the finite volume (FV) approach. For simplicity, we assume an equidistant discretization in two space dimensions with mesh widths ∆x1 , ∆x2 and a constant time step ∆t. A conservative time-explicit finite volume scheme for (12) has the formal structure 3 3 2 2 ˜ κ+1 = Qκ − ∆t F1 1 −F1 1 − ∆t F2 1 −F2 1 (14) H(∆t) : Q jk jk j− 2 ,k j,k− 2 ∆x1 j+ 2 ,k ∆x2 j,k+ 2 ˜ κ+1 and the important discrete conservation property = j,k Qjk satisfies κ j,k Qjk for vanishing boundary fluxes. Such a scheme can easily be constructed by applying the idea of operator splitting also to (12) and by using two quasi-one-dimensional FV schemes successively, e.g. 2 3 1 κ 1 κ κ ˜ κ+ 2 = Qκ − ∆t F1 (Qκ , . . . , Q )−F (Q , . . . , Q ) , Q jk j−ν+1,k j+ν,k j−ν,k j+ν−1,k jk ∆x1 2 3 1 1 1 1 1 ˜ κ+ 2 )−F2 (Q ˜ κ+ 2 , . . . , Q ˜ κ+ 2 ˜ κ+ 2 − ∆t F2 (Q ˜ κ+1 = Q ˜ κ+ 2 , . . . , Q ) , Q jk jk j,k−ν+1 j,k+ν j,k−ν j,k+ν−1 ∆x2 (∆t)

(∆t)

i.e. H(∆t) = X2 X1 . With this definition the entire splitting method reads (∆t) (∆t) Qκ+1 = S (∆t) X2 X1 (Qκ ). The later method is formally only first-order accurate, but it usually gives very satisfactory results, if high-resolution shock(∆t) capturing schemes are employed for the operators Xn . Formally secondorder accurate splitting methods are possible [32], but they lead to similar results in most practical cases and we have observed only minor improvements for typical detonation structure simulations [5]. For the upwind method

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R. Deiterding, G. Bader

formulated in Algorithm 1, the described splitting is stable under the CourantFriedrichs-Levy (CFL) condition & ∆t ' ∆t ≤1, (15) CCFL := max Sj+ 12 ,k , Sj,k+ 12 j,k ∆x1 ∆x2 where Sj+ 12 ,k , Sj,k+ 12 denote the maximal signal speeds in both space directions according to step (S12) in Algorithm 1. 3.2 High-resolution Upwind Method (∆t)

The operators Xn can be implemented effectively with a single quasi-onedimensional scheme that allows the canonical exchange of the velocities un and the indices j and k. The method should achieve a higher order of accuracy in smooth solution regions and should approximate discontinuities on the basis of the characteristic information (upwinding) without spurious overshoots. Today, several excellent textbooks are available, e.g. [32], which discuss the construction of quasi-one-dimensional high-resolution methods for supersonic hydrodynamics in great detail and we therefore sketch the basic components of our particular method only briefly. Our high-resolution scheme is built around a first-order Godunov-type method that solves the Riemann problem between two neighboring cell values Ql and Qr approximately, which we describe exemplary for the x1 -direction. The method is based on an extension of Roe’s linearized Riemann solver for Euler equations for a single polytropic gas for multiple thermally perfect species by Grossman and Cinella [13] that corresponds to the steps (S1) to (S7) in Algorithm 1. The structure of the Roe-averaged right eigenvectors ˆrm is given in (7). In (S8), (S9) the two intermediate states of the linearized RP are evaluated and the intrinsic problem of unphysical total densities and internal energies near vacuum due to the Roe linearization, cf. [9], is circumvented by switching in case of an unphysical approximation to the simple, but extremely robust Harten-Lax-Van Leer (HLL) Riemann solver. If Roe’s flux approximation is applied in step (S10), violations of the entropy condition are generally avoided by adding an appropriate amount of numerical viscosity [15]. A natural choice for the parameter η for Euler equations is η = 12 (|u1,r − u1,l | + |cr − cl |), cf. [27]. In one space-dimension, (16) need only be applied to ι = 1, 3 and s¯2 = s2 can be used, but two- and threedimensional detonation simulations usually require the extension of (16) to ι = 2. The shock of typical detonation waves is extraordinarily strong and its approximation is often corrupted by the carbuncle phenomenon, a multi-dimensional numerical crossflow instability that occurs at strong grid-aligned shocks or detonation waves Fig. 2. H-correc[26]. The carbuncle phenomenon can be avoided com- tion between the cells pletely by applying (16) to all characteristic fields and (j, k) and (j, k + 1).

High-resolution Simulation of Detonations with Detailed Chemistry (S1) Calculate ρˆ :=

√

ρl ρr and vˆ :=

√ √ ρl vl + ρr vr √ √ ρl + ρr

77

for un , Yi , H, hi , T .

K

cv with cˆ{p/v} = i=1 Yˆi cˆ{p/v}i and (S2) Compute γˆ := cˆp /ˆ Tr 1 c{p,v}i (τ ) dτ . cˆ{p/v}i = Tr − Tl T l & 2 ' ˆ i + γˆ R Tˆ. γ − 1) uˆ2 − h (S3) Calculate φˆi := (ˆ Wi '1/2 & K ˆ ˆ ˆ (S4) Calculate cˆ := γ − 1)(ˆ u2 − H) . i=1 Yi φi − (ˆ

(S5) Use ∆Q = Qr − Ql and ∆p to compute the wave strengths

∆p ∆p ∓ ρˆcˆ∆u1 , a1+i = ∆ρi − Yˆi 2 , aK+n = ρˆ∆un . 2ˆ c2 cˆ K+d rm , W3 = aK+d+1 ˆ rK+d+1 . r1 , W2 = am ˆ (S6) Calculate W1 = a1 ˆ a1,K+d+1 =

m=2

(S7) Evaluate s1 = u ˆ1 − cˆ, s2 = u ˆ1 , s3 = u ˆ1 + cˆ.

(S8) Evaluate ρ⋆l/r , u⋆1,l/r , e⋆l/r , c⋆1,l/r from Q⋆l = Ql + W1 and Q⋆r = Qr − W3 .

(S9) If ρ⋆l/r ≤ 0 or e⋆l/r ≤ 0 set s1 = min(u1,l − cl , u1,r − cr ), s3 = max(u1,l + cl , u1,r + cr ), use HLL ﬂux ⎧ f (Ql ) , 0 < s1 , ⎪ ⎪ ⎪ ⎨ s3 f (Ql ) − s1 f (QR ) + s1 s3 (Qr − Ql ) F(Ql , Qr ) = , s1 ≤ 0 ≤ s3 , ⎪ s3 − s1 ⎪ ⎪ ⎩ 0 > s3 , f (Qr ) ,

and go to (S12). , (S10) Evaluate Roe ﬂux F(Ql , Qr ) = 21 f (Ql ) + f (Qr ) − 3ι=1 |˜ sι |Wι with entropy enforcement formula |sι | , |sι | ≥ 2η , (16) |¯ sι | = |s2ι |/(4η) + η , |sι | < 2η . l K Yi , Fρ ≥ 0 , ⋆ (S11) With Fρ := i=1 Fi replace Fi by Fi = Fρ · Yir , Fρ < 0 . (S12) Evaluate maximal signal speed by S = max(|s1 |, |s3 |).

Algorithm 1. Hybrid Roe-HLL scheme for detonation simulation.

evaluating η in a multi-dimensional way. In all computations of Sec. 5 we have successfully utilized the “H-correction” of Sanders et al. [27] for this purpose. For instance in the x2 -direction it takes the form 6 5 η˜j,k+ 21 = max ηj+ 21 ,k , ηj− 21 , k , ηj, k+ 12 , ηj− 21 , k+1 , ηj+ 12 , k+1

in the two-dimensional case, see Fig. 2. Step (S11) ensures the positivity of the mass fractions Yi , if the Roe approximation is applied [21]. The HLL scheme can be proven to be positivity preserving in Yi and does not require this step.

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R. Deiterding, G. Bader

A detailed derivation of the entire Roe-HLL scheme and thorough numerical comparisons with various standard methods can be found in [5]. The hybrid Riemann solver is extended to a high-resolution method with the MUSCL-Hancock variable extrapolation technique by Van Leer [34]. The technique uses a five-point stencil with ν = 2. In contrast to the Euler equations for a single polytropic gas, the extrapolation for the Euler equations of Sec. 2.1 can not be formulated completely in conservative variables, because the solvability of the nonlinear equation (5) can not be guaranteed for an extrapolated vector of state. We recommend to apply the MUSCL extrapolation to ρ, p, Yi and ρun and to derive a thermodynamically consistent extrapolated vector of state from these. See [5] for details. 3.3 Integration of Reaction Terms The numerical treatment of chemical reaction terms with the method of fractional steps requires the solution of the ODE ∂ρi = Wi ω˙ i (ρ1 , . . . , ρK , T ) , ∂t

i = 1, . . . , K

with initial condition ρi (0) = ρYi0 , i = 1, . . . , K in every FV cell. The total density ρ in this cell, the specific energy E and the velocities un remain unchanged during the integration, what corresponds to a reaction in an adiabatic constant volume environment. ODEs arising from chemical kinetics are usually stiff and we employ a semi-implicit Rosenbrock-Wanner method by Kaps and Rentrop of fourth order with automatic step-size adjustment [18]. The computational expensive reaction rate expressions (10) are evaluated by a mechanism-specific routine, which is produced by a source code generator on top of the Chemkin-II library [19] in advance. The code generator implements the formulas of Sec. 2.2 without any loops and inserts the constants f /r f /r f /r νji , Aj , −Ej directly into the code. 3.4 Evaluation of the Temperature The FV method for thermally perfect gas-mixtures sketched in Sec. 3.2 and the reaction term integration described in Sec. 3.3 require the computation of the temperature T from a discrete vector of state Q by solving (5). As (5) has a unique temperature solution for all admissible vectors of state and ϕ(·, T ) can be shown to be a strict monotone function in T (see [5]), the efficient solution of (5) is straight-forward: We start the solution procedure with a standard Newton iteration that is initialized with the temperature value of the preceding time step. If the Newton method does not converge in a reasonable number of steps, a standard bisection technique is employed. In order to speed up further the polynomial evaluation of the temperaturedependent properties cpi (T ) and hi (T ), look-up tables for all species are constructed during the startup of the computational code. These tables store

High-resolution Simulation of Detonations with Detailed Chemistry 0.4

3000

6e-6

2000

4e-6

79

4 Pts/lig 1 Pts/lig

1000

0.1

0

2O2

0.2

YH

T [K]

3

ρ [kg/m ]

0.3

ρ (left axis) T (right axis) 15

10

298 5

0

2e-6

0 5

0

Fig. 3. ZND solution for a self-sustaining hydrogen-oxygen detonation (dCJ ≈ 1627 m/s, lig ≈ 1.404 mm) and representation of the mass fraction of H2 O2 on diﬀerent meshes (right). The points represent the value in the center of a ﬁnite volume. The abscissas display the distance behind the detonation front in mm.

cpi (T ) and hi (T ) for all integers in the valid temperature range and intermediate values are interpolated. 3.5 Meshes for Detonation Simulation Numerical simulations of detonation waves require computational meshes, which are able to represent the strong local flow changes due to the reaction correctly. In particular, the shock of a detonation wave with detailed kinetics can be very sensitive to changes of the reaction behind and, if the mesh is too coarse to resolve all reaction details correctly, the Riemann Problem at the detonation front is changed remarkably leading to a wrong speed of propagation. We make a simple discretization test in order to illustrate, how fine computational meshes for accurate detonation simulations in fact have to be. The left graph of Fig. 3 displays the flow fields of ρ and T according to the ZND detonation model for the frequently studied H2 : O2 : Ar CJ detonation with molar ratios 2 : 1 : 7 at T0 = 298 K and p0 = 6.67 kPa for a hydrogenoxygen reaction mechanism extracted from a larger hydrocarbon mechanism assembled by Westbrook [35]. The mechanism uses 34 elementary reactions for the 9 species H, O, OH, H2 , O2 , H2 O, HO2 , H2 O2 and Ar. Throughout this paper only this mechanism has been employed. The right graph of Fig. 3 displays the exact distribution of YH2 O2 discretized with different FV grids. Even a resolution of 4 finite volumes per induction length (4 Pts/lig ) is not sufficient to capture the maximum of the intermediate product H2 O2 correctly. This requires at least 5 to 6 Pts/lig , but in triple points even finer resolutions are required. The discretization of typical combustors with such fine uniform grids can easily exceed 109 FV cells in the two- and 1012 cells in the three-dimensional case. As multi-dimensional detonations are intrinsically unstable (compare Sec. 1), numerical simulations have to be instationary and usually involve several ten thousand time steps. Consequently, uniform meshes are far too expensive and the application of a sophisticated dynamically adaptive mesh refinement technique is indispensable.

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R. Deiterding, G. Bader

Fig. 4. The AMR method creates a hierarchy of rectangular subgrids.

4 Adaptive Mesh Refinement In order to the supply the required temporal and spatial resolution efficiently, we employ the blockstructured adaptive mesh refinement (AMR) method after Berger and Colella [2, 3] which is tailored especially for hyperbolic conservation laws on logically rectangular FV grids (not necessarily Cartesian). We have implemented the AMR method in a generic, dimension-independent object-oriented framework in C++. It is called AMROC (Adaptive Mesh Refinement in Object-oriented C++) and is free of charge for scientific use [6]. The adaptive algorithm has been realized completely decoupled from a particular FV method. All what the algorithm requires are specific implementations of the operators H(·) and S (·) on a single rectangular grid G, where H(·) has to utilize ν auxiliary cells (ghost or halo cells) around G to define discrete boundary conditions. The entire framework has been validated extensively on a large number of hydrodynamic standard test cases. See [5] and [6] for results. 4.1 Berger-Collela AMR Method Instead of replacing single cells by finer ones, as it is done in cell-oriented refinement techniques, the Berger-Collela AMR method follows a patch-oriented approach. Cells being flagged by various error indicators (shaded in Fig. 4) are clustered with a special algorithm [1] into non-overlapping rectangular grids Gl,m that define the domain of an entire level l = 0, . . . , lmax by 7 Ml Gl := m=1 Gl,m . Refinement grids are derived recursively from coarser ones and a hierarchy of successively embedded levels is thereby constructed, cf. Fig. 4. All mesh widths on level l are rl -times finer than on level l − 1, i.e. ∆tl := ∆tl−1 /rl and ∆xn,l := ∆xn,l−1 /rl with rl ∈ N, rl ≥ 2 for l > 0 and r0 = 1, and a time-explicit FV scheme (in principle) remains stable under a condition like (15) on all levels of the hierarchy. The recursive integration order visualized in the left sketch of Fig. 5 is an important difference to usual unstructured adaptive strategies and is one of the main reasons for the high efficiency of the approach.

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Fig. 5. Left: Recursive integration order. Right: Sources of ghost cell values.

The numerical scheme is applied on level l by calling the single-grid routines H(∆t) , S (∆t) in a loop over all subgrids Gl,m . The execution of the numerical loop in UpdateLevel() in Alg. 2 requires the previous setting of the ghost cell values. Three types of boundary conditions have to be considered in the sequential case, see right sketch of Fig. 5. Cells outside of the root domain G0 are used to implement physical boundary conditions. Ghost cells in Gl have a unique interior cell analogue and are set by copying the data value from the grid, where the interior cell is contained (synchronization). On the root level no further boundary conditions need to be considered, but for l > 0 also internal boundaries can occur. They are set by a conservative time-space interpolation from two previously calculated time steps of level l − 1. Beside a general data tree that stores the topology of the hierarchy (cf. Fig. 4), the AMR method requires at most two regular arrays assigned to each subgrid which contain the discrete vector of state Q for the actual and updated time step. In the Algorithms 2 and 3 we denote by Ql (t) and Ql (t + ∆tl ) the unions of these arrays on level l. The regularity of the input data for the numerical routines allows high performance on vector and super-scalar processors and cache optimizations. Small data arrays are effectively avoided by leaving coarse level data structures untouched, when higher level grids are created. Values of cells covered by finer subgrids are overwritten by averaged fine grid values subsequently. The later operation leads to a modification of the numerical stencil on the coarse mesh and requires a special flux correction in cells abutting a fine grid. The correction replaces the coarse grid flux along the fine grid boundary by a sum of grid fluxes and ensures the discrete conservation property for the hierarchical method. See [2] or [5] for details. The basic recursive AMR algorithm is formulated in Alg. 2. Except the regridding procedure, all operations have already been explained. New refinement grids on all higher levels are created by calling Regrid() from level l. Level l by itself is not modified. To consider the nesting of the level domains already in the grid generation, Alg. 3 starts at the highest refineable level lc , where 0 ≤ lc < lmax . The refinement flags are stored in grid-based integer arrays N ι . A clustering algorithm [1] is necessary to create a new refinement

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Regrid(l) AdvanceLevel(l) For ι = lc Downto l Do Repeat rl times Flag N ι according to Qι (t) Set ghost cells of Ql (t) ˘ ι+1 from N ι If time to regrid Generate G Regrid(l) ˘ l+1 , . . . , G ˘ lc +1 Ensure nesting of G UpdateLevel(l) If level l + 1 exists For ι = l + 1 To lc + 1 Do Set ghost cells of Ql (t + ∆tl ) ˘ ι (t) from G ˘ι Create Q ι−1 AdvanceLevel(l + 1) ˘ ι (t) Interpolate Q (t) onto Q Average Ql+1 (t + ∆tl ) onto ι ι ˘ Copy Q (t) onto Q (t) Ql (t + ∆tl ) ˘ ι (t) Set ghost cells of Q l n,l+1 Correct Q (t + ∆tl ) with δF ι ι ˘ Q (t) := Q (t) t := t + ∆tl Alg. 2. Recursive AMR algorithm.

Alg. 3. Regridding procedure.

˘ ι+1 on basis of N ι until the ratio between flagged and all cells in every new G ˘ ι+1,m is above a prescribed threshold 0 < ǫtol < 1. grid G The reinitialization of the hierarchical data structures is done in the second ˘ ι (t). Cells in newly refined regions loop of Alg. 3 utilizing auxiliary data Q ˘ ˘ ι ∩ Gι are copied. Gι \Gι are initialized by interpolation, values of cells in G As interpolation requires the previous synchronized reorganization of Qι−1 (t), recomposition starts on level l + 1. 4.2 Parallelization In the AMROC framework, we follow a rigorous domain decomposition approach and partition the AMR hierarchy from the root level on. We assume a parallel machine with P identical nodes and split the root domain G0 into P non-overlapping portions Gp0 , p = 1, . . . , P by G0 =

P 8

p=1

Gp0

with Gp0 ∩ Gq0 = ∅ for p = q .

The key idea now is that all higher level domains Gi are required to follow the decomposition of the root level, i.e. Gpl := Gl ∩ Gp0 .

(17)

Condition (17) can cause the splitting of a subgrid Gl,m into multiple subgrids on different processors. Under requirement (17) we estimate the work on an arbitrary subdomain Ω ⊂ G0 by / . l l max 1 (18) rκ . Nl (Gl ∩ Ω) W(Ω) = l=0

κ=0

Herein, Nl (·) denotes the total number of FV cells on level l in the given domain. The product in (18) is used to consider the time step refinement.

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A nearly equal distribution of the work necessitates Lp :=

P · W(Gp0 ) ≈ 1 for all p = 1, . . . , P . W(G0 )

(19)

In AMROC, decompositions Gp0 with similar workload are found at runtime as the hierarchy evolves by a hierarchical partitioning algorithm based on a generalization of Hilbert’s space-filling curve [25]. The space-filling curve defines an ordered sequence on the cells of the root level that can easily be split in load-balanced portions. As such curves are constructed recursively, they are locality-preserving and therefore avoid an excessive data redistribution overhead in the final loop of Alg. 3. The second goal in designing an efficient parallelization strategy for distributed memory machines, the minimization of the communication costs, is already considered in condition (17) in a natural way. Together with the use of ghost cells this condition allows an almost local execution of Alg. 2. The only parallel operations that have to be incorporated are the parallel ghost cell synchronization and the application of flux correction terms across processor borders. See [5] for implementation details. Analogous to Alg. 2 the regridding procedure of Alg. 3 is hardly affected by the parallelization as long as a repartitioning of the hierarchy is only allowed at root level time steps, which is usually sufficient in practice. New refinements ˘ p can be found local, only a global concatenation of the new topology G ι+1 ˘ ι+1 = 7 G ˘p G p ι+1 is mandatory to ensure the correct proper nesting of the new hierarchy and to create a new load-balanced root level distribution Gp0 with the partitioner. Finally, the data distribution of parts of Qι (t) to other processors must be incorporated.

5 Numerical Results The self-sustaining CJ detonation of Sec. 3.5 is an ideal candidate for fundamental detonation structure simulations, because it produces extremely regular detonation cell patterns [29]. The application of the numerical methods of Sec. 3 within the parallel AMROC framework allowed a two-dimensional cellular structure simulation that is four-times higher resolved (44.8 Pts/lig ) than earlier calculations [24, 8, 11]. Only recently Hu et al. presented a similarly resolved calculation for the same CJ detonation on a uniform mesh [16]. Unfortunately, no technical details are reported for this simulation. In our case, the calculation ran on a small Beowulf-cluster of 7 Pentium III-850 MHz-CPUs connected with a 1 Gb-Myrinet network and required 2150 h CPU-time. On 24 Athlon-1.4 GHz double-processor nodes (2 Gb-Myrinet) of the HEidelberg LInux Cluster System (Helics) our approach allowed a sufficiently resolved computation of the three-dimensional cellular structure of a hydrogen-oxygen detonation. The maximal effective resolution of this calculation is 16.8 Pts/lig

84

R. Deiterding, G. Bader Slip line Shock Contact Head of reaction zone

M

Head of reaction zone

K

Mach stem shock

H L I

F

G

D B

C

A

Primary triple point moving downwards

Transverse wave Head of reaction zone

E

lig

Incident shock

Fig. 6. Flow structure around a triple point before the next collision. Left: isolines of YOH (black) on schlieren plot of u2 (gray).

and the run required 3800 h CPU-time. Our adaptive results are in perfect agreement with the calculations of Tsuboi et al. for the same configuration obtained on a uniform mesh on a super-scalar vector machine [33]. Further on, we present successful simulations of diffracting two-dimensional hydrogenoxygen detonations that reproduce the experimentally measured critical tube diameter of 10 detonation cells. These computations demonstrate the advantages in employing a dynamically adaptive method impressively and used approximately 4600 h CPU-time on the Helics. 5.1 Two-dimensional Cellular Structure We extend the one-dimensional ZND detonation of Fig. 3 to two space dimensions with u2 = 0 and initiate transverse disturbances by placing a small rectangular unreacted pocket with the temperature 2086 K behind the detonation front, cf. [24] or [5]. After an initial period of ≈ 200 µs very regular detonation cells with λ ≈ 3 cm and oscillation period ≈ 32 µs can be observed in computations with a resolution finer than 7 − 8 Pts/lig (see [5] for a mesh refinement study). We exploit this regularity and simulate only a single cell. The calculation is done in a frame of reference attached to the detonation and requires just the computational domain 10 cm × 3 cm. The adaptive run uses a root level grid of 200 × 40 cells and two refinement levels with r1,2 = 4. A physically motivated combination of scaled gradients and heuristically estimated relative errors is applied as adaptation criteria. See [5] for details. Two typical snapshots with the corresponding refinement are displayed in Fig. 10. The adaptive computation uses between 340, 000 and 390, 000 FV cells, while a uniform grids with the same effective resolution would require 2, 048, 000 cells. About 3554 root level time steps (CCFL ≈ 0.95) to tend = 800 µs were calculated.

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The high resolution of the simulation now admits a remarkable refinement of the triple point pattern introduced in Sec. 1. As the two transverse waves form a perfectly regular flow, it suffices to zoom into a single triple point and to analyze the wave pattern between two triple point collisions in detail. Fig. 6 displays the flow situation around the primary triple point A that is mostly preserved during the last 7 µs before a collision. An analysis of the flow field uncovers the existence of two minor triple points B and C along the transverse wave downstream of A. While B can be clearly identified by a characteristic inflection, the triple point C is much weaker and very diffused. B is caused by the interaction of the strong shock wave BD with the transverse wave. The slip line emanating from B to K is clearly present. C seems to be caused by the reaction front (which can be interpreted as a diffused contact discontinuity) and generates the very weak shock wave CI. Downstream of BD a weaker shock wave EF shows up. It is refracted in the point F, when it hits the slip line BK. From F to G this minor shock is parallel and close to the transverse wave, what results in a higher pressure increase in the region FG than in the region EF. Unreacted gas crossing the transverse wave between B and C therefore shows a shorter induction length than gas entering through AB. The minor shock is refracted and weakened by the reaction front at point G and forms the shock GH that is almost parallel to CI. The downstream line of separation between particles passing through incident or Mach Stem shock is the slip line AD. Along its extension DEL the movement of A results in a shear flow between the reaction zones behind the Mach stem and downstream of BD. In the actual picture the contact discontinuity LM seems to originate in this shear flow region, but a complete instationary analysis uncovers that it propagates constantly downstream. The collision of two triple points in the reinitiation of a detonation cell leads to the formation of an unreacted region behind the detonation front that burns in less than a microsecond. The burning generates upstream traveling shock waves that prevent the appearance of the flow field in Fig. 6 at an earlier stage. The strongest of these shocks hits the Mach stem from behind and forms an additional triple point with the contact discontinuity LM. In Fig. 6 shock and intermediate triple point have already vanished in A, but the contact discontinuity LM is still left behind. A detailed hydrodynamic analysis of the intermediate phase from reinitiation to the almost stable situation of Fig. 6 can be found in [5]. 5.2 Three-dimensional Cellular Structure We utilize the regular oscillating solution of the preceding section as initial condition for the three-dimensional simulation and disturb the oscillation in the x2 -direction with an unreacted pocket in the orthogonal direction. We use a computational domain of the size 7 cm × 1.5 cm × 3 cm that exploits the symmetry of the initial data in the x2 -direction, but allows the development of a full detonation cell in the x3 -direction. The AMROC computation uses a

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R. Deiterding, G. Bader 2

3D 2D

1.8 d*/dCJ

1.6 1.4 1.2 1 0.8 670

675

680

685

690 t [µs]

695

700

705

710

Fig. 7. Comparison of the temporal development of the detonation velocity along the line of symmetry through the middle of the detonation cells in the two- and three-dimensional simulation, cf. Figs. 10 and 11.

two-level refinement with r1 = 2 and r2 = 3 on a base grid of 140 × 12 × 24 cells and utilizes between 1.3 M and 1.5 M cells, instead of 8.7 M cells like a uniformly refined grid (3431 root level time steps with CCFL ≈ 0.95 to tend = 800 µs). After a simulation time of ≈ 600 µs a regular cellular oscillation with identical strength in x2 - and x3 -direction can be observed. In both transverse directions the strong two-dimensional oscillations is present and forces the creation of rectangular detonation cells of 3 cm width. The transverse waves form triple point lines in three space-dimensions. During a complete detonation cell the four lines remain mostly parallel to the boundary and hardly disturb each other. The characteristic triple point pattern can therefore be observed in Fig. 11 in all planes perpendicular to a triple point line. Unlike Williams et al. [36] who presented a similar calculation for an overdriven detonation with simplified one-step reaction model, we notice no phase-shift between both transverse directions. In all our computations for the hydrogen-oxygen CJ detonation only this regular three-dimensional mode, called “rectangularmode-in-phase” [14], or a purely two-dimensional mode with triple point lines just in x2 - or x3 -direction did occur. A direct comparison of the temporal development of the detonation velocity in the two- and three-dimensional case in Fig. 7 along lines through the center of the graphics in Figs. 10 and 11 shows that both simulations reproduce the same oscillation period of approximately 32 µs, but the detonation appears to be remarkably higher overdriven during reinitiation in the three-dimensional case. 5.3 Structure of Diffracting Detonations Experiments have shown that the behavior of planar CJ detonations propagating out of tubes into unconfinement is determined mainly by the width of the tube. For square tubes the critical tube width has been found to be of the order of 10-times the cell width, i.e. 10λ [23]. For widths significantly below 10λ the process of shock wave diffraction causes a pressure decrease

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Fig. 8. Density distribution on four reﬁnement levels at tend = 240 µs for w = 10λ.

at the head of the detonation wave below the limit of detonability across the entire tube width. Hydrodynamic shock and reaction front decouple and the detonation decays to a shock-induced flame. This observation is independent of a particular mixture. While the successful transmission of the detonation is hardly disturbed for tubes widths ≫ 10λ, a backward-facing re-ignition wave reinitiates the detonation in the partially decoupled region for widths of ≈ 10λ and creates considerable vortices. We are interested in the decoupling of shock and reaction and also in the reignition phenomenon. Therefore, we simulate the two-dimensional diffraction of the H2 : O2 : Ar CJ detonation for the tube widths w = 8λ and w = 10λ. A periodically reproduced snapshot of the regular oscillating detonation propagating into unreacted gas at rest is used as initial condition. This is a reasonable idealization for the flow situation in real detonation tubes directly before the experimental setup. The symmetry of the problem is exploited by simulating just one half. The adaptive simulations utilize a base grid of 508 × 288 cells and use four levels of refinement with r1,2,3 = 2, r4 = 4. The calculations correspond to a uniform computation with ≈ 150 M cells and have an effective resolution of 25.5 Pts/lig in the x1 -direction (with respect to the initial detonation). Both runs are stopped 240 µs after the detonation has left the tubes (730 root level time steps with CCF L ≈ 0.8), when the flow situations of interest are clearly established. The enormous efficiency of the refinement is visualized in Fig. 8 for the setup with w = 10λ. At tend the calculation shown in Fig. 8 uses ≈ 3.0 M cells on all levels, where ≈ 2.4 M cells are inside one of the 2479 grids of the highest level (ǫtol = 0.8). A comparison of the flow fields in both setups after 240 µs in Fig. 9 clearly shows the extinction of the detonation for w = 8λ and the re-ignition wave for w = 10λ. The reappearance of triple points at the detonation front in the

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Fig. 9. Schlieren plots of ρ for a detonation diﬀracting out of the two diﬀerent tubes. Left: detonation failure for the width w = 8λ, right: reinitiation for w = 10λ.

lower left plot in Fig. 9 is a characteristic indicator for the preservation of the detonation throughout the diffraction. It is interesting to note, that the re-ignition wave by itself is a detonation. The triple point track for w = 10λ (not shown) uncovers that it has developed out of the transverse wave of an initial triple point.

6 Conclusions We have described an efficient solution strategy for the numerical simulation of gaseous detonation waves with detailed chemical reaction. All temporal and spatial scales relevant for the complex process of detonation propagation were successfully resolved. The achieved resolutions are significantly finer than in earlier publications [24, 8, 11] and provide similar insight into the formation and propagation of transient detonation structures like recent large-scale simulations on uniform meshes [16, 33]. Beside the application of the time-operator splitting technique and the construction of a robust high-resolution shock capturing scheme, the key to the high efficiency of the presented simulations is the generic implementation of the blockstructured AMR method after Berger and Collela [2] in the AMROC framework [6]. AMROC provides the required high local resolution dynamically and follows a parallelization strategy that is tailored especially for the emerging generation of distributed memory architectures. All presented results have been achieved on Linux-Beowulf-clusters of moderate size in a few days real time, what demonstrates that advances in computational fluid dynamics do not necessarily require large super-computers, but integrated approaches, which combine fast and accurate discretizations with sophisticated techniques from computer science. Acknowledgement. This work was supported by the DFG high priority research program “Analysis and Numerics of Conservation Laws”, grant Ba 840/3-3.

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Fig. 10. Temperature and schlieren plots of the density on reﬁnement regions in the ﬁrst (left) and second half (right) of a detonation cell. (See also color ﬁgure, Plate 9.)

Fig. 11. Schlieren plots of ρ (upper row) and YOH (lower row) in the ﬁrst (left) and second (right) half of detonation cell, mirrored at x2 = 0 cm, 5.0 cm < x1 < 7.0 cm. The plots of YOH are overlaid by an isosurface of ρ that visualizes lig . (See also color ﬁgure, Plate 10.)

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References 1. J. Bell, M. Berger, J. Saltzman, and M. Welcome. Three-dimensional adaptive mesh reﬁnement for hyperbolic conservation laws. SIAM J. Sci. Comp., 15(1):127–138, 1994. 2. M. Berger and P. Colella. Local adaptive mesh reﬁnement for shock hydrodynamics. J. Comput. Phys., 82:64–84, 1988. 3. M. Berger and J. Oliger. Adaptive mesh reﬁnement for hyperbolic partial differential equations. J. Comput. Phys., 53:484–512, 1984. 4. R. Courant and K. O. Friedrichs. Supersonic flow and shock waves. Applied mathematical sciences, volume 21. Springer, New York, Berlin, 1976. 5. R. Deiterding. Parallel adaptive simulation of multi-dimensional detonation structures. PhD thesis, Brandenburgische Technische Universit¨at Cottbus, Sep 2003. 6. R. Deiterding. AMROC - Blockstructured Adaptive Mesh Reﬁnement in Objectoriented C++. Available at http://amroc.sourceforge.net, Mar 2004. 7. Y. N. Denisov and Y. K. Troshin. Structura gazovoi detonatsii v trubakh (Structure of gaseous detonations in tubes). Zh. Eksp. Teor. Fiz., 30(4):450–459, 1960. 8. C. A. Eckett. Numerical and analytical studies of the dynamics of gaseous detonations. PhD thesis, California Institute of Technology, Pasadena, California, Sep 2001. 9. B. Einfeldt, C. D. Munz, P. L. Roe, and B. Sj¨ogreen. On Godunov-type methods near low densities. J. Comput. Phys., 92:273–295, 1991. 10. W. Fickett and W. C. Davis. Detonation. University of California Press, Berkeley and Los Angeles, California, 1979. 11. T. Geßner. Dynamic mesh adaption for supersonic combustion waves modeled with detailed reaction mechanisms. PhD thesis, Math. Fakult¨ at, University Freiburg, 2001. 12. E. Godlewski and P.-A. Raviart. Numerical approximation of hyperbolic systems of conservation laws. Springer Verlag, New York, 1996. 13. B. Grossmann and P. Cinella. Flux-split algorithms for ﬂows with nonequilibrium chemistry and vibrational relaxation. J. Comput. Phys., 88:131–168, 1990. 14. M. Hanana, M. H. Lefebvre, and P. J. Van Tiggelen. Pressure proﬁles in detonation cells with rectangular or diagonal structure. In Proc. of 17th Int. Colloquium on the Dynamics of Explosive and Reactive Systems, Heidelberg, Jul 1999. 15. A. Harten. High resolution schemes for hyperbolic conservation laws. J. Comput. Phys., 49:357–393, 1983. 16. X. Y. Hu, B. C. Khoo, D. L. Zhang, and Z. L. Jiang. The cellular structure of a two-dimensional H2 /O2 /Ar detonation wave. Combustion Theory and Modelling, 8:339–359, 2004. 17. N. N. Janenko. Die Zwischenschrittmethode zur L¨ osung mehrdimensionaler Probleme der mathematischen Physik. Springer-Verlag, Berlin, 1969. 18. P. Kaps and P. Rentrop. Generalized Runge-Kutta methods of order four with stepsize control for stiﬀ ordinary diﬀerential equations. Num. Math., 33:55–68, 1979. 19. R. J. Kee, F. M. Rupley, and J. A. Miller. Chemkin-II: A Fortran chemical kinetics package for the analysis of gas-phase chemical kinetics. SAND89-8009, Sandia National Laboratories, Livermore, California, Sep 1989.

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20. R. J. Kee, F. M. Rupley, and J. A. Miller. The Chemkin thermodynamic data base. SAND87-8215B, Sandia National Laboratories, Livermore, California, Mar 1990. 21. B. Larrouturou. How to preserve the mass fractions positivity when computing compressible multi-component ﬂows. J. Comput. Phys., 95:59–84, 1991. 22. B. Larrouturou and L. Fezoui. On the equations of multi-component perfect or real gas inviscid ﬂow. In Carasso et al., editor, Proc. of Second Int. Conf. on Nonlinear Hyperbolic Equations - Theory, Numerical Methods, and Applications, Aachen 1988, Lecture Notes in Mathematics 1402, pages 69–98. Springer-Verlag Berlin, 1989. 23. J. H. S. Lee. Dynamic parameters of gaseous detonations. Ann. Rev. Fluid Mech., 16:311–336, 1984. 24. E. S. Oran, J. W. Weber, E. I. Stefaniw, M. H. Lefebvre, and J. D. Anderson. A numerical study of a two-dimensional H2 -O2 -Ar detonation using a detailed chemical reaction model. J. Combust. Flame, 113:147–163, 1998. 25. M. Parashar and J. C. Browne. On partitioning dynamic adaptive grid hierarchies. In Proc. of the 29th Annual Hawaii Int. Conf. on System Sciences, Jan 1996. 26. J. J. Quirk. Godunov-type schemes applied to detonation ﬂows. In J. Buckmaster, editor, Combustion in high-speed flows: Proc. Workshop on Combustion, Oct 12-14, 1992, Hampton, pages 575–596, Dordrecht, 1994. Kluwer Acad. Publ. 27. R. Sanders, E. Morano, and M.-C. Druguett. Multidimensional dissipation for upwind schemes: Stability and applications to gas dynamics. J. Comput. Phys., 145:511–537, 1998. 28. J. Smoller. Shock waves and reaction-diffusion equations. Springer-Verlag, NewYork, 1982. 29. R. A. Strehlow. Gas phase detonations: Recent developments. J. Combust. Flame, 12(2):81–101, 1968. 30. R. A. Strehlow and F. D. Fernandez. Transverse waves in detonations. J. Combust. Flame, 9:109–119, 1965. 31. D. R. Stull and H. Prophet. JANAF thermodynamical tables. Technical report, U. S. Departement of Commerce, 1971. 32. E. F. Toro. Riemann solvers and numerical methods for fluid dynamics. Springer-Verlag, Berlin, Heidelberg, 2nd edition, 1999. 33. N. Tsuboi, S. Katoh, and A. K. Hayashi. Three-dimensional numerical simulation for hydrogen/air detonation: Rectangular and diagonal structures. In Proc. of the Combustion Institute, volume 29, pages 2783–2788, 2003. 34. B. van Leer. Towards the ultimate conservative diﬀerence scheme V. A second order sequel to Godunov’s method. J. Comput. Phys., 32:101–136, 1979. 35. C. K. Westbrook. Chemical kinetics of hydrocarbon oxidation in gaseous detonations. J. Combust. Flame, 46:191–210, 1982. 36. D. N. Williams, L. Bauwens, and E. S. Oran. Detailed structure and propagation of three-dimensional detonations. In Proc. of the Combustion Institute, pages 2991–2998. 26, 1997. 37. F. A. Williams. Combustion theory. Addison-Wesley, Reading, Massachusetts, 1985.

Numerical Linear Stability Analysis for Compressible Fluids Andreas S. Bormann Institut f¨ ur Verfahrenstechnik, Technische Universit¨ at Berlin, Germany [email protected]

Summary. The Rayleigh-B´enard problem and the Taylor-Couette problem are two well-known stability problems that are traditionally treated with linear stability analysis. In the vast majority of these stability calculations the ﬂuid is considered to be incompressible [Cha61, DR81]. Only with this assumption and simpliﬁcation is possible to conduct a linear stability analysis analytically. In order to calculate the stability limits of a compressible ﬂuid by use of a linear stability analysis therefore in this work a numerical linear stability analysis is presented. The numerical stability analysis is based upon the equations of balance for mass, momentum and energy that are completed with the constitutive equations by Navier-Stokes and Fourier. The algorithm allows to calculate the regions of stability for arbitrary one-dimensional and stationary basic states. This numerical stability analysis is used to calculate the stability region for the Rayleigh-B´enard problem. The main result is that the critical Rayleigh number does not have a constant value, as calculations involving the Boussinesq approximation suggest misleadingly, but that the value of the critical Rayleigh number depends strongly on the thickness of the ﬂuid layer. Furthermore, an empirically found relationship between the critical Rayleigh number and the thickness of the ﬂuid layer is presented (14). Its eﬃciency is successfully veriﬁed with the results of the numerical linear stability analysis. The results for the critical Rayleigh number show clearly that the compressibility of a ﬂuid must not be neglected in the stability analysis of the Rayleigh-B´enard problem. Secondly, the more complicated Taylor-Couette problem is treated with the numerical linear stability analysis. In contrast to the traditional stability analysis by Taylor [Tay23], the ﬂuid is considered to be compressible and includes the temperature as a ﬁeld variable. The eﬀectiveness of the numerical linear stability analysis is manifested by the good agreement of the comparison with experimental results. In addition to that, temperature eﬀects are studied and are compared with experiments.

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1 Numerical linear stability analysis The approach to the numerical stability analysis is documented in detail in [Bor03] and therefore will be presented here in a shortened form. The approach consists of five steps: • Calculation of the one-dimensional and stationary basic state • Linearization of the compressible and viscous Navier-Stokes and Fourier balance equation system with respect to the basic state • Discretization of the linearized system • Establishment of a characteristic eigenvalue problem • Calculation of the eigenvalues and determination of the stability properties First, the one-dimensional and stationary basic state has to be calculated. For this purpose, the constitutive equations of Navier-Stokes and Fourier are inserted into the balance equations for mass, momentum and energy which results typically in a system similar to

∂ψA (xi , t) ∂ 2 ψA (xi , t) ∂ψA (xi , t) = f ψA (xi , t), , , ... (1) ∂t ∂xi ∂xi ∂xj with

ψA (xi , t) = {ρ(xi , t), v1 (xi , t), v2 (xi , t), v3 (xi , t), T (xi , t)} .

(2)

Usually, the boundary conditions are combined up in a non-dimensional number α, e.g. the Rayleigh number Ra or the Taylor number T a, which may be treated as a parameter for the basic state. The calculation of the basic state gives ψ A = ψ A (x1 , α). (3) Linearization of (1) with respect to (3) is done in the typical manner by the use of the ansatz (4) ψA (xi , t) = ψ A (x1 , α) + ψ9A (xi , t). The term taking account for the disturbance is defined by ψ9A (xi , t) = ψ:A (x1 ) exp {i(k2 x2 + k3 x3 ) + σt} ,

(5)

where k2 and k3 are positive real wave numbers and σ is a complex constant that governs the temporal behavior of the disturbance. The resulting system of differential equations reads dψ:B (x1 ) d2 ψ:B (x1 ) , + DDAB σ ψ:A (x1 ) = AAB ψ:B (x1 ) + DAB dx1 dx21

with

AAB = f1 (ψ A (x1 , α),

dψ A (x1 , α) d2 ψ A (x1 , α) , , α, k2 , k3 ), dx1 dx21

DAB = f2 (ψ A (x1 , α),

dψ A (x1 , α) d2 ψ A (x1 , α) , , α, k2 , k3 ), dx1 dx21

DDAB = f3 (ψ A (x1 , α),

dψ A (x1 , α) d2 ψ A (x1 , α) , α, k2 , k3 ). , dx1 dx21

(6)

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This set of equations is often too complicated to be solved analytically. Mainly, this is due to the coefficients A, D and DD that depend on x1 . One possibility to deal with that problem is to discretize (6) within the domain x1 ǫ{0, d} over ASt points which will lead to an algebraic system. The discrete values of the fields are defined by ρ:(i) = ρ:(x1 )|x1 =(i−1)h ,

v:1

(i)

= v:1 (x1 )|x1 =(i−1)h ,

i = {1, .., ASt},

= v:3 (x1 )|x1 =(i−1)h , # # = T:(x1 )# ,

i = {1, .., ASt},

v:2 (i) = v:2 (x1 )|x1 =(i−1)h , v:3

(i)

T:(i)

i = {1, .., ASt},

x1 =(i−1)h

with h =

i = {1, .., ASt}, i = {1, .., ASt},

d . ASt − 1

With these definitions and the use of difference quotients for the derivations in (6) one obtains the eigenvalue problem (MY Z − σδY Z )φ:Z = 0, with

Y, Z = {1, .., 5ASt},

(7)

2

MY Z = f (ψ A , dψ A , d ψ A , α, k2 , k3 ).

where the vector φ:Z is given by 5 (1) (1) (1) φ:Z = ρ:(1) , v:1 , v:2 , v:3 , T:(1) , . . . ,

(8)

6 (ASt) (ASt) (ASt) :(ASt) , ρ:(ASt) , v:1 , v:2 , v:3 , T

with Z = {1, .., 5 ∗ ASt).

The matrix MY Z in (7) may be given conveniently by the following relations # # DAB ## DDAB ## M5(i−1)+A, 5(i−2)+B = + , 2h #x1 =(i−1)h h2 #x1 =(i−1)h # DDAB ## , M5(i−1)+A, 5(i−1)+B = AAB |x1 =(i−1)h − 2 h2 #x1 =(i−1)h (9) # # DDAB ## DAB ## + , M5(i−1)+A, 5i+B = − 2h # h2 # x1 =(i−1)h

with

x1 =(i−1)h

A, B = {1, .., 5} and i = {2, .., ASt − 1} .

The inclusion of the boundary conditions for the discretization points 1 and ASt is not trivial and therefore must be elaborated for a given problem individually. Details for the Rayleigh-B´enard problem and the Taylor-Couette problem are explained in [Bor01, Bor03].

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The eigenvalues σ of (7) must now be calculated in order to describe the stability behavior of the chosen basic state in (3). Of particular interest are the real parts of the eigenvalues. If one of the real parts of the calculated eigenvalues is greater than zero then instability is to be expected because a disturbance would grow in time (see ansatz (5)). Stability, on the other hand, can only be achieved if for all combinations of k2 and k3 the real parts of the eigenvalues are less than zero. In order to find the critical non-dimensional number αcrit for a given stability problem one has to calculate the eigenvalue problem (7) for a fixed combination of k2 and k3 in an iterative way where α has to be varied until αkrit (k) ⇔ max(Re(σ = f (α, k2 , k3 ))) = 0.

(10)

Then, this calculation has to be repeated for other combinations of k2 and k3 in order to find the minimum of αkrit (k) αkrit = min(αkrit (k))

∀ k.

(11)

αkrit is the typical critical parameter of a stability problem. It describes the point of neutral stability; if the basic state has an α that is less than αkrit then stability of the basic state is maintained. In the other case where α is greater than αkrit stability of the corresponding basic state is lost. It may be easily seen that the main problem of this numerical linear stability analysis is the calculation of the eigenvalues σ. Especially when the number of discretization points ASt is high, there is a need for a reliable and fast eigenvalue solver. The ARPACK routine package allows a very fast and efficient calculation of some eigenvalues of large scale eigenvalue problems [LSY98]. It can therefore be used successfully for the calculations of the results in the next two sections.

2 The compressible Rayleigh-B´ enard problem 2.1 Introduction and motivation The Rayleigh-B´enard problem concerns a fluid layer that is bounded by two plates and the lower plate is heated. If the temperature gradient in the fluid at rest exceeds a critical value, convection will occur and stability is lost. For the determination of the state of stability a non-dimensional parameter was introduced [Ray16], that is called the Rayleigh number Ra =

gα|∆T |d3 . κc ν

(12)

Here g denotes the gravitational acceleration, d the thickness of the fluid layer, |∆T | the temperature difference between top and bottom of the layer. α, κc ,

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and ν are the coefficients of volume expansion, thermometric conductivity and kinematic viscosity. One way to determine the critical value of the Rayleigh number is to use a linear stability analysis [Cha61], where, in order to simplify the equations for the linear stability analysis, usually the Boussinesq approximation is applied. According to this approximation the density of a fluid is assumed to depend on the temperature only, while the dependence on the pressure, i.e. the compressibility κT , is neglected. The result of such a linear stability analysis is a constant value for the critical Rayleigh number and critical wave length kcrit that depends only on the boundary conditions. Table 1. Critical Rayleigh number and critical wave length kcrit . rigid-rigid boundaries:

I RaBouss = 1707.8 crit

I kcrit = 3.177

rigid-free boundaries:

II RaBouss = 1100.7 crit

II kcrit = 2.682

free-free boundaries:

III RaBouss = 657.5 crit

III kcrit = 2.221

Unfortunately, from a thermodynamic point of view the Boussinesq approximation contradicts the second law of thermodynamics because thermodynamic stability requires the inequality [Mue85] α2 ≦

ρcp κT . T

(13)

If κT is neglected as it is in the Boussinesq approximation, then also the volume expansion coefficient α must vanish. Another hint that the results of Table 1 should not be used for compressible fluids is given by recent experiments of Kogan& Meyer [KM01]. They confirm experimentally that the value of the critical Rayleigh number is dependent on the layer thickness. Furthermore a quite simple relation for the calculation of the value of the critical Rayleigh number of a compressible fluid may be found by adding the critical Rayleigh number for an incompressible and viscous fluid to the critical Rayleigh number of a compressible and inviscid fluid according to the Schwarzschild criteria ⎧ ⎫ ⎨ 1707.8 ⎬ α2 ρ2 0 T0 g 2 d4 . (14) Racrit = 1100.7 + ⎩ ⎭ κη 657.5 Although this semi-empirical relation seems to be too simple at first sight, it coincides very well with the experimental results of [KM01] and the results of the numerical linear stability analysis that are presented in the next section. The derivation of (14) is explained in detail in [Bor03]

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Fig. 1. Comparison of critical Rayleigh number of the numerical linear stability analysis (symbols) with the equation (14) (drawn through lines). The dashed lines indicate the results of the critical Rayleigh number involving the Boussinesq approximation.

2.2 Results A stability analysis for compressible fluids was performed as outlined in Section 1 for both, liquids and gases. A detailed description of the calculations is found in [Bor01, Bor03]. The selected liquids were water, ether (DEE), and mercury. As a gas argon was chosen. All calculations refer to a temperature T1 = 298 K and a pressure of p1 = 1 bar for the liquids, pressures of 10 bar down to 0.001 bar for argon. In Figure 1 the results of the stability analysis for the selected liquids and the gas argon are presented for the different boundary conditions. The abscissa represents the layer thickness d in meters while the ordinate gives the critical Rayleigh number. The marked points represent the results of the numerical linear stability analysis while the dashed lines indicate the constant value for the critical Rayleigh number for a given boundary problem that is obtained by use of the Boussinesq approximation. The drawn out lines are calculated from equation (14) For small layer thicknesses the critical Rayleigh number approximates asymptotically the constant value of the Rayleigh number of the Boussinesq case, while the critical Rayleigh number for large layer thicknesses is strongly dependent on the layer thickness.

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This behavior may be understood from the fact that a large fluid layer exerts a non negligible pressure on the fluid layer at the bottom. The fluid becomes compressed and density is larger than in the Boussinesq case where a change of density due to pressure is neglected. Now, in order to achieve a critical density gradient for instability, the temperature gradient - and with it the thermal expansion - has to be increased beyond the one of an incompressible fluid. This implies that the Rayleigh number must increase with an increasing layer thickness. The critical wave numbers for the different boundary cases resulted in the same values as already calculated by use of the Boussinesq approximation in Table 1. This observation is confirmed by [JH99] The results of the linear stability analysis for compressible fluids show clearly that the stability behavior of a the Rayleigh-B´enard problem is characterized by two parameters instead of only the constant Rayleigh number for the Boussinesq case. For a linear stability analysis for compressible fluids the value of the critical Rayleigh number becomes dependent on the thickness d of the fluid layer. The simple ansatz (14) agrees perfectly with the results of the numerical linear stability analysis and should therefore be used instead of the Rayleigh number obtained by use of the Boussinesq approximation. One important remark concerning the value of the isothermal compressibility: the value of the compressibility κT does not affect at all the deviation of the critical Rayleigh number for a compressible fluid from the critical Rayleigh number calculated with the Boussinesq approximation at all. This can be seen clearly in equation (14). Even for a vanishing small value of the compressibility κT the value of the critical Rayleigh number will depend strongly on the layer thickness. This means that the often read justification for the use of the Boussinesq approximation for water - its low compressibility - is not valid.

3 The compressible Taylor-Couette problem 3.1 Introduction and motivation The geometry of the Taylor-Couette problem consists of two coaxial, freely rotating cylinders. Up to a critical angular velocity of the cylinders the fluid flow between these cylinders is a one-dimensional shear flow. If the angular velocities exceed the critical limit, the stability of the shear flow is lost and another stationary flow can be observed - the Taylor vortices. The task of a stability analysis for the Taylor-Couette problem is to find the critical angular velocity for the inner cylinder Ω1,crit for a given angular velocity of the outer cylinder Ω2 . Another important question is the determination of the critical wave number with which the height of the Taylor vortices may be calculated.

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Taylor calculated the critical Ω1 − Ω2 relations for a incompressible and isothermal fluid and compared them successfully with his own experimental results [Tay23]. Because of the very good agreement of analytical and experimental results his work is often taken as a paradigm for analytical fluid dynamics. For his calculations he had to restrict himself to the ”narrow-gap” geometry. In that case the distance between the two cylinders has to be much smaller than the radii of the cylinders. Chandrasekhar overcame this restriction for the case that the outer cylinder has twice the radius of the inner cylinder. For this ”wide-gap” geometry he calculated the critical Ω1 − Ω2 relations for an incompressible and isothermal fluid. Donnelly and Fultz experimentally confirmed the calculations by Chandrasekhar [Cha58, DF60]. In contrast to the Rayleigh-B´enard problem, where the critical nondimensional wave is a constant, numbers are strongly dependent on the angular velocities. Another difficulty arises from the fact that the basic state of the compressible and temperature dependent Taylor-Couette problem cannot not be calculated analytically but only numerically. These facts complicate the linear stability analysis and therefore the Taylor-Couette problem may be used to prove the efficiency of the numerical linear stability analysis. Furthermore the numerical linear stability analysis is capable of calculating the critical Ω1 − Ω2 relations not only for a narrow-gap and wide-gap geometry of the two cylinders but for any arbitrary geometry. Another important advantage of the numerical linear stability analysis is the consideration of the temperature field. Effects due to viscous heating or applied temperature gradients may be investigated more conveniently. 3.2 Results A numerical linear stability is performed as outlined in Section 1. For details about the exact algorithm and entries in the matrix of the characteristic eigenvalue problem see [Bor03]. In order to compare the results with experiments, for the narrow-gap geometry the experimental data of Taylor was used [Tay23]. He performed experiments with water at room temperature for three different sets of geometry of the cylinders: the radius of the outer cylinder R2 was fixed to 4.035 cm while the radius of the inner cylinder R1 had the values 3 cm, 3.55 cm and 3.8 cm. Taylor carried out his experiments in the following manner: First the angular velocity of the outer cylinder Ω2 was adjusted. Then he increased the angular velocity of the inner cylinder Ω1 very slowly until the basic state of the shear flow lost its stability and the Taylor vortices were present. The transition of the one dimensional shear flow into the vortex flow marks the critical Ω1 − Ω2 relation. In Figure 2 the results of the numerical linear stability analysis, Taylor’s experiments and Taylor’s calculations are compared for the three different geometries of the cylinders.

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Fig. 2. Comparison of the results of the critical Ω1 −Ω2 -relations for the narrow-gap geometry of the numerical linear stability analysis (nLSA) with Taylor’s experimental and analytical results.

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It can be seen clearly that there is a very good agreement of the results for positive Ω1 − Ω2 relations for all three sets of experiments. Especially for faster angular velocities, the numerical linear stability delivers good results, better than those of Taylor’s own calculations. For negative Ω2 the results of the critical inner angular velocity Ω1 of the numerical linear stability analysis also show a good agreement but deviate with higher angular velocities from the experimental results. On the one side, this might be due to the high angular velocities, where the measurement of the experiments could only be accomplished with some difficulties. On the other side, the ansatz (5) only allows for axis-symmetric secondary flows while Taylor observed some slightly non-axis-symmetric flows at high angular velocities. After having proved the effectiveness of the numerical linear stability analysis for the narrow-gap geometry, that analysis was applied to the widegap geometry as well. While Chandrasekhar had to develop a linear stability analysis for this geometry from scratch [Cha58], in the numerical linear stability analysis only the radii of the cylinders had to be changed. Donnelly&Fultz [DF60] performed experiments with water and viscous oils for the wide-gap geometry. The inner cylinder was fixed to a radius of R1 = 3.1432 cm while the outer cylinder had the radius R2 = 6.2864 cm. The results of the numerical linear stability analysis, the experiments by Donnelly and Fultz and Chandrasekhar’s calculations are plotted in Figure 3. Once again there is a very good agreements of the results of the numerical linear stability analysis with the experimental results. Because of the low angular velocities a deviation from the experimental results was not expected and is not observed. The critical wave numbers have been calculated as well for both sets of geometries and they show a good agreements with the experiments for both cases. Details are to be found in [Bor03]. The influence of temperature on the critical Ω1 − Ω2 relation has been studied not as thoroughly as the isothermal basic state. Mainly, two scenarios were investigated. First, the influence of an applied temperature gradient was investigated by Snyder&Karlsson [SK64] and Sorour&Coney [SC79]. Snyder&Karlsson found that for the narrow-gap geometry an applied temperature gradient increases the stability region slightly, i.e. for a given Ω2 a slightly higher critical inner angular velocity Ω1 could be measured. This behavior could be verified qualitatively with the numerical linear stability analysis. Unfortunately the data base of the experiments did not allow for an exact quantitative comparison. For faster angular velocities both, Snyder&Karlsson and Sorour&Coney observed non-axi-symmetric secondary flows. The critical Ω1 − Ω2 relation for these flows could not be calculated with the numerical linear stability analysis because of its strictly axi-symmetric model for disturbances (5). Secondly, the influence of viscous heating is of some importance for the critical Ω1 − Ω2 relation in the Taylor-Couette problem. There has not been much research on this topic until recently. Al-Mubaiyedh et al. [ASK02] came

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Fig. 3. Comparison of the critical Ω1 − Ω2 -relation of the numerical linear stability analysis (nLSA) for the wide-gap geometry with the experiments by Donnely&Fultz and the calculations by Chandrasekhar.

to the theoretical result, that for fixed outer angular velocity Ω2 viscous heating should lead to a critical Ω1 that is less than the critical Ω1 for an isothermal flow. Also, according to them, the decrease of the critical inner angular velocity runs monotonically with the increase of the temperature in the fluid due to viscous heating. White&Muller [WM02] recently performed experiments on this topic with water, mixtures of water and glycerin and some oils. They could confirm the predictions made Al-Mubaiyedh et al. although not quantitatively. Unfortunately, also the experimental data of White&Muller do not contain the necessary information that would allow to perform a quantitative comparison of the numerical linear stability analysis with their experimental results. At least, qualitatively the numerical linear stability analysis shows the same behavior as reported by White&Muller and Al-Mubaiyedh et al.: With increasing viscous heating the critical inner angular velocity drops monotonically.

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4 Conclusion A numerical stability analysis is presented and proven to be a reliable and effective method for the calculation of stability of one-dimensional, stationary, compressible and viscous basic states. For the investigated stability problems - the Rayleigh-B´enard problem and the Taylor-Couette problem - the results are in good agreement with experimental results. For the Taylor-Couette problem the advantage of the numerical linear stability analysis over the traditional incompressible and isothermal calculation is to be found in the arbitrary choice of cylinder geometries and the consideration of the temperature field, which allows for a much broader range of applications. The results for the Rayleigh-B´enard problem show clearly that the often used Boussinesq approximation is not suitable for that problem. It could be proven that the value of the critical Rayleigh number is not a constant but a function that is strongly dependent on the thickness of the fluid layer. Furthermore a semi-empirical relation for the critical Rayleigh number for compressible fluids is presented (14), which should be used instead of the well know constant values of the critical Rayleigh number for the determination of the stability limit for that problem.

References [ASK02] Al-Mubaiyedh, U.A., Sureshkumar, R., Khomami, B.: The eﬀect of viscous heating on the stability of the Taylor-Couette ﬂow. J. Fluid Mech., 462, 111–132 (2002) [Bor01] Bormann, A.S.: The onset of convection in the Rayleigh-B´enard problem for compressible ﬂuids. Cont. Mech. Therm., 13, 9–23 (2001) [Bor03] Bormann, A.S.: Lineare Stabilit¨ atsanalyse kompressibler Fluide. Diss., Technische Universit¨at Berlin, http://edocs.tuberlin.de/diss/2003/bormann andreas.pdf (2003) [Cha58] Chandrasekhar, S.: The stability of viscous ﬂow between rotating cylinders. Proc. Roy. Soc. A, 246, 301–311 (1958) [Cha61] Chandrasekhar, S.: Hydrodynamic and hydromagnetic stability. At the Clarendon Press, Oxford (1961) [DF60] Donnely, R., Fultz, D.: Experiments on the stability of viscous ﬂow between rotating cylinders. II. Visual observations. Proc. Roy. Soc. A., 258, 101–123 (1960) [DR81] Drazin, P.G., Reid, W.H.: Hydrodynamic stability. Cambridge Universtity Press, Cambridge (1981) [JH99] Jeng, J., Hassard, B.: The critical wave number for the planar B´enard problem is unique. Int. J. Non Linear Mech., 34, 221–229 (1999) [KM01] Kogan, A.B., Meyer, H.: Heat transfer and convection onset in a compressible ﬂuid: 3 He near the critical point. Phys. rev. E, 63, 4635–4638 (2001)

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[LSY98] Leboucq, R.B., Sorensen, D.C., Yang, C.: ARPACK Users guide: Solution of large scale eigenvalue problems with implicitly restarted Arnoldi methods. SIAM, Philadelphia(PA) (1998) [Mue85] M¨ uller, I: Thermodynamics. Pitman Advanced Publishing Program, Boston (1985) [Ray16] Strutt, J.W. (Lord Rayleigh): On the convection currents in a horizontal layer of ﬂuid when the high temperature is on the under side. Phil. Mag., 2, 833–844 (1916) [SC79] Sorour, M.M., Coney, J.E.R.: The eﬀect of temperature gradient on the stability of ﬂow between vertical, concentric, rotating cylinders. J. Mech. Eng., 21, 403–409 (1979) [SK64] Snyder, H.A., Karlsson, S.K.F.: Experiments on the stability of Couette motion with a radial themal gradient. Phys. Fluids, 7, 1669–1706 (1964) [Tay23] Taylor, G.I.: Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. Roy. Soc. A, 223, 289–343 (1923) [WM02] White, J.M., Muller, S.J.: Experimentals studies on the stability of newtonian Taylor-Couette ﬂow in the presence of viscous heating. J. Fluid Mech., 462, 133–159 (2002)

Simulation of Solar Radiative Magneto-Convection M. Sch¨ ussler1 , J.H.M.J. Bruls2 , A. V¨ ogler1 , and P. Vollm¨ oller1,3 1

2

3

Max-Planck-Institut f¨ ur Sonnensystemforschung Max-Planck-Str. 2, 37191 Katlenburg-Lindau, Germany [email protected] Kiepenheuer-Institut f¨ ur Sonnenphysik Sch¨ oneckstr. 6, 79104 Freiburg, Germany [email protected] Present address: VAW, ETH-Zentrum, 8092 Z¨ urich, Switzerland

1 Introduction The term ‘magneto-convection’ summarizes the variety of processes arising from the dynamic interaction between convective motions and magnetic fields in an electrically conducting medium. Magneto-convective processes play an important role in many astrophysical systems; their effects can be best studied in the case of the Sun, where the relevant spatial and temporal scales of the phenomena can (in principle, at least) be observed. The generation of magnetic flux in the Sun by a self-excited dynamo process and the various spectacular phenomena of solar activity, like sunspots, coronal loops, flares, and mass ejections all are, directly or indirectly, driven by magneto-convective interactions. The large length scales of the typical convective flow structures on the Sun lead to high (hydrodynamic and magnetic) Reynolds numbers, so that the magneto-convective processes typically involve nonlinear interactions and formation of structures and patterns. Fig. 1 illustrates typical regimes of magneto-convection near the visible solar surface, differing mainly in the amount of magnetic flux per unit area (i.e., average magnetic field strength) and the orientation of the field. In the ‘quiet’ Sun, some magnetic flux is concentrated in bright magnetic elements, isolated patches with field strengths exceeding 1000 G (corresponding to 0.1 Tesla). In magnetically active regions, such magnetic elements densely populate the dark convective downflow network and decrease the size of convective upflows (‘granules’ in solar physics lingo). In the dark core of a sunspot (the so-called umbra), the strong vertical magnetic field is space-filling and largely suppresses the convective energy transport. The less dark, striated periphery of a sunspot (the penumbra) har-

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bors a magnetic field which is strongly inclined with respect to the vertical direction.

Umbra

Plage

Penumbra

Quiet Sun Fig. 1. Magnetic structure on the visible solar surface and diﬀerent regimes of magneto-convection. There is only a small amount of magnetic ﬂux in quiet regions and the convective pattern (hot plasma rising in bright ‘granules’, cooled by radiation and ﬂowing back into the interior in the network of dark intergranular lanes) is best visible. In areas with more magnetic ﬂux (plage regions), the ﬂux becomes assembled in small ﬂux concentrations, which appear bright because of locally enhanced transparency of the atmosphere. At even higher levels of magnetic ﬂux density, the convection is largely suppressed by the magnetic ﬁeld and sunspots form. They have a dark core (the umbra) with almost vertical magnetic ﬁeld and a surrounding region of inclined ﬁeld (the penumbra), whose striated appearance and mode of energy transport are not well understood (Image taken with the German Vacuum Tower Telescope on Tenerife, Spain, operated by the Kiepenheuer-Institut, Freiburg; courtesy O. von der L¨ uhe)

Realistic numerical simulations of solar magneto-convection represent a considerable computational challenge. There is an extended range of length scales between the dominant scale of the convective flow pattern (the granulation) of about 103 km and the dissipation scales of the order of a few km and less. The plasma is strongly stratified with pressure scale heights down

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to about 100 km and even a restricted simulation has to cover a density ratio of the order of 100. Convective velocities reach the sound speed, so that full compressibility is mandatory. Solar convection is strongly affected by partial ionization effects and over some height range near the surface the major part of the convective energy flux is transported in the form of latent heat. Consequently, the ionization state of the most abundant species (foremost hydrogen) has to be monitored in the course of the simulation and the related energetics have to be incorporated into the equation of state, which then cannot be written as a closed expression but has to be specified in the form of a numerical table. The energetics of the lower solar atmosphere is strongly affected by radiative energy transport. In this region, radiative heating or cooling of the plasma cannot be considered as a local (diffusive) process since the mean free path of photons is comparable to or larger than the dominant spatial scale of the flow patterns and the pressure scale height. Consequently, the radiative transfer equation for the specific intensity of radiation has to be integrated along a large number of rays of various angles in order to determine the radiation incident on each grid cell. In order to correctly represent the temperature field in the solar atmosphere, the frequency dependence of the radiation has to be taken into account. This further complicates the problem because about a million spectral lines contribute to the energy balance in the solar photosphere. Another complication is related to the boundary conditions. Because of the strong stratification and large size of the solar convection zone, the computational box for any realistic simulation of solar convection can only cover a tiny fraction of the whole convection zone and solar surface. Therefore, we have numerical boundaries where physically no boundaries are. For the side boundaries one can assume periodic conditions if the box is much larger than the dominant scale of the flow, but for the top and bottom boundaries the situation is less clear in a gravitationally stratified medium. Particularly at the bottom the assumption of a closed boundary would be quite unrealistic: in compressible convection, the downflows are narrow, fast and coherent over many scale heights. It is thus necessary to develop appropriate ‘open’ boundary conditions, which permit the free in- and outflow of matter while maintaining the total mass in the computational box and allowing for the correct amount of convective energy transport. In order to meet the computational and methodological challenges, our ANumE project had three major goals: 1. Develop and evaluate methods to treat frequency-dependent radiative energy transport in simulations of magnetohydrodynamic processes, 2. develop combined compressible MHD/radiative transfer codes with partial ionization and open boundaries, and

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3. perform realistic simulations and analyze them with spectral and polarization diagnostics in order to compare with observations and measurements of the Sun. The whole project is carried out in close cooperation with our colleagues A. Dedner, M. Wesenberg, Christian Rohde, and D. Kr¨ oner from the Institute for Applied Mathematics (IAM) of the University of Freiburg. It proved to be fruitful to develop two codes more or less in parallel. One development (mainly by the IAM group) concerned a code based upon MHD Riemann solvers for non-ideal gases, an unstructured grid with adaptive grid refinement and dynamical load balancing for parallel computation. The second code was developed in cooperation with F. Cattaneo, Th. Emonet, and T. Linde from the University of Chicago; it is less sophisticated (fixed structured grid, 4th-order finite-differences and time stepping) and thus was available in 3D somewhat earlier than the IAM code, so that the developed methods and modules for non-grey radiative transport (RT) could be incorporated and tested, including first full simulation runs. A 2D version of the IAM code has been already combined with the RT; for the 3D version this is currently been done and first results are expected in 2005. This paper is organized as follows. We briefly describe the basic equations of radiative MHD in Sec. 2. The developments of numerical methods for radiative transfer are discussed in Sec. 3: RT on unstructured grids in Sec. 3.2 and the treatment of the frequency dependence in Sec. 3.3. Results of simulations are presented in Sec. 4: first results of 2D simulations with the IAM code combined with RT are shown in Sec. 4.1 while 3D results with the MPAe-Chicago code and non-grey RT are given in Sec. 4.2. We conclude with a brief outlook in Sec. 5.

2 Equations of radiative magnetohydrodynamics The magnetohydrodynamic (MHD) approximation can be used to describe a collision-dominated, electrically well-conducting, quasi-neutral plasma. These conditions are fairly well fulfilled in the convection zone and lower atmosphere of the Sun. Starting from the Maxwell equations (in Gaussian units), ∇×B =

1 ∂E 4π j+ , c c ∂t

∇×E = −

1 ∂B , c ∂t

(1) (2)

∇ · E = 4πǫ ,

(3)

∇·B = 0,

(4)

with the electric field E, the magnetic field B, current density j, and electrical charge density ǫ, the MHD approximation is obtained under the assumptions

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that all material speeds and phase velocities are small compared to the speed of light and that the plasma is charge-neutral owing to a sufficiently high electrical conductivity. These conditions entail that the electrical field is small compared to the magnetic field and that the displacement current (second term on the right-hand side of (1)) can be neglected. Using Ohm’s law in a medium locally moving with velocity v, 1 j = σ(E + v×B) , c

(5)

where σ is the electrical conductivity, we derive a single equation for the time evolution of the magnetic field, the induction equation, viz. ∂B = ∇× (v × B) − ∇× (η∇ × B), ∂t

(6)

with the magnetic diffusivity defined as η = c2 /4πσ. The induction equation governs the time evolution of the magnetic field for a given velocity field, v. The first term on the r.h.s. describes the inductive effect of the velocity field, the second term accounts for diffusion of magnetic field due to the finite conductivity of the plasma. The order-of-magnitude estimate for the ratio of these terms gives the magnetic Reynolds number, Rm = vL/η, where v is a typical speed and L is a typical length scale of the flow under consideration. Estimates for the photosphere and upper convection zone give magnetic Reynolds numbers of the order of 105 − 106 , so that the diffusion term is almost negligible in these regions. In the high-Rm regime, Alfven’s theorem of flux-freezing applies: magnetic field lines are transported by the fluid as if frozen in and fluid motions relative to the magnetic field are possible only along the direction of field lines. The rest of the equations of the MHD approximation are the equations of hydrodynamics with appropriate magnetic terms in the momentum equation (the Lorentz force) and the energy equation (the Joule dissipation term in the case of a non-vanishing magnetic diffusivity). The continuity equation ∂̺ + ∇ · (̺v) = 0 ∂t

(7)

represents mass conservation. The equation of motion is written in momentum conservation form: |B|2 BB ∂̺v + ∇ · ̺vv + p + (8) 1− = ̺g + ∇ · τ . ∂t 8π 4π Here p is the gas pressure and g the vector of gravitational acceleration. vv and BB are dyadic products and 1 is the 3 × 3 unit matrix. The magnetic force (Lorentz force) has been split into the gradient of the magnetic pressure, pmag = |B|2 /8π, and the term −∇·(BB/4π), which represents a tension along magnetic field lines. The last term on the r.h.s. of (8) is the viscous force, written as divergence of the viscous stress tensor, τ .

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The energy equation is written in the form |B|2 ∂e 1 + ∇· v e+p+ B(v · B) − ∂t 8π 4π =

(9)

1 ∇ · (B × η∇ × B) + ∇ · (v · τ ) + ∇ · (K∇T ) 4π

+ ̺(g · v) + Qrad . for the total energy density per unit volume, e, which is the sum of internal, kinetic and magnetic energy densities. T is the temperature and K the thermal conductivity. Qrad is the radiative source term which accounts for heating and cooling of the plasma by radiation. This term is discussed in detail in Sec. 3. In order to close the system of MHD equations, we have to specify a relation between pressure, p, density, ̺, and internal energy of the gas, eint . At temperatures typically encountered in the photosphere and upper convection zone, the solar plasma is partly ionized and the simple thermodynamical relations for an ideal gas do not apply. Owing to the corresponding changes in the thermodynamical properties of the matter up to 2/3 of the enthalpy flux is transported by latent heat and buoyancy driving of convective motions is strongly enhanced. Under the given conditions in the solar photosphere and the uppermost part of the convection zone it is sufficient to consider only the first ionization of the eleven most abundant elements in the Sun. The internal energy per mass unit εint = eint /̺ can be written as 3 1 ∗ (ne + na ) kT + ni χi , 2̺ ̺

(10)

' 1 3kT & xi νi χi , 1+ xi νi + 2µa m0 µa m0

(11)

εint =

where the sum runs over the particle species, n∗i is the number density of ionized particles of type i, and χi is the corresponding ionization energy. na = ni is the number density of nuclei, and ne the number density of electrons. Defining the ionization degree, xi = n∗i /ni , and the relative abundance, νi = ni /na , (10) can be rewritten as εint =

where µa is the mean molecular weight of the neutral gas (µa = 1.29 for solar composition) and m0 is the atomic mass unit. The ionization degrees, xi , are determined by the set of Saha equations 3/2 ui1 (T ) µa m0 2 (2πme kT ) xi xi νi = exp (−χi /kT ) . 1 − xi ui0 (T ) ̺ h3

(12)

The temperature dependence of the partition functions ui1 , ui0 can be obtained from the literature. For temperatures exceeding about 16, 000 K, the

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elements are almost fully ionized and the temperature dependence can be neglected. In order to obtain the temperature from ̺ and eint , the nonlinear system of equations (11) and (12) needs to be solved iteratively. Once the temperature is known, the gas pressure follows from the perfect gas equation: ' ̺ & p = (ne + na ) kT = (13) 1+ xi νi kT. µa m0

3 Numerical methods for radiative transfer 3.1 The radiative source term The photosphere is the region where most of the radiation leaves the Sun and where radiation takes over from convection as the dominant mechanism of energy transport. The energy exchange between gas and radiation determines the temperature structure of the photosphere and is responsible for the entropy drop which acts as the main driver of convection. Therefore any realistic simulation must include the radiative energy exchange rate, Qrad , as a source term in the energy equation. Since the mean free path of photons increases strongly as the atmosphere becomes transparent in the photosphere, radiative transfer at this height becomes essentially non-local and a diffusion approximation of radiative energy transport as adequate for the solar interior cannot be applied. The starting point for determining Qrad is the (time- and frequencydependent) radiative transfer equation (RTE hereafter), 1 ∂ + Ω · ∇ Iν = κν ̺(Sν − Iν ) . (14) c ∂t The specific intensity, Iν , is defined such that the amount of energy dErad transported by radiation along direction Ω in the frequency interval (ν, ν +dν) across an area element dS into a solid angle dω in a time interval dt is dErad = Iν (x, Ω, t)(Ω · dS) dω dν dt .

(15)

Sν is the source function and κν is the frequency-dependent absorption coefficient of the material. Since the travel time of a photon through the photosphere is much shorter than any other relevant timescale, the radiation field can be assumed to adjust quasi-instantaneously to any change of the thermodynamical state of the gas, i.e. the time derivative in (14) can be neglected and we obtain: (16) Ω · ∇Iν = κν ̺(Sν − Iν ) . Defining the optical depth of a path element ds as dτν = κν ̺ ds the RTE for a given direction can be written in the form

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dIν (Ω) = Sν − Iν (Ω) , dτν which has the formal solution

Iν (τν ) = Iν (0) e−τν +

τν

Sν (tν ) e−(τν −tν ) dtν .

(17)

(18)

0

In the case of local thermal equilibrium (LTE), the source function is given by the Planck function Sν = Bν . The essential premise of LTE is that elastic collisions between particles represent the dominant interaction on a microscopic level. Then the velocity distribution of particles is Maxwellian and the ionization states and population numbers of atomic, ionic, and molecular energy levels are determined by Saha-Boltzmann statistics corresponding to the local temperature. Significant departures from LTE must be expected in the upper photosphere, especially in strong spectral lines, for which scattering dominates over thermal emission and disturbs the detailed energy balance of LTE. This effect can be neglected as long as these lines do not contribute significantly to the total (frequency-integrated) energy exchange rate, Qrad . Since in LTE the source function is independent of the radiation field, (18) can be integrated in a straightforward manner. The numerical treatment of radiative transfer is based on this formal solution. Once the radiation field is known, the radiative energy flux, Fν = Iν (Ω) Ω dω , (19) and the average intensity, Jν =

1 4π

Iν (Ω) dω ,

can be calculated. The radiative heating rate then follows either from Qrad = − (∇ · Fν ) dν

(20)

(21)

ν

or from the equivalent expression Qrad = 4πκ̺

ν

κν (Jν − Bν ) dν .

(22)

Consequently, for each cell in the computational grid, the numerical determination of the radiative heating rate, Qrad , requires a sequence of integrations: 1. Spatial integration of the equation of radiative transfer (16) along a number of directions to determine the respective specific intensity Iν (Ω), 2. angular integration of Iν (Ω) to determine the radiative energy flux, Fν , or the mean intensity Jν , and 3. frequency integration to determine the radiative heating rate from (21) or (22).

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3.2 Radiative transfer on unstructured grids In the course of our ANumE project in cooperation with the IAM of the University of Freiburg, we have carried out two studies on the spatial and angular integrations (steps 1 and 2 in the preceding section) in unstructured grids. The first study [1] concentrated on the modification of the short-characteristic formal solver [2] for the case of unstructured grids, while the second study [3] led to the development of a new class of adaptive solvers. Ignoring the frequency dependence (to be discussed in Sec. 3.3) for the time being and thus dropping the index ν in all frequency-dependent quantities, a short-characteristic solver starts from a discretized form of the formal solution (18) of the RTE. Consider the situation shown on the left panel of Fig. 2: given the incident intensity Ii+1 at grid point i + 1, characterized by optical depth τi+1 measured along the ray in the direction −Ω starting from τ (s = ∞) = 0, the intensity Ii is given by τi+1 Ii = Ii+1 e−∆τi + S(τ ) e−(τ −τi ) dτ, (23) τi

where ∆τi = τi+1 − τi =

si

κ(s)ρ(s) ds.

(24)

si+1

Accurate numerical evaluation of the integral in (23) would in general be cumbersome, but after approximating S(τ ) by a linear or quadratic function of τ , it can be written analytically as ∆I =

τi+1 τi

S(τ ) e−[τ −τi ] dτ =

i+1

Wj S j ,

(25)

j=i−1

where the coefficients Wj depend only on the optical depth intervals, ∆τi and ∆τi−1 . Approximating κ(s) by a linear or a quadratic function, (24) can also be replaced by a simple analytic expression in terms of the function values at the points i + 1, i, and i − 1. Details of the method are described in [1]. The next step towards the determination of QR is the integration of I(Ω) over 4π steradian of solid angle in order to obtain the mean intensity, J, or the radiative flux, F. This integration is expressed as a quadrature sum over a discrete set of directions. The choice of the directions Ωm and their weights wm is subject to a few mandatory normalization criteria for the lowest moments of the intensity. In particular, the mean intensity J = wm Im and the flux F = 4π wm Im Ωm should obtain their correct values for an isotropic radiation field. A sensible criterion is the invariance under rotation around the z−axis over multiples of 90◦ . Another desirable criterion, although difficult to achieve, is that the directions should be distributed as evenly as possible over the entire sphere. An exact construction procedure of the angular quadrature has been given in [4]. It can be shown that, for a given quadrature,

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the discretization errors in F are markedly smaller than those in J. Errors are typically in the percent range for a quadrature with 3 directions per octant; significant improvements can only be achieved at a high computational cost.

C

z,s

τ i

i+1 ∆τ i

Ω

E

i-1 ∆τ i-1

Ω

A D B

Fig. 2. Left: geometry for the short-characteristic solution of the radiative transfer equation in direction Ω. Right: radiative transfer on a cell from a triangular grid. The speciﬁc intensity on corner C can be computed once the values on A and B are available for interpolation on point D (from [1]).

As an example for the adaptation of the short-characteristic method to a finite-volume scheme on an unstructured grid we consider the planar case with a triangular grid. The most obvious way to obtain QR would be through (20) by computing J directly for each cell center, but this straightforward approach requires excessive and poorly-defined interpolation. It is easier, more efficient, and more accurate to compute the intensities I at the vertices of the cells and afterwards use either J or F to compute the cell-average of QR required by a finite-volume method. First consider the radiative transfer problem within a single triangle (right panel of Fig. 2), where we want to compute IC (Ω). The procedure is as follows: starting from C, tracing the ray in the upwind direction, locate point D where the ray enters the cell. IC can be computed by means of a short-characteristic integration over the interval DC provided that the incident intensity ID is known. In general, point D does not coincide with a vertex of the grid and ID has to be interpolated from IA and IB , which must therefore be known beforehand: this requires that the intensities at the vertices are computed in the proper order. The incident intensities at the periodic side boundaries are obtained by iteration: from a suitable starting guess for the incident intensities at the vertices of the inflow boundary, compute the intensities at the corresponding vertices on the opposite boundary — which should be exactly

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the same — and use those as update for the incident intensities. This process is repeated until the intensities have converged. Using iteration for the initialization, however, is tricky because the convergence rate of this iteration procedure can be slow in the optically thin part of the atmosphere. However, once the incident intensities have been initialized correctly, iteration can well be used to update the incident intensities after each time step; the outflow intensities at time tn provide a good initial guess for the incident intensities at time tn+1 , so that few if any iterations are needed. The radiative transfer problem for the entire computational domain then simplifies to a sequence of radiative transfer problems on single triangles. The simplest short-characteristic integration method uses κC and κD (the values of the opacity at point C and D, respectively) to define a linear relation κ(s) on the interval DC. Together with the relation for ρ(s) this yields an analytic expression for ∆τDC . That, in turn, is used together with SC and SD to define a linear relation S(τ ) to evaluate ∆I from (23). This approach has limited applicability though: given the strongly non-linear dependence of κ on T , a small cell-to-cell variation of T already leads to significant nonlinear variation of κ, so that the optical path length ∆τDC is misrepresented. Straight application of the short-characteristic method with quadratic S(τ ) is not possible, since it requires a point downwind from point C in order to define the necessary quadratic functions for κ(s) and S(τ ). Such a point would be located outside of the triangle and violate the local character of the method. However, without significant adverse effects the method can be reformulated to use an auxiliary point E exactly halfway between D and C to define those quadratic functions. This quadratic approach significantly increases the maximum allowable cell-to-cell variation of T at a given accuracy level of I. We have implemented radiative transfer routines to compute the radiative heating rate, Qrad for various 2D model situations on a triangular grid [1]. Qrad can be computed from the mean intensity, J, or from the radiation flux, F. We have studied the accuracy of the short-characteristic radiative solver on such a grid and the accuracy of the angular integration required to compute J and F. It turned out that QJR has severe accuracy problems in the optically thick regions while QF R is stable and accurate there but may fail completely in optically thin layers. Therefore, the best solution is a combination of QJR and QF R with a selector based on the optical path lengths. In a subsequent study in the course of the ANumE project, Dedner and Vollm¨ oller [3] have compared a number of methods for the solution of the equation of radiative transfer on a triangular planar grid. These included the discontinuous-Galerkin finite element method [5], the long-characteristic method [6], and the short-characteristic method. All these methods can be generalized to the 3D case and to different grid geometries in a straightforward way. As a result of this study, a new class of methods was developed, the extended-short-characteristic (ESC) solvers, which combine the finite-element and short-characteristic approaches. In the case of triangles, the first-order version, ESC1, requires the determination of the intensity on the three node

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A

F

E

C

B

D

E

D

ff

A

cc

B

C

A

ee

cc

F dd

B

Fig. 3. To solve the RT equation on a single element using the extended-shortcharacteristic (ESC) method it is necessary to calculate the intensity for diﬀerent points, depending on the desired order of the method (1 or 2) Left: in the ESC1 method the intensity at point C has to be computed on triangles with only one inﬂow edge along the characteristic starting at point cc. Middle and right: in the ESC2 method the intensities have to be computed at point F (two inﬂow edges) or E, C, D (one inﬂow edge) using the corresponding inﬂow intensities at point ﬀ or at points ee, cc and dd respectively (from [3]).

points, while the second-order ESC2 method additionally includes the midpoints of the edges (see Fig. 3). The ESC methods combine the idea of local ansatz functions for the solution from the finite element framework with the idea of solving local 1D initial value problems along characteristics. Variation of the ansatz functions allows to develop schemes of higher order. The method can be adapted to specific applications by using ODE solvers for RT equation which depend on the stiffness of the underlying ODE. Indeed it was found that Runge-Kutta solvers are superior to the classical formal solution approach with respect to the error-to-runtime ratio. To speed up the higher order ESC method, an adaptation strategy including the variation of the order from grid cell to grid cell is relatively simple to implement because the computation of the intensity coefficients is independent of the chosen ansatz function. In three space dimensions, the coefficients for the discrete solution are also given by the same ODE as in 2D, so that this part of the module can be used without modifications. Details about this study and the ESC methods can be found in [3]. 3.3 Frequency-dependent radiative transfer Since the total absorption coefficient (opacity) in the solar atmosphere comprises the effect of the order of 106 atomic and molecular spectral lines, a number of roughly 106 -107 frequency points is required to model the detailed frequency dependence. While this direct approach is feasible in calculating 1D static models, the computational cost is intolerable in time-dependent 2D or 3D simulations. The most radical simplification of the problem is achieved by

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the grey approximation, replacing the frequency-dependent opacity by an averaged value, e.g. the Rosseland mean. The grey approach in combination with the diffusion approximation may be appropriate in the optically dense subphotospheric regions, where the radiation field is in local equilibrium with the gas, it is however unsatisfactory in the optically thinner regions where the radiation transfer becomes nonlocal. Here it is not only necessary to treat the full radiative transfer but also to incorporate line opacities, since line-blanketing effects have a considerable impact on both the photospheric dynamics and the emergent intensities. It is well known that the inclusion of line opacities in calculations of stellar model atmospheres strongly modifies the resulting temperature profiles, leading to considerably cooler outer layers, while the temperature is raised in deeper regions (the line cooling and backwarming effects). The effect of non-grey radiative transfer on the results of our MHD simulations has been studied in [7] and [8]. L

∆λ i-1

∆λ i

∆λ i+1 λ

L k+1

l

τ ref Lk

l -1

τ ref

∆λ i

λ

τ ref

Fig. 4. Left: Sorting of wavelengths according to a discretization of opacity. The hatched areas mark those parts of the wavelength interval ∆λi for which the opacity lies in the interval [Lk , Lk+1 ]. Right: Schematic illustration of the τ -sorting procedure. The wavelength intervals ∆λi are sorted to opacity bins according to the height where τ = 1 is reached for that wavelength interval, which is indicated by bold arrows (from [9]).

For reasons of computational feasibility in 2D/3D simulations, one has to resort to an appropriate statistical treatment of the line opacities that conserves the non-grey character of the radiation transport while drastically reducing the computational expense. In the context of time-dependent threedimensional simulations, for which the radiative transfer must be solved for every timestep, the only feasible approach – given the computing resources of today – is the opacity binning approach, also called multi-bin or multi-group method [10, 11, 12]. The basic idea of this method is to sort frequencies into 4–6 (non-contiguous) groups according to the geometrical depth in a 1D reference atmosphere at which optical depth unity at that frequency is reached. For each of these frequency groups, a separate RT equation with group-integrated source function and opacity is solved and the respective intensities are then

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added to obtain the total intensity. Under the assumption of local thermodynamic equilibrium (LTE), the (frequency-dependent) source function Sν is equal to the Planck function, Bν . The opacities can then be obtained from pre-compiled tables without having to solve the system of rate equations for each gridpoint and timestep during a simulation run. While the opacity binning approach has been tested in the case of 1D static model atmospheres [13], the application in the context of time-dependent 2D/3D simulations is accompanied by new sources of errors not encountered in the static 1D case like, for instance, strong lateral variations of the atmospheric properties and the occurrence of steep velocity gradients and shocks. In order to gain confidence in the applicability of this approach in simulations, we have performed in [9] a series of tests of the multigroup method for several 1D and 2D cases with the solution based on opacity distribution functions [14] serving as the reference solution. In a plane-parallel atmosphere, a substantial improvement of the radiative heating rates in comparison to the grey case can be achieved with the multigroup method already with a moderate number of frequency groups. In order to test whether this result also holds for situations with substantial lateral variations, we have considered two cases: a magnetic flux sheet embedded in a non-magnetic atmosphere and a snapshot from a 2D simulation of solar surface convection. Magnetic flux sheet We consider a simple model of a magnetic flux concentration, a 2D flux sheet. Using a 1D model atmosphere, the stratification in the interior of the sheet is shifted downwards by 200 km relative to the surrounding atmosphere, resulting in a strong lateral variation of the thermodynamic quantities. The width of the sheet as a function of height is determined by magnetic flux conservation together with the condition of total (magnetic plus gas) pressure equilibrium between the interior and the exterior of the sheet. At the height z = 0, corresponding to the visible surface (continuum optical depth unity) in the exterior, the flux sheet has a width of 150 km. The sheet is fanning out with increasing height as the magnetic pressure necessary to balance the jump in gas pressure decreases. At the interfaces between the interior and the exterior of the sheet, the atmospheric parameters are smoothed horizontally over a distance of a few tens of kilometers by way of a Gaussian error function. Owing to the mirror symmetry of the sheet, the calculations can be restricted to one half of the flux sheet, with symmetrical boundary conditions imposed on the sheets symmetry axis (located at x = 0). A Cartesian grid with 201 × 161 grid points and horizontal and vertical resolutions of 2.5 km and 5 km, respectively, was used. At equal geometrical height, density and temperature within the flux sheet are lower than the corresponding values in its surroundings. Consequently, the flux sheet is more transparent and thus subject to radiative heating from the

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Fig. 5. Horizontal proﬁles of Qrad at two diﬀerent heights in a 2D atmosphere with an embedded ﬂux sheet. x = 0 corresponds to the symmetry axis of the sheet. The curves indicate the reference solution (obtained using opacity distribution functions), the grey solution (1 bin, Rosseland opacity) and the solutions for opacity binning with 3 and 5 bins, respectively (from [9]).

hot ‘walls’ of surrounding plasma. This can be seen in Fig. 5, which shows the radiative heating rate, Qrad , as a function of the horizontal coordinate for two geometrical heights. The grey (Rosseland opacity) and multi-group results are compared with the reference solution based upon an opacity distribution function (ODF) description of the spectrum with effectively nearly 4000 frequency points. Corresponding to the horizontal temperature gradient, a heating peak inside the sheet and a stronger cooling region outside form near the sheet boundary. Below z ≃ −150 km, the outside atmosphere and most of the boundary region are optically thick and the radiative transfer is essentially grey, so that Qrad is well represented even without a detailed treatment of the frequency dependence. The maximum relative errors near the heating and cooling peaks range between 5 and 10 percent. At z = 100 km (left panel of Fig. 5), both multi-group solutions qualitatively reproduce the ODF case while the grey case is much less accurate. The heating peak at the boundary is shifted outwards by approximately 10 km, the peak value being reduced by 15 percent with respect to the ODF case, while the small cooling dip outside is not captured at all. At a still greater height of 250 km (right panel of Fig. 5), both the 3- and 5-bin approximations yield acceptable results, though neither captures the full extent of the cooling peak outside the sheet boundary. The grey solution, on the other hand, does not even approximately reproduce the reference solution. The heating peak inside the sheet has vanished; instead, heating takes place immediately outside the sheet, where the reference solution shows considerable cooling. This behavior can be explained by the fact that, on the basis of the Rosseland mean, the interior of the sheet is transparent at a height of 250 km; accordingly, the

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interaction between gas and radiation is very weak within the sheet, resulting in small values of Qrad . In the optically thicker regions immediately outside the sheet, radiation originating from deeper, hotter regions at the opposite sheet boundary and crossing the sheet almost unattenuated, leads to a net heating effect. This example clearly demonstrates how important a nongrey approach to radiative transfer can be in optically thin regions in order to obtain accurate values of Qrad . It also shows that the multi-group approach leads to a reasonably adequate description of the non-grey effects.

Fig. 6. Horizontal proﬁles of Qrad for a snapshot from a 2D simulation of solar convection at heights of 100 km (left panel) and 500 km (right panel) above the visible solar surface (from [9]).

Convection simulation As a further step towards more realistic situations we tested the opacity binning models with a snapshot from a 2D simulation of solar surface convection [15], which has no relation to the one-dimensional reference atmosphere used for sorting the frequencies into groups.. The vertical and horizontal extent of the computational domain is 1400 × 1400 km2 with a grid resolution of 35 km. Fig. 6 shows Qrad in horizontal cuts at heights of approximately 100 and 500 km above the visible solar surface, respectively. Similar to results for the flux sheet, all solutions (including the grey case) agree reasonably well with each other in the deeper layers, although the errors of the grey case become more pronounced towards the horizontal boundaries of the domain. The differences between the 3- and 5- bin solutions are only marginal. At 500 km height, the grey solution completely fails to reproduce the reference solution while the 5-bin solution excellently matches the reference curve. In summary we can conclude that our test calculations have shown that the multi-group approach yields a good approximation to the frequency-integrated

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radiative heating rate at moderate computational cost, which makes it the method of choice in multidimensional time-dependent MHD simulations. The advantage over the grey approximation is particularly pronounced in situations which deviate strongly from the one-dimensional plane-parallel case. In these cases the radiative transfer is complicated by lateral heating and cooling effects. Grey radiative transfer often fails to capture these effects, which leads to qualitatively wrong heating rates in the upper photosphere. This is particularly relevant if magnetic field concentrations are included, since the partial evacuation of these structures leads precisely to the kind of lateral inhomogeneities which are not well modeled by the grey approach. The test calculations with snapshots from numerical simulations have shown that the good performance of the multi-group method does not strongly depend on the choice of the reference atmosphere, which underlines the applicability of this method in realistic multi-dimensional simulations.

4 Simulations results for solar magneto-convection In the course of the ANumE project, we have developed two codes for the simulation of radiative MHD. The MHD part of the first code (called IAMMPAe code in what follows) is due to our partners at the IAM (University of Freiburg) and incorporates Riemann solvers for real gases and unstructured grids. We have implemented the ESC solvers for radiative transfer in the 2D version of this code, calculated the tables of the (non-ideal) equation of state, and introduced a realistic open boundary condition at the bottom of the computational box. This code has then be used to follow the formation of magnetic flux concentrations in the solar atmosphere and to analyze their oscillation properties [16]. Some of these results are summarized in Sec. 4.1. The MHD part of the second code has been jointly developed with colleagues from the University of Chicago. We have introduced a shortcharacteristic radiative transfer solver including a non-grey treatment based upon the multi-group method [7], partial ionization, and an open lower boundary. Results from simulations runs with this code (the MURAM1 code) are given in Sec. 4.2. 4.1 Convective intensification of magnetic flux Observations show that the majority of the magnetic flux through the solar atmosphere is assembled in magnetic field concentrations with a field strength of 1500 G and above [17]. Such field strengths exceed the value Beq corresponding to equipartition between magnetic energy density and kinetic energy density of the convective flows by at least a factor of three. Consequently, the concentration of the magnetic flux cannot be solely due to the passive advection of 1

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magnetic field lines by converging convective motions until the Lorentz force impedes further transport. It has been suggested that thermodynamic effects play an important role for the intensification of the magnetic field beyond the equipartition limit, Beq . The suppression of the horizontal convective motion once the field strength approaches Beq throttles the energy supply to the magnetic regions. Since at the solar surface the radiative energy loss into free space continues, this leads to a substantial cooling of the gas. Under the influence of gravity and pressure forces, the cool and dense gas sinks down. The downflow is further accelerated by the convectively unstable (strongly superadiabatic) stratification below the solar surface. This leads to a partial evacuation of the upper part of the forming flux concentration, which becomes laterally compressed by the external pressure until the field has grown strong enough to reestablish lateral pressure equilibrium. The combination of processes of flux advection by horizontal flow, suppression of convection, radiative cooling, downflow and compression is called convective collapse or convective intensification [18, 19, 20, 21]. We have used the 2D version of the adaptive IAM-MPAe code to study the convective intensification of magnetic flux in a region of 12, 000 km horizontal size extending in the vertical direction between 300 km above and 900 km below the visible solar surface. The computational setup is to start with a slightly perturbed plane-parallel convectively unstable stratification and let non-magnetic convection develop until a statistically stationary situation is reached. Then a homogeneous vertical magnetic field of 100 Gauss is introduced and its development followed in the course time. Fig. 7 (to be found in the section with color pages at the end of this volume) shows a sequence of snapshots of the magnetic field (field lines) and the temperature field (color coding), which clearly demonstrates the concentration of the magnetic field into a few intense flux sheets of kilogauss field strength located in convective downflow regions. The simulations nicely confirm the theoretical concepts of flux expulsion and convective collapse. Within a few minutes, most of the magnetic flux is transported by the converging horizontal flows to the cool downflow region (flux expulsion). Suppression of the convective energy transport and ongoing radiative cooling leads to downflow of the gas in the flux concentrations. The reduced internal pressure leads to lateral compression by the external gas pressure, resulting in a strong intensification of the field strength to kilogauss values. About 20 minutes after the introduction of the magnetic field, the flux concentrations have merged into three large flux sheets, which govern the surrounding flow pattern with strong downflows surrounding the flux sheets. After about 30 minutes this quasi-stationary situation has fully developed. Figure 8 shows the temporal evolution of the average field strength in magnetic flux sheet I in Fig. 7 near the visible solar surface. Two stages of field amplification can be seen: first, a rapid concentration of the field to about 1 kG by flux expulsion and radiative cooling within the first 3-4 minutes after the introduction of the magnetic field, which is followed by a second, slower

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Fig. 7. Time evolution of magnetic ﬁeld (ﬁeld lines are shown in black) and temperature (color coded) from a 2D simulation run of solar magneto-convection near the visible solar surface (located at z ≃ 100 km height). A homogeneous vertical ﬁeld of 100 Gauss has been introduced at t = 0 after a statistically stationary convection pattern has evolved. Within a few minutes, most of the magnetic ﬂux is transported by the converging horizontal ﬂows to the cool downﬂow region (ﬂux expulsion). Owing to the suppression of the convective energy transport, the gas in the ﬂux concentrations cools and sinks; lateral compression by the external gas pressure then leads to a strong intensiﬁcation of the ﬁeld strength, which reaches kilogauss values. About 20 minutes after the introduction of the magnetic ﬁeld, the ﬂux concentrations have merged into three large ﬂux sheets (labeled I,II,III), which start to determine the surrounding ﬂow pattern with strong downﬂows surrounding the ﬂux sheets. After about 30 minutes a quasi-stationary situation has developed. The velocity ﬁeld in this state is shown in the form of velocity vectors in the last panel (from [16]). (See also color ﬁgure, Plate 11.)

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Fig. 8. Temporal evolution of the magnetic ﬁeld strength for the magnetic ﬂux sheet I in Fig. 7 at a geometrical height roughly corresponding to the visible solar surface. The initial ﬁeld of 100 G is intensiﬁed to about 1 kilogauss within a few minutes by the combined action of ﬂux expulsion by the convective ﬂow and radiative cooling. The continuing downﬂow within the magnetic ﬂux concentration leads to further evacuation and ampliﬁcation of the ﬁeld to values around 2 kilogauss.

increase of the field due to a persistent downflow and evacuation of the upper layers of the flux sheet. The second process leads to field strengths around 2 kG. 4.2 Formation of dynamic magnetic structure The MURAM code The study of the full dynamics of the solar magnetic structure requires realistic simulations in three dimensions. As an intermediate step before the full completion of the 3D version of the IAM-MPAe code by implementation of a non-grey radiative transfer module, we have developed the MURAM code. This code allowed us to carry out simulation runs with a full non-grey radiative transfer module in 3D. The MURAM code solves the MHD and RT equations on a three-dimensional regular Cartesian grid with constant grid spacing. The spatial derivatives are discretized with 4th-order centered differences on a 53 point stencil. Time stepping is explicit with a 4th-order RungeKutta solver. The scheme is stabilized by the application of shock-resolving diffusion and hyperdiffusivity [22], which prevent the build-up of energy at scales comparable to the size of the grid cells. These artificial diffusivities assume significant values only near discontinuities and in regions of unresolved waves while those regions which are well resolved remain largely unaffected by diffusion. For the equation of state the instantaneous ionization equilibrium for the first ionization of the 11 most abundant elements is considered.

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The relevant thermodynamic quantities are stored in tables from which the required values are interpolated during a simulation run. The bottom boundary conditions implemented in the MURAM code permit free in- and outflow of matter and maintain a constant mass in the box as well as a fixed energy flux through the system. In the present version of the code, the top of the domain is closed with stress-free boundary conditions for the horizontal velocity components; the implementation of a more realistic transmitting upper boundary is under development. The magnetic field is assumed to be vertical at the top and bottom boundaries, the footpoints of fieldlines are allowed to move freely. The horizontal directions are taken to be periodic in all variables. The code is parallelized by means of domain decomposition. The computational domain is divided into a three-dimensional array of subdomains, each of which is endowed with two layers of ghost cells at its boundaries as required by the 4th-order spatial discretization scheme. We use message passing (MPI) for parallel computers with distributed memory. The radiative transfer equation is solved for each frequency set determined by opacity binning and for each direction using the short characteristic scheme with linear or parabolic interpolation of opacity and source function as well as linear interpolation of density. In the context of the domain decomposition used for parallelization, the short characteristic scheme requires an iteration for each ray direction and each frequency set. For a given ray direction the scheme starts in each subdomain at those boundaries through which the radiation enters (the ‘upwind’ boundaries). The intensity values at these boundaries are assumed to be known. Then the traversal of the subdomain proceeds in the downwind direction, systematically moving away from the upwind boundaries, thus making sure that the upwind intensities required for the interpolation are always known. However, on those upwind boundaries of a subdomain which do not coincide with the top or bottom boundary of the computational box, the intensities are a priori unknown. Therefore, the scheme is iterated until convergence at the boundaries is obtained. After each iteration the intensities at a given upwind boundary are updated with the new values provided by the neighboring subdomain. We found that 2 to 3 iteration steps per frequency set and direction are usually sufficient, if one chooses as initial guess for the intensities on the upwind boundaries a linear extrapolation of the values of the previous two time steps. More details about and further results obtained with the MURAM code can be found in [7, 23, 24, 25]. Simulation of a solar plage region A plage region (cf. Fig. 1) is a strongly magnetized part of the solar atmosphere outside sunspots (but often in their vicinity) with a horizontally averaged field strength of about 200 G. We have carried out a simulation run in a

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computational box corresponding to a height of 1400 km and 6000× 6000 km2 width on the Sun, discretized with a spatial grid of 100 × 288 × 288 points. 600 km

τ=1

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Fig. 9. Geometrical setup of the simulation runs with the MURAM code. The vector B0 indicates the vertical homogeneous magnetic ﬁeld introduced into the hydrodynamic convection at the beginning of the magnetic phase.

Similarly to the 2D runs described in the previous section, the simulations with the MURAM code were started as non-magnetic convection. After the convection had fully developed and reached a statistically stationary state (about one hour solar time after the start of the simulation), a homogeneous vertical initial magnetic field of field strength B0 was introduced. Here we show results from a run with B0 = 200 G, corresponding to a solar plage region. Within a few minutes of simulated time (approximately one turnover time of the convection) most of the magnetic flux has been transported to the downflow lanes of the convective granulation pattern and intensified to kilogauss field strength. For a snapshot taken about 2 hours solar time after the start of the magnetic phase, Fig. 10 shows the vertical magnetic field, vertical velocity, and temperature distributions on a horizontal plane corresponding roughly to the visible solar surface. In addition, the frequency-integrated intensity (brightness) is shown on the lower right panel. The magnetic map shows sheet-like magnetic structures extending along convective downflow lanes, while larger structures with diameters of up to 1000 km appear at the vertices where several downflow lanes merge. Typical field strengths in these field concentrations are between 1500 and 2000 G. The network of magnetic structures is organized on a ‘mesoscale’ which typically comprises 4–6 convective upflow regions (granules). While this magnetic pattern is rather stable (it evolves on a time scale of hours), the smallscale pattern of the field concentrations is highly time-dependent, with magnetic flux being constantly redistributed within the magnetic network. In the intensity map shown in Fig. 10, the larger flux concentrations appear dark owing to the reduced efficiency of convective energy transport. There is a considerable small-scale variation of the intensity within the pore-like flux concentrations, which is related to localized hot upflows in regions of reduced

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Fig. 10. Snapshot from the simulation of a solar plage region (B0 = 200 G) with the MURAM code. Brightness map (lower right) and horizontal cuts near the visible solar surface of vertical magnetic ﬁeld component (upper left), vertical velocity component (upper right) and temperature (lower left). Light and dark shades indicate higher and lower values, respectively. The velocity plot shows convective upﬂows shaded in light grey separated by intergranular downﬂow lanes. In the magneticﬁeld plot, the strong sheet- and pore-like magnetic ﬁeld concentrations appear in white.

field strength. In the thin sheets, lateral heating effects in combination with the depression of the level of optical depth unity lead to a brightening with respect to the surrounding downflow regions (see Fig. 11). Fig. 12 shows some statistical properties of the simulation run from a series of statistically independent snapshots (i.e., with a temporal cadence exceeding the granule lifetime). A horizontal slice consisting of 8 grid layers (corresponding to a thickness of 112 km), which include the visible solar surface, served as the basis for the analysis.

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Fig. 11. Radiative properties of a sheet-like magnetic structure in a convective downﬂow lane. Left: vertical ﬁeld strength (top) and brightness (bottom). The vertical line indicates the position of the cut shown on the right panel. Right: grey-shading of the temperature distribution in a vertical cut, together with the radiative ﬂux vectors. There is an inﬂux of radiation from the hot walls of the ﬂux sheet into its partially evacuated cooler interior. The radiative heating leads to enhanced brightness (shown at the top), so that the ﬂux sheet appears as a bright structure within the darker downﬂow lane.

The probability distribution function (PDF) for the magnetic field, signed with the orientation of its vertical component is shown in the upper left panel of Fig. 12. It shows a superposition of two components. Most of the volume considered is occupied by weak field, the probability density dropping off approximately exponentially with increasing field strength. The distribution reveals a pronounced local minimum at B = 0, indicating that the magnetic field, albeit being mostly weak, permeates the whole volume and field-free regions are nearly absent. Superimposed on this exponential distribution is a Gaussian “bulge” (the high field strength wing showing the characteristic parabolic shape on a logarithmic scale) with a maximum around 1500 G, which reflects the sheet- and pore-like structures in the network of concentrated magnetic field. The correlation diagram (joint PDF) of magnetic field strength and inclination angle of the field vector with respect to the horizontal plane given in the upper right panel of Fig. 12 shows that most of the strong field above the kilogauss level is vertical and upward directed (which is the orientation of the homogeneous initial field), presumably as the result of buoyancy forces acting on the partially evacuated magnetic structures. The inclination angle of weak

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Fig. 12. Statistical properties of a layer of about 110 km thickness around the visible solar surface. Upper left: probability distribution (PDF) of the ﬁeld strength, signed with the vertical orientation of the ﬁeld vector. Upper right: joint PDF of ﬁeld strength and the inclination angle of B with respect to the horizontal, theta(B). Lower left: joint PDF of ﬂow velocity, signed with its vertical orientation, and ﬁeld strength. Lower right: joint PDF of the inclination angles of the ﬂow, theta(v), and of the magnetic ﬁeld, theta(B). The grey-scaling indicates the probability density on a logarithmic scale.

fields is much more evenly distributed. With decreasing field strength a slight preference for upward directed fields is observed. The joint PDF of the vertical magnetic field and the flow velocity signed with the vertical orientation of the flow in the lower left panel of Fig. 12 (positive velocities correspond to upflows) shows the effect of strong fields on the fluid motions: while flow velocities up to 8 km s−1 can be found in weak field regions, the amplitudes of fluid motions are reduced in magnetic structures with field strengths above 1000 G. Fluid motions are not completely suppressed, however, since the predominantly vertical fields leave vertical fluid motions largely unaffected. Downflows are preferred inside strong field features.

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The lower right panel of Fig. 12 shows the joint PDF of the inclination angles of magnetic field vector and flow vector with respect to the horizontal plane. The pronounced diagonal lobe indicates that in most of the volume considered flow field and magnetic field are more or less aligned. In addition to this component, one observes a strong correlation of (strong) vertical magnetic field with downflows. Further simulations and studies We have already carried out a number of further simulations with the MURAM code and used the results to study various physical processes on the Sun. Among these investigations are: 1. A study of the quantitative effects of the frequency-dependent radiative transport on the simulation results. It turns out that the non-grey treatment leads to significantly smaller temperature fluctuations, particularly in the upper photosphere. This leads to brightness contrasts in agreement with observational values [8]. 2. A parameter study of magneto-convection with different amounts of vertical magnetic flux in the box ranging from B0 = 10 G (‘quiet’ Sun), B0 = 50 G, B0 = 200 G (the plage region described in the foregoing section), to B0 = 800 G. As the average field strength increases, the magnetic flux concentrations become larger and the field increasingly affects and eventually dominates the convective motions (V¨ ogler et al., publication in preparation) 3. Simulations of larger magnetic structures in the solar photosphere. Such darkish ‘pores’ with diameters of a few Mm represent the transition from small, bright flux concentrations to dark sunspots. The questions investigated are: how is a pore held together, how is it affected by energy transport along the field, how does it interact with the surrounding convection? Further studies concern the atmospheric structure (temperature, pressure) within a pore and its observational signatures (Cameron et al., publication in preparation). 4. A comparison of the fractal dimension of the magnetic field pattern in the simulations with observations of the magnetic structure in the solar photosphere. There is a remarkable agreement between simulation and observation, increasing the confidence in the ‘realism’ of the simulations [26]. 5. A study of the brightness of magnetic flux concentrations when observed in spectral bands dominated by molecular lines, particularly in Fraunhofer’s ‘G band’ with many spectral lines from the CH molecule. Through the comparison of synthetic images from simulation results with actual observations, the physical mechanism leading to the brightness contrast of magnetic structures could be unveiled. The evacuated and heated magnetic flux concentrations show lower abundances of CH, leading to less

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absorption in spectral lines and thus larger brightness. These results put the indirect approach of studying of the magnetic field dynamics in the solar atmosphere by observing bright features in the G band on a firm physical basis [27]. 6. An investigation of the evolution of an initial field with mixed polarity, i.e., half of the initial magnetic field pointing upward and the other half downward. This permits the temporal decay of unsigned magnetic flux by reconnection of field lines and cancellation of opposite polarity flux. By running simulations with various horizontal distributions of the initially vertical field (half-by-half, two-by-two,...) and determining the decay rate, it is possible to determine values of the ‘turbulent’ magnetic diffusivity, which is a very important quantity for studies of the long-term evolution of the magnetic flux at the solar surface. Furthermore, the results of such simulation runs are compared with observations of the magnetic field dynamics in regions of mixed polarity on the Sun (V¨ ogler et al., publication in preparation). 7. The various tools developed for spectroscopic and polarimetric analysis to compare the simulation results with actual solar measurements have been applied to various magneto-convection simulations. The analysis of these results and the comparison with observations reveals rich diagnostic information in good agreement with measurements, which is used to identify the physical processes behind the observational phenomena (Shelyag et al., publication in preparation).

5 Summary & Outlook Our work on the ANumE project(s) has proven to be very fruitful. We have developed and tested methods to incorporate radiative transfer into MHD simulations and to take account of the frequency dependence of radiation in a stellar atmosphere. State-of-the-art codes have been developed and successfully applied to simulate magneto-convection in the solar atmosphere. With the MURAM code we have a fully working 3D code for realistic simulations, which has already produced a wealth of useful results. This code is our present ‘workhorse’ and several ongoing projects exploit its rich possibilities and analyze data produced by this code, often in direct comparison with observational results. Further development of the MURAM code will include introducing a transmitting boundary condition at the top and using a non-uniform grid in the vertical direction with a grid spacing proportional to the average pressure. This will allow us to extend the computational box in the vertical direction. Work on the 3D version of the IAM-MPAe code still continues. Comparison runs with the MURAM code will allow us to optimize the grid refinement strategy and will also provide a mutual test of both codes. The IAM-MPAe code will then be used to attack the challenging physical questions of the

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higher solar atmosphere, in particular the chromosphere, which is dominated by interacting MHD shock waves. The IAM-MPAe code with its Riemann solvers and adaptive mesh refinement is perfectly suited for such kind of environment. Acknowledgement. We would like to thank Andreas Dedner, Dietmar Kr¨ oner, Christian Rohde, and Matthias Wesenberg, our partners from the Institute of Applied Mathematics (IAM), University of Freiburg, for a fruitful cooperation during the ANumE programme. We are grateful to Fausto Cattaneo, Thierry Emonet, and Timur Linde from the University of Chicago for their contributions to the development of the MURAM code. We thank the coordinator of ANumE, Prof. G. Warnecke and his staﬀ for much support and for the management of the programme. Last, but not least, we thank the Deutsche Forschungsgemeinschaft (DFG) for generous support of our project Schu 500/7 and all the DFG staﬀ involved with ANumE and with our project in particular for their eﬃcient, friendly, and non-bureaucratic dealing with our sometimes non-standard requests.

References 1. Bruls, J. H. M. J., Vollm¨oller, P., Sch¨ ussler, M.: Computing radiative heating on unstructured spatial grids. Astron. Astrophys, 348, 233–248 (1999) 2. Kunasz, P. B., Auer, L.: Short characteristic integration of radiative transfer problems: formal solution in two-dimensional slabs. J. Quant. Spectrosc. Radiat. Transfer, 39, 67–79 (1988) 3. Dedner, A., Vollm¨ oller, P.: An Adaptive Higher Order Method for Solving the Radiation Transport Equation on Unstructured Grids. Journal of Computational Physics, 178, 263–289 (2002) 4. Carlson, B. G.: The numerical theory of neutron transport. In B. Alder, S. Fernbach and M. Rotenberg (eds) Methods in Computational Physics 1, pp. 1–42 (1963) 5. Lesaint, P., Raviart, P. A.: On a ﬁnite element method for solving the neutron transport equation. In: C. de Boor (ed) Mathematical Aspects of Finite Elements in Partial Diﬀerential Equations, Academic Press, New York, pp. 89–123 (1974) 6. Mihalas, D., Auer, L. H., Mihalas, B. R.: Two-dimensional radiative transfer. I - Planar geometry. Astrophys. J, 220, 1001–1023 (1978) 7. V¨ ogler, A.: Three-dimensional simulations of magneto-convection in the solar photosphere. PhD thesis, University of G¨ottingen (2003) 8. V¨ ogler, A.: Eﬀects of non-grey radiative transfer on 3D simulations of solar magneto-convection. Astron. Astrophys, 421, 755–762 (2004) 9. V¨ ogler, A., Bruls, J. H. M. J., Sch¨ ussler, M.: Approximations for non-grey radiative transfer in numerical simulations of the solar photosphere. Astron. Astrophys, 421, 741–754 (2004) 10. Nordlund, A.: Numerical simulations of the solar granulation. I - Basic equations and methods. Astron. Astrophys, 107, 1–10 (1982) 11. Ludwig, H.-G., Jordan, S., Steﬀen, M.: Numerical simulations of convection at the surface of a ZZ Ceti white dwarf. Astron. Astrophys, 284, 105–117 (1994)

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12. Skartlien, R.: A Multigroup Method for Radiation with Scattering in ThreeDimensional Hydrodynamic Simulations. Astrophys. J, 536, 465–480 (2000) 13. Ludwig, H.-G.: Nichtgrauer Strahlungstransport in numerischen Simulationen stellarer Konvektion. PhD thesis, University of Kiel, Germany (1992) 14. Kurucz, R. L.: Model atmospheres for G, F, A, B, and O stars. Astrophys. J. Suppl. Ser, 40, 1–340 (1979) 15. Ploner, S. R. O., Solanki, S. K., Gadun, A. S.: The evolution of solar granules deduced from 2-D simulations. Astron. Astrophys, 352, 679–696 (1999) 16. Vollm¨ oller, P.: Untersuchung der Wechselwirkung von Magnetfeldkonzentrationen und konvektiven Str¨ omungen mit dem Strahlungsfeld in der Photosph¨ are der Sonne. PhD thesis, University of G¨ottingen (2001) 17. Solanki, S. K.: Small scale solar magnetic ﬁelds - an overview. Space Sci. Rev, 63, 1–188 (1993) 18. Spruit, H. C., Zweibel, E. G.: Convective instability of thin ﬂux tubes. Sol. Phys, 62, 15–22 (1979) 19. Parker, E. N.: Hydraulic concentration of magnetic ﬁelds in the solar photosphere. VI - Adiabatic cooling and concentration in downdrafts. Astrophys. J, 221, 368–377 (1978) 20. Roberts, B., Webb, A. R.: Vertical motions in an intense magnetic ﬂux tube. Sol. Phys, 56, 5–35 (1978) 21. Sch¨ ussler, M.: Theoretical Aspects of Small-Scale Photospheric Magnetic Fields. In J. O. Stenﬂo (ed) Solar Photosphere: Structure, Convection and Magnetic Fields, IAU Symposium 138, Kluwer, Dordrecht, pp. 161–179 (1990) 22. Caunt, S. E., Korpi, M. J.: A 3D MHD model of astrophysical ﬂows: Algorithms, tests and parallelisation. Astron. Astrophys, 369, 706–728 (2001) 23. V¨ ogler, A., Shelyag, S., Sch¨ ussler, M., Cattaneo, F., Emonet, Th., Linde, T.: Simulation of solar magneto-convection. In: N. E. Piskunov, W. W. Weiss, and D. F. Gray (eds) Modelling of Stellar Atmospheres, ASP Conf. Series, Astronomical Society of the Paciﬁc, San Francisco, pp. 157–168 (2003) 24. V¨ ogler, A., Sch¨ ussler, M.: Studying magneto-convection by numerical simulation. Astron. Nachr./AN, 324, 399–404 (2003) 25. Sch¨ ussler, M.: MHD simulations: what’s next? In: J. Trujillo Bueno and J. & S´ anchez Almeida (eds) Third International Workshop on Solar Polarization, ASP Conf. Ser., Astronomical Society of the Paciﬁc, San Francisco, pp. 601–613 (2003) 26. Janßen, K., V¨ ogler, A., Kneer, F.: On the fractal dimension of small-scale magnetic structures in the Sun. Astron. Astrophys, 409, 1127–1134 (2003) 27. Sch¨ ussler, M., Shelyag, S., Berdyugina, S., V¨ ogler, A., Solanki, S. K.: Why Solar Magnetic Flux Concentrations Are Bright in Molecular Bands. Astrophys. J, 597, L173–L176 (2003)

Riemann Problem for the Euler Equation with Non-Convex Equation of State including Phase Transitions Wolfgang Dahmen, Siegfried M¨ uller, and Alexander Voß Institut f¨ ur Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, D-52056 Aachen, Germany dahmen|mueller|[email protected] Summary. An exact Riemann solver is developed for the investigation of nonclassical wave phenomena in BZT ﬂuids and ﬂuids which undergo a phase transition. Here we outline the basic construction principles of this Riemann solver employing a general equation of state that takes negative nonlinearity and phase transition into account. This exact Riemann solver is a useful validation tool for numerical schemes, in particular, when applied to the aforementioned ﬂuids. As an application, we present some numerical results where we consider ﬂow ﬁelds exhibiting non-classical wave phenomena due to BZT ﬂuids and phase transition.

1 Introduction The dynamics of compressible flows has a strong influence on the design of aircraft and turbomachinery. In many applications the fluid is modeled by a perfect gas. In the range of perfect gas theory only two types of waves are possible and allowed by the entropy inequality, the compression shock and the centered expansion fan or rarefaction wave. A compression shock is a discontinuity where the pressure of the fluid increases while the shock is passing, whereas in a rarefaction wave the pressure decreases and the wave shape forms a fan. However, the perfect gas model is no longer appropriate when dealing with so-called BZT fluids characterized by a large heat capacity, e.g., high molecular fluorcarbons and hydrocarbons. These fluids exhibit a region in the phase space where the isentropes in the pressure-volume plane are nonconvex, see Figure 2. It is also referred to as region of negative nonlinearity. From experiments, see [BBKN83], it is known that there may occur nonclassical effects such as expansion shocks and compression fans, as well as waves composed of adjacent shocks and rarefaction parts. In short, most of the classical inequalities and effects are reversed. In addition, below the critical point in the phase space, see Figure 2, liquid and vapor co-exist in a so-called

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mixture region. One well known example is the heating of water, where at a certain temperature and pressure the water starts to vaporize and further heating does not result in hotter water but in more vapor. The temperature of the steam increases again, when all water has vaporized. Due to the phase change at the phase boundary new phenomena such as liquefaction shocks, shock splitting, rarefaction shocks, complete evaporation shocks and liquid-evaporation waves have been observed, cf. [DTMS79, TK83, TCK86, TCM+ 87]. In this range we need an equation of state (EOS) which is able to model the negative nonlinearity of BZT fluids as well as phase transition. In order to investigate these phenomena analytically, we consider the exact solution to a Riemann problem for compressible fluid flow where we employ a general EOS. ’Exact’ means, the Riemann solution is based on wave curve analysis rather than on numerical approximations. Such an exact Riemann solver is helpful in many respects. First of all, it may serve as a validation tool for numerical schemes. Moreover, it can be used to determine initial conditions for experiments exhibiting effects such as, for instance, a liquefaction shock. The general concept of constructing an exact Riemann solution is well-known, namely to construct the nonlinear wave curves corresponding to the nonlinear characteristic fields and to determine the intersection states in the pressurevelocity plane, cf. [MP89]. However, it has been an open problem so far how to construct the wave curves itself in this general setting including phase transition. If only negative nonlinearity is taken into account, their construction is well–known due to Wendroff [Wen72b] and Liu [Liu75]. In case of phase transition the wave curves are no longer smooth curves but they suffer kinks at the phase boundaries. These kinks cause jumps in the characteristic speeds which makes the construction of the waves much more difficult. Within the present project we extend for the first time the general concept of the wave composition and construction to this nonsmooth case. Here we confine to some basic principles motivated by a characteristic example. More details as well as analytical results concerning this subject can be found in [Voß04]. Realizing an exact Riemann solver for the Euler equations equipped with an EOS, including phase transition, requires a number of modules also needed for numerical schemes, such as an efficient evaluation of physical quantities, e.g., temperature and energy, for states inside the mixture region. For this purpose the C++-library xrms has been developed, see [Voß04]. This library has been incorporated into the QUADFLOW solver, cf. [BGMH+ 03]. The outline of the paper is as follows. First of all, we introduce retrograde and BZT fluids in Section 2 and summarize some characteristic features. In order to model these fluids we present a simplified real gas model in Section 3, based on the van der Waals EOS where we incorporate Maxwell’s construction principle to resolve the unphysical region of ellipticity. Here the states in the mixture region are modeled in thermodynamical equilibrium. In Section 4 we outline the basic construction principles of the Riemann solver employing a

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general EOS, taking negative nonlinearity and phase transition into account. Finally, we present some numerical results in Section 5 where we consider flow fields in BZT fluids exhibiting non-classical wave phenomena.

2 Retrograde and BZT Fluids The material properties are characterized by the EOS. Certain constraints on the EOS are imposed by the principle of thermodynamics. Here, we confine ourselves to thermodynamical equilibrium, i.e., the internal specific energy e = e(v, s) of an equilibrium state is related to the specific entropy s and the specific volume v. According to the fundamental thermodynamic identity the internal specific energy is characterized by de = −p dv + T ds,

(1)

where the pressure p and the temperature T are defined by the partial derivatives of e (2) p(v, s) := −ev (v, s), T(v, s) := es (v, s). In contrast to regular fluids such as air, non-regular fluids exhibit anomalies in the T-s diagram and in the p-v diagram. These fluids are characterized by a large molar heat capacity, e.g., high–molecular hydrocarbons and fluorcarbons. In the literature, we distinguish between retrograde fluids and BZT-fluids. 2.1 Retrograde Fluids Retrograde fluids are characterized by an overhang of the saturated-vapor curve in the T-s-diagram. For a regular fluid the entropy increases with decreasing temperature along the saturated-vapor curve. In contrast to this, retrograde fluids exhibit some part of the saturated-vapor curve where the entropy decreases with decreasing temperature, see Figure 1. This implies that for regular fluids condensation takes place on isentropic expansion whereas for retrograde fluids it takes place on isentropic compression, at least in these particular regions. By means of the characteristic heat capacity cv = ~

c0v (Tc ) R

retrograde fluids can be distinguished from regular fluids. Here ~ cv is defined by the ideal-gas heat capacity at the thermodynamic critical temperature Tc normalized by the gas constant R. It was found by Lambrakis [Lam72] that fluids exhibit the overhang whenever the characteristic heat capacity exceeds cv ≥ 11.2. ~

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T

(E)

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cr.pt.

cr.pt. (D)

(C)

(C)

(D)

(F) (B)

(B) (F*)

(F)

(A)

(A)

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(a) regular ﬂuid

S

(b) retrograde ﬂuid Fig. 1. Clausius-Rankine-cycles.

2.2 BZT Fluids BZT fluids are a subgroup of retrograde fluids. They are named after Bethe (1942), Zel’dovich (1946), Thompson (1971). For BZT fluids the curvature of the saturation curve is so intense that it leads to a concave bending of the isentropes near the critical point in the p-v-diagram, see Figure 2, i.e., the isentropes are locally non-convex. This is in contrast to the behavior of the most common fluids where isentropes are convex. A measure for the bending is the fundamental derivative of gas dynamics v pvv (v, s) v3 (3) Γ := 2 pvv (v, s) = 2c 2 −pv (v, s) which becomes negative in a region near the critical point. Here, c denotes the speed of sound defined by c2 := −v2 pv (v, s).

(4)

Note, that the so-called region of negative nonlinearity, i.e., Γ < 0, is located in the vapor phase at the curve of saturated vapor, see Figure 2. Another feature of BZT fluids is that isentropes inside the mixture region may cross the curve of saturated vapor such that the isentropes suffer a nonconvex kink. This is different for other fluids whose isentrope will never cross the saturated vapor curve. For more details on BZT fluids see, for instance, Thompson [Tho91].

3 Physical Model For our investigations the fluid is modeled by the van der Waals EOS. This gives a qualitatively good representation of the fluid in various important

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1.6

s 0 : convex isentrope s 1 : non-convex isentrope s 2 : isentrope with kink Γ = 0 : locus of inflection points of isentropes

p/pc 1.4

Γ >0

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liquid

Γ <0

vapor

1

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mixture region 0.6

s2

0.4 0

0.5

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2.5

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Fig. 2. Isentropes of a BZT Fluid in p-v diagram.

regions, namely the regimes of high compression, the area with negative nonlinearity near the critical point regarding BZT fluids and the mixture region to allow for phase transition. 3.1 Van der Waals Equation of State A vital assumption of the ideal gas model is that the particles are noninteracting mass points, i.e., an appreciable force acts on them only during a collision. Furthermore, the volume of the particles is negligible compared to the total volume occupied by the gas. This is no longer valid if the density is so high that the distance between two particles is of the order of their interaction diameter. Such extreme densities can be reached during the collapse of a bubble. In this case, the range of validity of the ideal gas model is exceeded. The van der Waals EOS is an extension of the ideal gas model with two material parameters a and b to take into account the attraction of particles and the reduction of free volume. The thermal EOS is determined by p(v, T) =

RT a − 2 v−b v

(5)

with pressure p, temperature T, specific volume v, specific gas constant R, internal pressure a/v2 and covolume b. The internal pressure is subtracted from the pressure to take into account that the attraction of particles diminishes the pressure. The covolume b reduces the volume v to the available free volume v − b. The caloric EOS reads T a (6) cv (T) dT − e(v, T) = e0 + v T0

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where cv denotes the heat capacity at constant volume, e the internal energy and e0 the heat of formation. Together with the fundamental identity of gas dynamics (1) we conclude s(v, T) = s0 +

T

T0

1 cv (T) dT . T

3.2 Mixture Region Since we want to apply the van der Waals EOS to gases and liquids as well, there is a need for describing what happens in the region of phase transition. In Figure 3 three isotherms are plotted in the p-v diagram, corresponding to (i) a temperature above the critical temperature Tc , (ii) the critical temperature (the critical isotherm) and (iii) a temperature below Tc . In the mixture region, liquid and gas are present in a continuously changing fraction where the fraction of liquid increases when the volume decreases. However, the original van der Waals equation shows an unphysical inclination in this area. This can be seen if we consider an isotherm for a temperature below the critical one. We observe two local extrema, called spinodal points. They correspond to the endpoints of supersaturated vapor (maximum) and overexpanded liquid (minimum), respectively. Between these two points the derivative pv (v, T) is positive and, hence, physically excluded. In order to provide physically meaningful data, we calculate the equilibrium pressure p ¯=p ¯(T), see Figure 3, as a function of the temperature using the Maxwell’s construction principle: 1.4

p/pc T>Tc

1.2

1

T=Tc

0.8

pe/pC

T
0.6

phase boundary 0.4

0.5

1

1.5

2

2.5

3

3.5

4

v/vc

Fig. 3. Isotherms of a BZT-Fluid in p-v diagram.

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143

¯ v2

¯ v1

p(v, T) dv = p ¯ (¯ v2 − v ¯1 ),

p(¯ v1 , T) = p(¯ v2 , T) = p ¯.

By means of this system we determine for any T ≤ Tc the equilibrium pressure p ¯ and the corresponding specific volumes v ¯1 and v ¯2 characterizing the phase boundary corresponding to saturated vapor and saturated liquid, respectively. In order to determine internal energy and entropy inside the mixture region we assume equilibrium. Hence, the corresponding states can be calculated by the convex combination of the equilibrium states at the phase boundary, see [Voß04] for details on the implementation. Note, that by this ansatz, we exclude the possibility of so-called metastable states.

4 The Riemann Problem of Gas Dynamics In order to investigate analytically nonclassical wave phenomena such as expansion shocks and compression waves as well as wave splitting, we consider the Riemann problem for gas dynamics. In Lagrangian coordinates this problem can be formulated as qt + f(q)x = 0 with piecewise constant initial data q(0, x) =

ql : x < 0 . qr : x > 0

Here the vector of conserved quantities q and the flux f(q) are determined by T

q = (v, u, E) ,

T

f(q) = (−u, p, pu) ,

with the specific volume v, the velocity u, pressure p, the total energy E = e + 0.5 u2 and the internal energy e. This system is closed by a general EOS p = p(v, s),

pv (v, s) < 0.

(7)

Here we assume that the isentrope is a strictly monotone function in order to ensure strict hyperbolicity of the underlying system of equations. According to standard assumptions of equilibrium thermodynamics we may express equivalently the pressure also in terms of specific volume and internal energy. By now, the Riemann problem is well understood for the Euler equations that model equilibrium hydrodynamics. For an overview we refer to the review article by Menikoff and Plohr [MP89]. The general construction principle for the Riemann solution is essentially based on the scale–invariance of the solution and the hyperbolicity of the governing equations of fluid motion. These properties require the solution to be composed of different waves in the time–space continuum which correspond to different characteristic velocities. Moreover, there exists a one–to–one correspondence between a single wave in

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the t–x diagram and the states in the phase space which can be connected by this wave. All of these states are lying on one-parametric curves and the solution of the Riemann problem is determined by the intersection points of the different curves each corresponding to a characteristic field in the phase space connecting the two initial states. Hence, the most crucial point in solving the Riemann problem analytically is the construction of the curves in phase space. To distinguish the respective settings in the course of the discussion we will consistently refer to one–parameter families of states in phase space as curves while speaking of waves in the x–t plane. 4.1 Characterization of Wave Curves In order to characterize the different curve types we first introduce the characteristic speeds λk (q), k = 1, 2, 3, as the kth eigenvalue of the Jacobian of f(q) which turn out to be % % λ1 = − −pv (v, s) = −c/v < λ2 = 0 < λ3 = + −pv (v, s) = c/v

where c denotes the sound speed defined in (4). The corresponding right eigenvectors are given by rk =

1 T (−1, λk , p + λk u) , λk

k = 1, 3

T

r2 = (−pe (v, e), 0, pv (v, e)) . Then the corresponding characteristic k-field is characterized by the variation of the characteristic velocity λk in the direction of the corresponding kth right eigenvector of the Jacobian of f(q), i.e., αk := ∇λk rk ,

k = 1, 2, 3.

These are determined by α1 = α3 = Γ/v, α2 = 0 where Γ denotes the fundamental derivative of gas dynamics defined in (3). Obviously, for a convex EOS, i.e., Γ > 0, we distinguish between two cases, namely, the k-field is linearly degenerated, i.e., αk (q) = 0 for all q, or the k-field is genuinely nonlinear, i.e., αk (q) = 0 for all q. In case of a non-convex EOS the nonlinear fields may locally degenerate, i.e., there are states q ¯ in q) = 0. phase space such that αk (¯ 4.2 Construction of an Exact Riemann Solution. Next we give an idea on how to construct a Riemann solution for the Euler equations if non-genuine nonlinear fields and phase transition come into play.

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To focus on the main principles we restrict our characterization to some special cases, clarified by means of a comprehensive example. A complete description containing all cases for the Euler equations augmented with an EOS of van der Waals-type can be found in [Voß04]. In particular, we skip a discussion on uniqueness and existence and assume, that there is always a unique entropy solution. 4.3 Wave Curves In general, the entropy solution of a Riemann problem for a hyperbolic system of m conservation laws can be constructed by considering the so-called wave curves Wk : R → Rm , k = 1, . . . , m. These are parameterized curves connecting some initial state qk = Wk (0) with all “reachable” states. Then the solution for a Riemann problem with initial data ql and qr is found, if there is a vector of parameters ξk , k = 1, . . . , m, connecting ql and qr , here from left to right, by means of the m wave curves, i.e., Wm (. . . W2 (W1 (ql , ξ1 ), ξ2 ), . . . , ξm ) = qr . In other words, each wave curve except the first one, starts at some intermediate state of the previous one and the crucial point is to find these states qk , respectively the parameters ξk , cf. Figure 4. In case of the Euler equations the problem can be reduced to the 1- and the 3-curve, because the 2-wave is always a contact discontinuity, where the pressure p and velocity u are constant. Therefore, the construction principle in this case simplifies as follows: Compute the 1-wave, i.e., collect all states which can be reached from ql . Simultaneously compute the 3-wave backwards, i.e., collect all possible intermediate states from where one can reach the end state qr . The intersection point of the two waves projected onto the p-u plane is the origin of W3 . Thus the parameters ξ1 and ξ3 and the solution respectively, are known.1

q 4=q r q 2=W 1(q 1,ξ1) W2

W3

W1 q 3=W 2(q 2,ξ2) q 1=q l

Fig. 4. Construction of the Riemann solution by means of wave curves. 1 For the sake of completeness, the parameter ξ2 can be reconstructed from the known intermediate states.

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W. Dahmen, S. M¨ uller, A. Voß 2

ξk=ξk

2

q k=q k+1

W 2k ξ

1 k

q 1k

W 1k 0

0

0=ξk q k=q k

Fig. 5. Wave curve partitions of Wk .

4.4 Computation of Wave curves At this point it remains to compute the wave curves. Using a convex EOS like the ideal gas law leads to wave curves composed of exactly one rarefaction and one shock branch. Here either side of the wave curve is referred to as a wave branch. By changing the EOS to a non-convex one, for instance, to the van der Waals EOS, applied to BZT fluids, regions in phase space arise where the sign of the fundamental derivative becomes negative and the nonlinear k-fields degenerate. Regarding the wave branches this may cause the break-down of the classical curve types rarefaction and shock and implies various facts: firstly, a wave branch can be composed of so-called wave parts, that is a section of the wave branch with one wave type. Then the parameter range of the branch [0, ξk ] is partitioned into smaller parts [ξkj , ξkj+1 ]. The situation is shown in Figure 5. Secondly, the classical wave types are not sufficient to continue the wave curves in an admissible way. Therefore, a new part type composite has to be introduced, taking sonic shocks and rarefactions (see below) into account. The concept of composite waves was introduced first by Wendroff [Wen72a], [Wen72b] and Liu [Liu75]. Thirdly, in case of composites away from the phase boundaries, the wave curves are no longer twice but only once differentiable and at phase boundaries only continuous. 4.5 Wave parts A wave curve, or, more precisely, a wave branch may be composed of several wave parts. Each part is of either the type rarefaction R, shock H or composite C. They are defined, in short, as follows. Rarefaction Curves For the nonlinear fields, here k = 1, 3, we define the parameterized curve Rk as the solution to the ODE ∂Rk (¯ q, ξ) = rk (Rk (ξ)), ∂ξ

Rk (¯ q, 0) = q ¯,

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147

i.e., as integral curve along the right eigenvector through a state q ¯. If λk is not decreasing along these curve, i.e., dλk /dξ ≥ 0, it is equivalent to the classical rarefaction curve. Such a curve is called admissible and it collects all states which can be connected to q ¯ by a centered rarefaction fan. Note, that when λk decreases along a wave part being part of a Riemann solution, thus would lead to (prohibited) folding of the solution in the x-t plane. Similarly, the usual parameterization ξ = x/t is not appropriate when dλk /dξ vanishes along Rk .2 In case of the Euler equations a rarefaction curve parameterized by the pressure is obtained by solving dq =

1 rk dp, λk

from a given state q ¯. Using the fundamental identity of gas dynamics (1), we conclude that the entropy is constant along Rk , as long as it is admissible. Hence, a projection of Rk onto the p–v plane depicts the isentropes. It can be shown, that the admissible branch of the rarefaction curve R1 corresponds to the expansion branch (dv > 0, dp < 0) in case of positive nonlinearity (Γ > 0) and the compression branch (dv < 0, dp > 0) in case of negative nonlinearity (Γ < 0). The roles of positive and negative nonlinearity are reversed for the 3-field. Moreover, pressure and velocity are monotone functions inside the compression (expansion) branch of the rarefaction curve. We emphasize that the above conclusions hold true as long as the rarefaction curve stays smooth. However, if the isentrope crosses the phase boundary, see Figure 2, then the rarefaction curve suffers a convex kink at the saturated-liquid curve and, in addition, in the case of BZT fluids a nonconvex kink at the saturated-vapor curve. Hence, the sound speed and the characteristic speed, respectively, jump at the phase boundary. Shock Curves The set of discontinuities satisfying the Rankine-Hugoniot conditions is given by the Hugoniot locus q) := {q : ∃ σ ∈ R s.t. Hk (¯

f(q) − f(¯ q) = σ (q − q ¯)} ,

where σ = σ(q, q ¯) denotes the shock speed. For the Euler equations in Lagrangian coordinates the Rankine–Hugoniot conditions for the nonlinear k-fields read σ2 = − 2

∆p , ∆v

∆e + (

∆p +p ¯)∆v = 0 . 2

(8)

As a technical remark: it is important for the construction principle to know the sign of the characteristic ﬁelds behind such a state with dλk /dξ = 0. Hence the parameterization must not depend on ξ = x/t.

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Here, we use the notation ∆A = A − A¯ for an arbitrary quantity A. In particular, σ = 0 or, equivalently, ∆u = 0, ∆p = 0. In order to determine the admissible branch of the Hugoniot curves, we consider the entropy jump. For a weak shock the entropy jump ∆s is proportional to the third power of the pressure jump ∆p, i.e., ∆s =

, c2 ¯ ¯ TΓ (∆p)3 + O (∆p)4 , 3 6v

see [Tho72]. According to the second law of thermodynamics, the entropy jump has to be positive, i.e., ∆s > 0, when the shock passes. Hence, the pressure jump has to be positive (Γ¯ > 0) and negative (Γ¯ < 0), respectively, to fulfill the entropy condition. From this we conclude that for Γ¯ > 0 a compression shock (∆p > 0, ∆v < 0) is admissible and for Γ¯ < 0 an expansion shock (∆p < 0, ∆v > 0). In order to analyze the uniqueness of the Riemann problem Smith [Smi79] introduced several conditions on the EOS referred to as strong, medium and weak condition, respectively. Menikoff and Plohr [MP89] verified that they are also important for discussing monotonicity of thermodynamic and hydrodynamic quantities along the Hugoniot locus. In particular, they proved that the following holds at a point on the compression branch of the Hugoniot locus, in the direction of increasing shock strength: (i) v decreases monotonically if the strong condition holds; (ii) e and u increase monotonically if the medium condition holds and (iii) p increases monotonically if the weak condition holds. So far, all materials known satisfy the weak condition. Therefore it makes sense to parameterize the wave curves by the pressure. Moreover, it can be proven that the entropy s is an extremum, if and only if the shock speed σ is an extremum. This does hold true at a point on a shock curve where it is smooth, see [MP89], Theorem 4.2. A variant is true even at points where the Hugoniot locus intersects the phase boundary. In the latter case the entropy s increases with shock strength if and only if the shock speed σ increases with shock strength, provided that the weak condition holds at a point on the compression branch of a shock curve, see [MP89], Theorem 4.5. Hence, the entropy as well as the shock speed vary monotonically along the admissible branch as is predicted by Liu’s extended admissibility relations in [Liu76]. It is extremal, if and only if the shock becomes sonic. This coincides with the state where the second law of thermodynamics is violated marking the end of the admissible branch. Hence, we can use the variation of the shock speed as admissibility criterion of the shock. Composite Curves As we discussed above the nonlinear fields may degenerate at states where the fundamental derivative of gas dynamics vanishes, i.e., Γ = 0. Beyond this state the rarefaction curve is no longer admissible. In order to continue the

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149

Fig. 6. Composite curve.

wave curve, Wendroff [Wen72b] introduced the notion of a composite curve. Later on, Liu extended this concept to general hyperbolic conservation laws, see [Liu75]. Following Liu’s definition, the composite locus is determined by Ck (Rk , q ¯) = { q : ∃ q∗ ∈ R, q ∈ H(q∗ ) and q is the first state on H(q∗ ) with λk (q∗ ) = σ(q∗ , q) }. Since the shock is as fast as the characteristic speed of state q∗ , each state q on Ck represents a wave in the x-t plane combined of a rarefaction up to q∗ and an adjacent shock from q∗ to q, see Figure 6. This locus is a one parameter family of states. In case that q ¯ corresponds to a state where Γ vanishes, it can be proven that the composite curve is tangentially attached to the rarefaction curve Rk at q¯, see [MV01]. The corresponding curve is admissible as long as (i) σ(q∗ , q) = λk (q) and (ii) q∗ is not the origin of Rk , cf. [Liu75]. 4.6 Admissible conditions Knowing the construction principle, as well as the mathematical conditions on the wave parts, we want to collect the conditions for changing the wave types during the computation of the wave branch, or, raise the question, how long is a wave part admissible? We shall see, that due to phase transition additional conditions arise. No Phase transition. As mentioned before, a necessary condition for a rarefaction part to be admissible is that the wave speed λk increases along the wave part. Otherwise the solution would fold in the x-t plane. Increasing λk is tantamount to a fixed sign of the fundamental derivative Γ and if the rarefaction part never crosses a region where Γ changes its sign, the wave type does not change. At a state with vanishing derivative of λk , i.e., Γ = 0, the rarefaction part ends and a new composite part begins. A similar speed condition has to hold along the shock part, namely, the shock speed σk has to decrease along the Hugoniot curve, which is the extended Liucondition. It reduces to the well–known Lax criteria in case of a convex EOS. As mentioned before, if σk does not decrease anymore, the wave curve must have entered a region of negative nonlinearity before. Then the shock part

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W. Dahmen, S. M¨ uller, A. Voß

ends here and the wave branch can be smoothly continued by a rarefaction part. The shock is called sonic because it travels with the same speed as the first state on the following rarefaction. The new type composite is a set of shock states. In the current context such a wave part starts at a state q ¯ where Γ = 0, and the wave part before must be a rarefaction part. Like a shock part, the composite part is admissible as long as the shock speed along the composite part is decreasing. If this is violated the next wave part is a rarefaction part again and the situation is similar to a sonic shock. It is known that the associated states at the preceding rarefaction part R are moving towards the origin of the rarefaction part. This leads to the other terminating condition for the composite part, i.e., the associated rarefaction state is the origin of R. Then the wave curve can be continued with a shock part where the shock origin is the origin of R. Including Phase transition. So far we described the non-convex situation where the wave curves are smooth. It becomes even more complicated if the wave curves crosses a phase boundary. The good news is that, like in the smooth non-convex case, three types of wave parts suffice to construct a Riemann solution. The bad one is that the discontinuity of the characteristic speed λk as well as the possible change of the sign of nonlinearity across a phase boundary causes additional terminating conditions for wave parts. This is directly related to the question how to continue the wave branch. These topics are treated in mathematical detail in [Voß04]. Here we can only give a summary of the results by means of a table containing the different cases and terminating conditions, see Table 1. As we mentioned above the monotonicity of the wave speed, denoted by s′ , i.e., λ′k or σk′ , along the curve, is one of the main criteria to continue or terminate a wave part. One can find it in the first part of Table 1, the admissible criteria. They are in effect while a wave part is computed. In some cases the wave speed s or its derivative s′ jumps while crossing the phase boundary and it makes sense to speak of a quantity before (·)− and behind (·)+ the boundary, depending on the parameterization. In the end there is a final terminating condition. It emerges if a wave takes over another wave. This happens, for instance, if a shock wave crosses a phase boundary, the jump in shock speed is positive and behind the boundary the wave curve continues with a second shock part, see case H[-] . Since the jump is positive and the shock speed is decreasing along the second part, it may happen, that the speed coincides with the last shock speed of the first shock part. In this case the solution cannot consist of two shocks anymore but the first shock part will be continued and it takes over the second one. Otherwise it would result in a folding of the second shock in the x-t plane. This means, there are terminating conditions depending on the history of the wave curve, respectively on wave speeds passed on previous wave parts. To handle these speeds appropriately the idea of a stack is helpful. Such a stack works like a book pile. One can put some book on top of it (push) or remove some from the top (pop). Now some conditions, namely, the one where the

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151

Table 1. Terminating conditions for wave parts. types of wave parts and conditions in eﬀect R H C

s′ > 0, curve is smooth s′ < 0, s = s[old] s′ < 0, s = s[old] , origin of corresponding R is not reached

abbreviations for terminating conditions

succ. remark

R[m] R[++] R[+-] R[-+] R[--] R[e]

monotonicity (m) of s broken, i.e., s′ = 0 phase boundary crossed, s′+ > 0, s+ > s− phase boundary crossed, s′+ > 0, s+ < s− phase boundary crossed, s′+ < 0, s+ > s− phase boundary crossed, s′+ < 0, s+ < s− external (e) condition fulﬁlled, i.e., a given state is reached or a maximum number of states is computed.

C R C H C

H[m] H[+] H[-] H[R] H[H] H[C] H[e]

monotonicity (m) broken, i.e., s′ = 0 phase boundary crossed, s′+ > 0 phase boundary crossed, s′+ < 0 overtaken by R, i.e., s = s[old] overtaken by H, i.e., s = s[old] overtaken by C, i.e., s = s[old] external (e) condition fulﬁlled

R R H C H C

push s− push s− push s− pop pop pop

C[m] C[+] C[-] C[H] C[C] C[b] C[e]

monotonicity (m) broken, i.e., s′ = 0 phase boundary crossed, s′+ > 0 phase boundary crossed, s′+ < 0 overtaken by H, i.e., s = s[old] overtaken by C, i.e., s = s[old] begin (b) of corresponding R reached external (e) condition fulﬁlled

R R H H C H

push s− push s− push s− pop pop

push s− push s−

wave speed jumps, push speed on the stack while all overtaking cases pop speed. Consequently, the current shock and composite speeds have to be compared to the speeds on the stack, and this is denoted by s = s[old] in the first part of the table and by s = s[old] in the second part. 4.7 Example We would like to give an example to demonstrate the interaction between different wave parts along one wave curve. Since in case of the Euler equations both nontrivial waves are treated similarly, we can confine the discussion to only one family, here k = 1. The EOS used is the van der Waals one and the fluid is the BZT fluid PP10. The wave curves are parameterized by the pressure.

152

W. Dahmen, S. M¨ uller, A. Voß

Figure 7(a) shows the wave curve projected into the phase diagram, i.e., the p-v plane. Here we can see which physical regions are crossed and which types of phases and nonlinearities are present at every state and part of the curve. Figure 7(b) specifies the wave speed along the curve. This clarifies the terminating condition for a wave part and in which way the wave curve is continued. Figure 8 reflects the evolution of the solution structure (here: pressure) along the wave curve in space at a constant time. Note, that the solution can consist only of one shock, even if the wave curve is composed of various wave parts of different type. To this end we fix one part of initial Riemann data, ql , and vary the right state qr = qir . In each case, we calculate the exact solution, and compare it with a numerical one, obtained by the WAF method. The wave curve starts at ξk0 near the critical point in the liquid phase and is calculated with decreasing pressure as the curve parameter, cf. Figure 7(a). The first part is the rarefaction part R1 . Here the wave speed increases along the curve, cf. Figure 7(b), up to the point ξk1 . It terminates, because the wave curve crosses the saturated-liquid phase boundary, that can be seen at best in the small zoomed window in Figure 7(a). Hence, the solution in the x-t plane, corresponding to the right state qr = q1 taken from R1 , is a rarefaction wave, cf. Figure 8(a). The wave is very fast and looks like a shock because the isentropes are very steep in this region, but it is still smooth. At the phase boundary the corresponding eigenvalue is discontinuous. Thus the speed jumps, cf. Figure 7(b), but to a higher value, implying that the wave curve is continued with a second rarefaction wave, R2 . Figure 8(b), calculated for qr = q2 , shows, that the discontinuity in the wave speed leads to a splitting of the two rarefaction parts. Inside the mixture region the isentrope remains convex and the rarefaction part R2 lasts up to the saturated-vapor boundary, that is at ξk2 . Now the wave speed is decreasing and the next wave part is the composite part C2 corresponding to the rarefaction part R2 from inside the mixture. We expect a Riemann solution for data qr = q3 to split into two waves and the right one to be composed of a rarefaction and an adjacent shock, like in Figure 8(c). The composite part ends at the state ξk3 , because we reach the beginning of the rarefaction part R2 with the corresponding rarefaction states. It can be seen from the wave speed diagram that this is exactly the case when the speed at ξk3 is equal to the speed of the first state on R2 . In a certain sense the gap in the speed is by-passed with a shock part H1 from ξk3 to ξk4 . This is in agreement with the observation that along the last composite part C2 the exact solutions of the type as shown in Figure 8(c) are combined of a shock with increasing strength and a vanishing rarefaction part. Here the shock part continues this evolution. In the x-t plane, the solution to qr = q4 is still split into two waves, but the splitting itself decays, i.e., the wave speeds are approaching, cf. Figure 8(d). In the end the speed of the shock part reaches the speed of the first rarefaction part R1 at ξk4 , cf. Figure 7(b).

Riemann Problem for the Euler Equation

153

Table 2. Wave parts, Example 1. parameter range wave part terminating condition ﬁgure [ξk0 , ξk1 ] R1 R[++] 8(a) 1 2 [ξk , ξk ] R2 R[--] 8(b) C2 C[b] 8(c) [ξk2 , ξk3 ] 3 4 [ξk , ξk ] H1 H[R] 8(d) C1 C[b] 8(e) [ξk4 , ξk5 ] H2 H[m] 8(f) [ξk5 , ξk6 ] R3 R[e] 8(g) [ξk6 , ξk7 ]

Now, if not terminated, the shock would be faster than the last rarefaction states, and this means a folding in the x-t plane. Hence we continue the wave curve with a composite part again, C1 , now corresponding to the first rarefaction part R1 . Such a solution is given for the Riemann data qr = q5 in Figure 8(e). Like before, the corresponding rarefaction R1 is eaten up by the composite states and we continue the wave curve from the point, where the speed of the first state of the rarefaction is reached, here at ξk5 , with a new shock part H2 . The speed is decreasing and a solution to one of these shock states qr = q6 is provided by Figure 8(f). Since the wave curve crossed the region of negative nonlinearity before, the shock may become sonic. This happens at ξk6 . The following rarefaction part R3 is adjacent to the last shock state and because there are no further anomalies in phase space the wave curve remains a rarefaction part for all parameter greater than ξk6 . Hence, the last possible wave curve type is shown in Figure 8(g).

wave curve phase bnd. Γ=0 wave par.ξjk i RP.q r=q r

1.1 0 1

1

ξk ξk ξ2 3 kξ q k 2 3 k ξ4k

s

wave speed wave par.ξjk i RP.q r=q r

2

ξk

-7500

pr

23

ξ3k

4

4

5

5

ξk

ξ0k 1 ξk 0.8 1

-10000

0.9

6

ξk

0.7

ξ2k ξ3

2

6

5 7

ξ5k

ξ0k q k

ξk 7

3

1.5

(a) Wave curve.

2

2.5

3

ξk

1

k

0.6 1

7

ξ4k

ξ1k

6

ξk

7

6

vr

1

1.5

(b) Wave speed. Fig. 7. Wave characteristics, Example 1.

2

2.5

3

vr

154

W. Dahmen, S. M¨ uller, A. Voß p

p exact WAF

ql

0.997

exact WAF

ql 0.995

0.99 0.996 0.985

q -0.65

-0.6

-0.55

-0.5

1 r

x

-0.45

(a) parts: R

q 2r -0.65

-0.6

-0.55

-0.5

x

-0.45

(b) parts: R-split-R

p

p exact

ql

1

exact

ql

WAF

0.995 0.99

WAF

0.98

0.985 0.96 0.98

q 4r

q 3r

0.975 -0.65

-0.6

-0.55

-0.5

x

-0.45

0.94 -0.65

-0.6

-0.55

-0.5

x

-0.45

(d) parts: R-split-H

(c) parts: R-split-R-sonic-H p

p exact

ql

1

exact

ql

1 WAF

WAF

0.9 0.95 0.8

q 5r

0.9 -0.65

-0.6

-0.55

(e) parts: R-sonic-H

-0.5

x

-0.45

q 6r

0.7 -0.65

-0.6

(f) parts: H Fig. 8. to be continued

-0.55

-0.5

x

-0.45

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155

sequel to Fig. 8. p exact

ql

1

WAF 0.9

0.8

0.7

q 7r 0.6 -0.65

-0.6

-0.55

-0.5

x

-0.45

(g) parts: H-sonic-R Fig. 8. Pressure in space for diﬀerent Riemann problems at constant time.

5 Numerical Results We consider two types of numerical experiments where phase transition occurs in the flow field. First, we investigate a liquefaction shock expanding from a shock tube into an observation chamber. Another configuration concerns the collapse of a planar bubble near a rigid wall. The two configurations are shown in Figure 9. The numerical simulations are performed by the solver QUADFLOW, see [BGMH+ 03]. This is a finite volume solver for block-structured grids. In each block we perform grid adaptation by means of a multiscale analysis. The convective fluxes are determined by solving quasi–one dimensional Riemann problems at the cell interfaces. Here we use the Roe Riemann solver adapted to real gas. The spatial accuracy is increased by a linear, multidimensional reconstruction of the conservative variables. In order to avoid oscillations in the vicinity of local extrema and discontinuities, limiters with TVD property are used. The time integration is performed by an explicit multistage Runge– Kutta scheme. 5.1 Expansion of a Liquefaction Shock The setting of this configuration is sketched in Figure 9(a). The fluid under consideration is the high-molecular fluorcarbon FC75. In a shock tube of radius r = 0.288 m a “liquefaction shock” is generated. Here the pre-shock state q0 and the post-shock state q2 are chosen such that they are lying on the same Hugoniot curve which crosses both phase boundaries connecting the state q2 in the liquid phase by the state q0 in the vapor phase. In Figure 10 the wave curve and the initial states are shown in the p-v-diagram. The liquefaction shock is running to the right and is expanding into the observation chamber. This results in a diffraction of the planar shock wave. It has been

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W. Dahmen, S. M¨ uller, A. Voß

0.08

Rigid Wall

y [m] radius: 0.01 m distance to wall: 0.0005m

0.06

7

pout = 5.73 * 10 N/m ρout = 200 kg/m3 Tout = 900 K

Mn0 = 1.86 p0 = 321400 [Pa]

T2 = 463 [K]

0.04

T0 = 430 [K]

u2 = 148 [m/s]

2

7

pin = 1.56*10 3 ρin = 200 kg/m Tin = 595K

u0 = 0 [m/s]

rigid wall

p2 = 1285600 [Pa]

0.02

Rigid Wall

0

(a) Expanding liquefaction shock

0

0.02

0.04 x [m] 0.06

0.08

(b) Planar bubble collapse

Fig. 9. Initial conﬁgurations.

investigated experimentally, see [Tho91]. According to Section 3 we apply the van der Waals EOS with Maxwell’s construction principle. The critical values of the fluid FC75 are vc = 0.0017 m3 /kg, Tc = 500.21 K, pc = 0.1607×107 Pa. Notable features of the axisymmetric, unsteady flow are the strong vortex and the Prandtl-Meyer expansion characteristics. An internal shock originates near the vortex center. This is depicted in Figure 12(d). The structure of the extreme expansion fan is referred to as the “Mach trumpet”, see [Tho91]. The main compression shock forms the front of the trumpet whereas the sides are formed by a rarefaction shock. The rarefaction shock is of the mixture-evaporation type. It is associated to the kink of the isentropes at the saturated-vapor boundary. To validate this we extract data along a particle path crossing the shock front and project the corresponding states in the p-v diagram, see Figure 11. The streamline is depicted from Figure 12(d). In addition, we present in Figure 12(e) the different regions in the flow field where the fluid undergoes a phase transition. At the phase boundaries the sound speed jumps due to the kink of the isentropes in the p-v diagram. These discontinuities can be depicted from the Mach contours in Figure 12(c). pr

wave curve phase boundary Γ=0

1

v/vc p/pc

12

0.9

11

0.8

10 0.7

9 8

q2

0.6 0.5

6 5

p/pc

v/vc

7

0.4

4

0.5

0.3

3 2

0.2

1 0.1

q0

1

2

3

4

5

6

7

8

9 10

0

vr

Fig. 10. Wave curve with initial data.

A

liquid 0

mixture

B

vapor 0.1

Cx

0.2

D

E

0.3

0

F

Fig. 11. Data extracted from streamline.

Riemann Problem for the Euler Equation

157

5.2 Bubble Collapse near a rigid Wall We consider a cylindrical bubble filled with water steam surrounded by water near a planar solid wall. The water is at high pressure and high temperature. The fluid inside the bubble is at low pressure (below vapor pressure) and lower temperature corresponding to wet steam, i.e., the state lies inside the mixture region. Both states are at constant volume. This configuration is sketched in Figure 9(b). Here the fluid water is modeled by the van der Waals EOS applying Maxwell’s construction principle modeling the mixture region. The critical values are vc = 0.0018016 m3 /kg, Tc = 647.4 K, pc = 0.2212 × 108 Pa. In the following we focus on the dynamics of the different occurring waves and their interactions. Characteristic instants are sketched in Figures 13(a) – (r) representing the pressure and density. Note that the color scaling differs for each figure due to the variation of the extrema. The initial conditions correspond to a Riemann problem where three types of waves occur: an inward running compression shock, an outward running rarefaction wave and a contact discontinuity. The outward running rarefaction is visible in the pressure as well as the density whereas the contact discontinuity is only seen in the density. At time t = 3.01 µs the rarefaction wave is reflected at the rigid wall. Due to a superposition of the rarefaction wave and the reflected rarefaction wave a low pressure region develops between the bubble and the wall. The reflected rarefaction wave is running over the contact discontinuity and the shock wave. Due to the pressure drop behind the reflected rarefaction wave the fluid is accelerated towards the wall and, hence, the bubble starts moving towards the wall and the bubble surface is deformed, see Figure 13(c)–(h). Since the acoustic impedance is about three times higher in the water phase than in the wet steam phase the rarefaction wave is running faster than the shock wave. At time t = 13.93 µs the inward running shock has reached the center of the bubble and is reflected, see Figure 13(g), (h). Behind the shock the pressure increases above vapor pressure and the wet steam vaporizes. The reflected shock wave is running over the contact discontinuity, see Figure 13(i), (j), and the fluid behind the shock is accelerated in outward direction. Therefore mass is transported away from the bubble center and a low pressure region develops near the bubble center, see Figure 13(k), (l). At time t = 28.30 µs parts of the reflected shock are reflected at the wall. Due to the curved shock front the angle condition is violated and, hence, a Mach stem is developing. The Mach stem becomes stronger with increasing time propagating away from the symmetry line, see Figure 13(k)–(r). Again the reflected shock wave is running over the bubble and the mass behind the wave accelerated in the direction of the moving shock front. Therefore the fluid is accelerated away from the wall which leads to an additional deformation of the bubble shape compensating for the previous deformation. Finally, the bubble shape becomes almost symmetric again. In particular, the bubble does not interact with the wall and we do not observe a collapse of the bubble.

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W. Dahmen, S. M¨ uller, A. Voß

Acknowledgement. The authors would like to thank Prof. Dr. Josef Ballmann and Dipl.-Phys. Sigrid Andreae, Lehr- und Forschungsgebiet f¨ ur Mechanik, RWTH Aachen, for the fruitful cooperation within the DFG-Priority Research Program Analysis and Numerics for Conservation Laws. In particular, the discussions on the physical modeling have been very helpful.

(a) Pressure contours

(b) Density contours

(c) Mach contours

(d) Streamlines and pressure ﬁeld

(e) Fluid phases: liquid (blue), wet steam (green), vapor (grey) Fig. 12. Expansion of a liquefaction shock. (See also color ﬁgure, Plate 12.)

Riemann Problem for the Euler Equation

p [N/m2]

y

x

(a) t = 3.01 µs

(b) t = 3.01 µs p [N/m2]

y

ρ [kg/m3]

y

x

x

(d) t = 6.66 µs

(c) t = 6.66 µs p [N/m2]

y

ρ [kg/m3]

y

x

x

(e) t = 11.50 µs

ρ [kg/m3]

y

x

159

(f) t = 11.50 µs Fig. 13. to be continued

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W. Dahmen, S. M¨ uller, A. Voß

sequel to Fig. 13. p [N/m2]

y

x

x

(h) t = 13.93 µs

(g) t = 13.93 µs p [N/m2]

y

ρ [kg/m3]

y

x

x

(i) t = 21.48 µs

(j) t = 21.48 µs p [N/m2]

y

ρ [kg/m3]

y

x

x

(k) t = 28.30 µs

ρ [kg/m3]

y

(l) t = 28.30 µs Fig. 13. to be continued

Riemann Problem for the Euler Equation

161

sequel to Fig. 13.

p [N/m2]

y

ρ [kg/m3]

y

x

x

(m) t = 30.30 µs

(n) t = 30.30 µs

p [N/m2]

y

y

3

ρ [kg/m ]

x

x

(p) t = 48.10 µs

(o) t = 48.10 µs

3

ρ [kg/m ]

2

p [N/m ] y

y

x

(q) t = 56.30 µs

x

(r) t = 56.30 µs

Fig. 13. Bubble collapse. (See also color ﬁgure, Plate 13.)

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References [BBKN83]

A.A. Borisov, Al.A. Borisov, S.S. Kutateladze, and V.E. Nakoryakov. Rarefaction shock wave near the crticial liquid vapor point. J. Fluid Mech., 126:59–73, 1983. uller, M. Hesse, Ph. Lamby, S. M¨ uller, [BGMH+ 03] F. Bramkamp, B. Gottschlich-M¨ J. Ballmann, K.-H. Brakhage, and W. Dahmen. H-Adaptive Multiscale Schemes for the Compressible Navier-Stokes Equations — Polyhedral Discretization, Data Compression and Mesh Generation. In J. Ballmann, editor, Numerical Notes on Fluid Mechanics, volume 84, pages 125 – 204. Springer, 2003. [DTMS79] G. Dettleﬀ, P.A. Thompson, G.E.A. Meier, and H.-D. Speckmann. An experimental study of liquefaction shock waves. J. Fluid Mech., 95(2):279–304, 1979. [Lam72] K.C. Lambrakis. Negative-Γ fluids. PhD thesis, Rensselaer Polytechnic Institute, 1972. [Liu75] T.-P. Liu. The Riemann problem for general systems of conservation laws. J. Diff. Eqns., 18:218–234, 1975. [Liu76] T.-P. Liu. The entropy condition and the admissibility of shocks. J. Math. Anal. Appl., 53:78–88, 1976. [MP89] R. Menikoﬀ and B.J. Plohr. The Riemann problem for ﬂuid ﬂow of real materials. Rev. Mod. Physics, 61:75–130, 1989. [MV01] S. M¨ uller and A. Voss. On the existence of the composite curve near a degeneration point. Preprint IGPM 200, RWTH Aachen, 2001. [Smi79] R.G. Smith. The Riemann problem in gas dynamics. Trans. Amer. Math. Soc., 249(1):1–50, 1979. [TCK86] P. Thompson, G. Carafano, and Y.-G. Kim. Shock waves and phase changes in a large-heat-capacity ﬂuid emerging from a tube. J. Fluid. Mech., 166:57–92, 1986. [TCM+ 87] P.A. Thompson, H. Chaves, G.E.A. Meier, Y.-G. Kim, and H.-D. Speckmann. Wave splitting in a ﬂuid of large heat capacity. J. Fluid Mech., 185:385–414, 1987. [Tho72] P. Thompson. Compressible Fluid Dynamics. Rosewood Press, Troy, New York, 1972. [Tho91] P. Thompson. Liquid–vapor adiabatic phase changes and related phenomena. In A. Kluwick, editor, Nonlinear waves in real fluids, pages 147–213. Springer, New York, 1991. [TK83] P. Thompson and Y.-G. Kim. Direct observation of shock splitting in a vapor-liquid system. Phys. Fluids, 26(11):3211–3215, 1983. [Voß04] A. Voß. Exact Riemann Solution for the Euler Equations with Nonconvex and Nonsmooth Equation of State. PhD thesis, RWTH Aachen, submitted 2004. [Wen72a] B. Wendroﬀ. The Riemann problem for materials with nonconvex equations of state, I: Isentropic ﬂow. J. Math. Anal. Appl., 38:454–466, 1972. [Wen72b] B. Wendroﬀ. The Riemann problem for materials with nonconvex equations of state, II: General ﬂow. J. Math. Anal. Appl., 38:640–658, 1972.

Radiation Magnetohydrodynamics: Analysis for Model Problems and Efficient 3d-Simulations for the Full System A. Dedner1 , D. Kr¨ oner1 , C. Rohde1 , and M. Wesenberg2 1

Mathematisches Institut der Albert-Ludwigs Universit¨ at Freiburg, Hermann-Herder Str. 10, 79104 Freiburg im Breisgau dedner|dietmar|[email protected] 2

Behr GmbH & Co. KG, Technologiecenter, Methodenentwicklung 3D Simulation, Heilbronner Str. 397, 70469 Stuttgart [email protected] Summary. The equations of compressible radiation magnetohydrodynamics provide a widely accepted mathematical model for the basic ﬂuid-dynamical processes in the sun’s atmosphere. From the mathematical point of view the equations constitute an instance of a system of non-local hyperbolic balance laws. We have developed and implemented numerical methods in three space dimensions on the basis of a ﬁnite volume scheme that allow the eﬃcient approximation of weak solutions. Key features are the use of eﬃcient Riemann solvers, a special treatment of the divergence constraint, higher-order schemes, the extended short characteristics method, local mesh adaption, and parallelization using dynamic load balancing. Moreover, methods to cope with the special nature of the atmosphere are included. In this contribution we give an overview of our work, highlight our most important results, and report on some new developments. In particular, we present a scalar model problem for which an almost complete analytical treatment is possible.

1 Introduction Since the possibilities for a direct observation of physical processes below and around the solar surface are limited, numerical simulations play an important role in obtaining a clearer understanding of solar phenomena. An example of a solar phenomenon not yet fully understood is the eleven year cycle in which the number of sun spots on the solar surface increases and decreases. Even the development of these sun spots themselves is an open question. One problem is that the filaments at the boundary of the sun spots are supposed to consist of magnetic fluxtubes that are formed about 2 · 105 kilometers below the solar surface in the lower convection zone of the sun. In this region direct observation is hardly possible so that the formation and evolution of the flux-

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tubes has to be studied by means of numerical simulations. These fluxtubes consist of a strong magnetic field which is aligned in a tube like structure. In these lower regions of the convection zone energy transport is achieved only through convective motion of the plasma. In the solar photosphere which is the region around the solar surface energy transport through radiation starts to play an important role in the energy balance of the plasma. To study for example magneto-convection in the photosphere the mathematical model has to be modified accordingly. In addition, the physical parameters are very different. While the supersonic flow in the photosphere can lead to shock waves, in the lower convection zone the Mach number is much smaller. The evolution of a compressible, electrically-conducting fluid as we find it in the scenarios described above is governed by the equations of magnetohydrodynamics. To take into account the energy exchange by radiation a family of radiation transport equations parameterized by the direction of radiation is coupled via the energy equation. We shall neglect relativistic effects and the effects of viscosity, resisticity, and heat conduction. In this case we are confronted with a (non-local) first-order system of balance laws which turns out to be hyperbolic. At least on the level of the most famous system of hyperbolic conservation laws, the equations of gas dynamics, the state-of-the-art numerical methods allow reliable and efficient computations in arbitrary space dimensions. It was the primary goal of our ANumE project to transfer these methods to the more complex equations of radiation magnetohydrodynamics. This transfer seemed to be managable since many of the above-mentioned methods (like local mesh adaption, construction of approximate Riemann solvers, parallelization,....) are independent of the underlying specific type of balance laws or, at least on a formal level, can be easily extended. In retrospect, this initial assumption was roughly correct. However, it turned out that in particular an efficient and sometimes even a reliable treatment of the system was not possible without deep changes or extensions of the basic methods described above. Especially in our cooperation with our partners from solar physics in the ANumE project of M. Sch¨ ussler many problems with the basic extensions of the above-mentioned methods became evident. While we concentrated mainly on purely mathematical issues of the applications, the developed methods and the software that we implemented have been used in the joint project to tackle real-world solar physics. In the following we cannot present all details of our numerical methods, but we want to give an overview of some key features which we list below together with bibliographical references; more details can also be found in [Ded03, Wes03]. Let us note that by a time-explicit treatment of the system it is possible to split the solution of the complete system of radiation magnetohydrodynamics into two modules; one for solving the magnetohydrodynamic part, and one for computing the radiation transport. • Most of the execution time of a Riemann solver based finite volume scheme for the MHD equations is spent on computing the numerical fluxes. We

Radiation Magnetohydrodynamics

•

• •

•

• •

•

•

165

developed a new Riemann solver, which after extensive tests turned out to be the most efficient one [DKRW01a, Wes02]. For a realistic treatment of the flow in the solar atmosphere it is necessary to consider a partially-ionized fluid. In this paper we describe both a strategy for directly extending Riemann solvers as well as an energy relaxation method to deal with general non-perfect fluids [DW01, DRW03a]. The equations for the magnetic field are accompanied by an additional constraint on its divergence which has to be taken into account on the discrete level [DKK+ 02, DKRW03, DRW03b]. For unstructured triangular grids in 2d we introduced a new reconstruction technique which yields a second-order extension of the base scheme. However, we found that in general local grid adaption provides a simpler way of enhancing the efficiency of a finite volume scheme [DRW03a, Wes03]. While a gravitationally-layered background atmosphere is stable as a solution of the hydrodynamical equations, this might not apply to its discretization which requires special attention in the numerical scheme [Ded03]. Typically, astrophysical problems are posed in unbounded domains. The selection of a bounded computational domain requires the derivation of suitable artificial boundary conditions [DKSW01a, DKSW01b]. While the evaluation of the numerical fluxes is the most time-consuming part of the approximation to the MHD equations, more than 70% of the computing time for the complete system is spent on the solution of the radiation transport equation. Therefore, we studied a new class of schemes that lead to an efficient discretization of the radiation transport equation [DV02]. The numerical methods have to be embedded in a powerful environment that allows to handle time-dependent locally-adapted meshes combined with parallel computing strategies for simulations in 3d [DRW99, DKRW01b, DRSW04]. For justifying the schemes and the mathematical model we performed rigorous analytical studies for simplified settings [DR02, DR04, DRW02]. Although this has not directly contributed to an enhancement of the efficiency of the schemes, it is the ultimate way to ensure their reliability.

Before we enter the topics of the list we will present the mathematical model and the base scheme in more detail in the remaining part of the introduction. 1.1 Mathematical Model The system of ideal magnetohydrodynamics (MHD) is derived by combining the Maxwell equations of electrodynamics and the Euler equations of gas dynamics. This system describes the motion of an electrically-conducting fluid in the presence of a magnetic field in a domain ΩT := Ω × (0, T ) ⊂ R3 × R+ . It consists of eight balance laws together with a set of algebraic relations and

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A. Dedner, D. Kr¨ oner, C. Rohde, and M. Wesenberg

a constraint equation on the magnetic field. The gravitational force which acts on the fluid is taken into account by source terms. The energy transport through radiation is modeled by means of the radiation source term Qrad (cf. [MM84]). ∂t ρ + ∇ · (ρu) = 0 T

∂t (ρu) + ∇ · (ρuu + P) = qρu T

T

∂t B + ∇ · (uB − Bu ) = 0 ∂t (ρe) + ∇ · (ρeu + Pu) = qρe + Qrad ∇·B=0 µ · ∇I + χI = χB

(conservation of mass),

(1a)

(cons. of momentum),

(1b)

(induction equation), (conservation of energy),

(1c) (1d)

(divergence constraint), (radiation transport).

(1e) (1f)

If the source terms on the right-hand side are neglected, the equations (1a)– (1d) form a hyperbolic system of conservation laws; the conserved quantities are the density ρ > 0, the momentum ρu ∈ R3 , the magnetic field B ∈ R3 , and the total energy density ρe > 0. The MHD system (1a)–(1d) is augmented by some algebraic relations. The total energy e is related to the internal energy ε > 0, the kinetic and the magnetic energy via 1 1 |B|2 . e = ε + |u|2 + 2 8πρ

(1g)

The pressure tensor combines the influence of the hydrodynamic and the magnetic pressure P=

p+

1 1 |B|2 I − BBT 8π 4π

(1h)

where I denotes the unit tensor. Equations of state (EOS) are used to close the system. The EOS defines the (hydrodynamic) pressure p > 0 as a function of the internal energy and the density. We also define the temperature θ > 0 in the same way p = p(ρ, ε)

(EOS for the pressure),

(2a)

θ = θ(ρ, ε)

(EOS for the temperature).

(2b)

The simplest equations of state are given for the case of a thermally and calorically perfect gas, i.e., if we use the equations p(ρ, ε) = (γ − 1)ρε c

and θ(ρ, ε) =

ε . cV

(3)

Here γ := cVp > 1 is the adiabatic exponent for the specific heat capacities cp and cV at a constant pressure and a constant volume, respectively. However, as we sketch in Sect. 3 our numerical schemes work for an arbitrary EOS, in particular for an EOS which models the partial ionization of the plasma in

Radiation Magnetohydrodynamics

167

the solar photosphere. For more details we refer to [DW01, DRW03a, Vol02, Ded03, Wes03]. The force of gravity leads to the following expressions for the source terms in the momentum and energy equations (1b) and (1d): qρu = ρg ,

qρe = ρg · u ,

where the vector-valued function g = g(x) is defined on Ω. To define the radiation source term Qrad (x, t) = χ(x, t) (I(x, t, µ) − B(x, t)) dµ

(4)

(5a)

S2

the transport equation (1f) for the radiation intensity I(x, t, µ) has to be added to the MHD system (1a)–(1d). Both χ and B are given functions of the density ρ and the temperature θ: χ(x, t) = ρ(x, t)κ(ρ(x, t), θ(x, t)) , 4

B(x, t) = B(θ(x, t)) = σθ(x, t) , (σ > 0) .

(5b) (5c)

By virtue of (2a) and (1g) the MHD system (1a)–(1d) can be written as a system of hyperbolic balance laws in the conserved variables U = (ρ, ρu, B, ρe)T ∈ U

(6)

with a suitable state space U ⊂ R8 . At time t = 0 the conserved variables are prescribed: , -T U(x, 0) = U0 (x) = ρ0 (x), (ρu)0 (x), B0 (x), (ρe)0 (x) ∈ U (x ∈ Ω).

Defining Qrad := (0, 0, 0, 0, 0, 0, 0, Qrad )T , q = q(U) = (0, qρu , 0, 0, 0, qρe )T the system (1) together with the initial conditions can be rewritten in the compact form ∂t U + ∇ · F(U) ∇·B µ · ∇I U(·, 0)

= q(U) + Qrad , =0, = χ(B − I) , = U0

(7)

for the flux vector F = (Fx , Fy , Fz )T which consists of the fluxes in the three coordinate directions. In accordance with (1e) the initial conditions have to satisfy ∇ · B0 = 0. Note that the flux in the induction equation (1c) can be rewritten as ∇ · (uBT − BuT ) = ∇ × (u × B), i.e., as the curl of a vector field. Therefore, taking the divergence of equation (1c) yields ∂t ∇ · B = 0. This shows that (1e) is a condition on the initial data of the magnetic field and not an additional elliptic constraint.

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1.2 Finite Volume Scheme We assume in the following that the computational domain Ω is decomposed into a grid T consisting of non-overlapping polygonal elements Ti for i ∈ I. Two neighboring elements Ti and Tj have a common interface Sij of codimension one. The index set of all neighboring elements of Ti is given by N(i). In the finite volume approach the conserved variables U are approximated by averaged values on elements of a grid T = {Ti : i ∈ I}. As a motivation we start from the integral form of the MHD system, i.e., the first line of (7) , q(U) + Qrad . (8) ∂t U + F(U) · n = Ti

∂Ti

Ti

Here n denotes the outer unit normal to ∂Ti . We define averaged values 1 1 Ui (t) := U(x, t), Qrad i (t) := Qrad (x, t) |Ti | Ti |Ti | Ti

for i ∈ I and a grid function Uh (x, t) := Ui (t)

for x ∈ Ti

(9)

which we use for approximating equation (8): d 1 1 Ui (t) = − F(Uh (·, t)) · n + q(Uh (·, t)) + Qrad i (t) . (10) dt |Ti | ∂Ti |Ti | Ti To arrive at a fully-explicit scheme which is suitable for simulations on a computer we introduce a few more approximation steps: • The computation of the radiation source term Qrad requires the approximation of the integral in (5a). On a given element Ti this leads to an approximation Qrad i (t). This is described in Sect. 5. • We use standard Runge–Kutta methods for approximating the time derivative and quadrature rules to approximate the integrals in (10). • Since Uh (·, t) is discontinuous across the cell boundaries Sij , the flux F(Uh (·, t))·n is not well-defined and has to be approximated by numerical flux functions gij (t) satisfying F(Uh (·, t)) · n = F(Uh (·, t)) · nij ≈ gij (t). (11) ∂Ti

j∈N(i)

Sij

j∈N(i)

This is described in Sect. 3.1. A first-order (piecewise-constant) approximation of U on Ω × [0, T ] is given by Uh (x, 0) = U0i for x ∈ Ti and Uh (x, t) = Un+1 i

(x ∈ Ti , tn < t ≤ tn+1 ) .

(12)

Radiation Magnetohydrodynamics

169

The values Un+1 at the time level tn+1 = tn + ∆tn are computed from the i n values Ui via = Uni − Un+1 i

∆tn gij (tn ) + ∆tn q(Uni ) + ∆tn Qrad i (tn ) . |Ti |

(13)

j∈N(i)

This method is formally first-order accurate in space and time. If the numerical fluxes gij are based on suitable piecewise-linear data as described in Sect. 3.4, an extension to (formal) second-order accuracy can be obtained by 1. applying (13) twice (with identical time step ∆tn ) for the updates n∗ n∗∗ Uni → Un∗ and i , Ui → Ui

2. defining Un+1 := i

1 n∗ (U + Un∗∗ ). i 2 i

2 Radiation Hydrodynamics: Analysis Starting point of this section is system (1). The existence and uniqueness of classical solutions, locally-in-time, for the initial value problem for (1) (with a fully time-dependent transport equation for the intensity) has been proven in [RZ03]. What about existence of solutions globally-in-time? If the radiation source term Qrad would not be present the equations in (1) were reduced to the equations for an ideal fluid. In this case global-in-time classical solutions cannot exist for all initial data even if the distance of the initial data to some constant state becomes small in, say W 1,∞ ([Liu77], see however [Gra98] for an interesting exception). It is a remarkable property of the coupled system (1) with Qrad that there exist global-in-time classical solutions close to equilibria (i.e., constant state vectors (ρ, ρu, ρe, I)T such that Qrad vanishes). This property has been observed numerically (cf. [Pom73]) and can be verified rigorously in one space dimension ([KNN98, RY]). We will not present the wellposedness theory for classical solutions of the full system (1). To exemplify the dissipative effects of radiation we restrict ourselves to a scalar model problem. In Sect. 2.1 we derive the model problem and present stability theorems, which are the basis for establishing error estimates for finite volume schemes. The latter topic is discussed in Sect. 2.2. 2.1 The Scalar Model Problem: Analysis Let us formally solve the linear equation (1f) for I(., µ) by the method of characteristics. We get an explicit formula for I(., µ) which we use in (5a) to compute Qrad . If we assume χ ≡ 1 the result in two space dimensions is

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A. Dedner, D. Kr¨ oner, C. Rohde, and M. Wesenberg

Fig. 1. The thick-line graphs display the numerical solution for (15), (16), (18) for a = 0.2 (left) and a = 1 (right). The thin-line graphs correspond to the solutions of the homogeneous Burgers’ problem and of the viscous problem (26), (16) with (18).

Qrad (x, t) := T[θ(., t)](x) :=

R2

φ(x) := exp (−|x|)/|x|.

φ(x − y)(B(θ(y, t)) − B(θ(x, t))) dy,

(14)

If we now lump the hydromechanical unknowns into some scalar function u : R2 × [0, T ] → R, T > 0, it is reasonable to consider the following problem as a model for the Cauchy problem for (1). ∂t u(x, t) + divf (u(x, t)) = T[u(., t)](x),

(x, t) ∈ ΩT ,

(15)

x ∈ R2 .

(16)

u(x, 0) = u0 (x),

By f = (f1 , f2 )T ∈ C 1 (R, R2 ) we denote the lumped flux. Actually, most of our results hold for more general operators. In the proof we only require that there is a function CT ∈ C 0 (R≥0 , R>0 ) and a constant CL > 0 such that T[w]L∞ (R2 ) + T[w]L1 (R2 ) ≤ CT (wL∞ (R2 ) )(wL1 (R2 ) + 1), T[w] − T[v]L1 (R2 ) ≤ CL w − vL1 (R2 ) .

(17)

Here w, v are in the space L1 (R2 ) ∩ L∞ (R2 ). More details on the derivation of (15) can be found in [Ded03, DR02, DR]. The equilibrium state of (15) is u ≡ 0. We conjecture that there are classical solutions which are defined globally-in-time provided the initial data (and its gradient) are small. This is confirmed by the results of a one-dimensional numerical experiment taken from [Ded03]. Fig. 1 shows an approximation of the time asymptotic solution for the initial data u0 (x) = sgn(−x)a

(x ∈ R)

(18)

with a = 0.2, 1. The chosen flux is f (u) = u2 /2 so that the Kruzkov-solution (cf. (20) below) of the homogeneous problem is the discontinuous function u(., t) = u0 for all t > 0. The numerical experiment shows that—as in the case of the full model (1)—we have to study at least two different scenarios:

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(i) global-in-time weak solutions for arbitrary initial data, (ii) global-in-time classical solutions for small initial data. In our discussion we shall focus on the uniqueness and stability of solutions for (15). First we discuss this issue for weak solutions where we rely on the concept of Kruzkov [Kru70]. For s > 0 let Ωs := R2 × (0, s). A weak solution of (15), (16) is a function u ∈ L∞ loc (ΩT ) such that we have u0 ξ(., 0) dx (19) uξt + f (u) · ∇ξ + T[u(., t)]ξ dx dt = R2

ΩT

for all ξ ∈ C0∞ (Rd × [0, T )). A Kruzkov solution of (15), (16) is a function u ∈ L∞ loc (ΩT ) such that E(u, κ, ω) ≥ − |u0 − κ|ω(., 0) dx (20) R2

holds for all ω ∈ C0∞ (R2 × [0, T ), R≥0 ) and all κ ∈ R. Here we used E(u, κ, ω) := |u − κ|ωt + (f (max{u, κ}) − f (min{u, κ})) · ∇ω dx dt ΩT + sgn(u, κ)T[u(., t)]ω dx dt. ΩT

Extending the results in [Roh98] for local source terms one can prove the following theorem. Theorem 1 (Stability of weak Kruzkov solutions). For i = 1, 2 let ui ∈ 2 L∞ loc (ΩT ) ∩ C(0, T ; BV (R )) be Kruzkov solutions of (15), (16) with initial 1 2 data u0i ∈ L (R ). Then there exists a constant C > 0 such that we have for t ∈ [0, T ) (21) u1 (., t) − u2 (., t)L1 (R2 ) ≤ Cu01 − u02 L1 (R2 ) . The constant C only depends on φ and T . Now we turn to classical solutions and show a stability result that is a counterpart to Thm. 1. First we define an entropy solution for (15), (16). A weak solution u ∈ L∞ loc (ΩT ) of (15), (16) is called an entropy solution if η(u)ωt + q(u) · ∇ω + η ′ (u)T[u(., t)]ω dxdt ≥ − η(u0 )ω(., 0) dx (22) ΩT

R2

holds for all ω ∈ C0∞ (Rd ×[0, T ), R≥0 ). Here η : R → R and q = (q1 , q2 )T : R → ;w 2 R2 are given by η(w) = w2 , qi (w) = 0 fi′ (w)η ˜ ′ (w) ˜ dw ˜ (i = 1, 2, w ˜ ∈ R).

Theorem 2 (Stability of smooth entropy solutions). We assume the simplified relation B(θ) = θ. For i = 1, 2 let ui ∈ L∞ (ΩT ) be entropy solutions of (15), (16) with u0 = u0i ∈ L2 (R2 ) ∩ L∞ (R2 ). Moreover, if u2 is a smooth

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function in W 1,∞ (R2 ) there exists a constant C > 0 such that we have for t ∈ [0, T ) 2 u1 (., t) − u2 (., t)L2 (R2 ) + H[u1 (., s) − u2 (., s)](x) dx ds (23) Ωt 2 ≤ Cu01 − u02 L2 (R2 ) . The constant C only depends on u2 L∞ (0,T ;W 1,∞ (R2 )) , u1 L∞ (ΩT ) , and T . The functional H in (23) is given by H[w](x) = φ(x − y)(w(x) − w(y))2 dy (w ∈ L2 (R2 ), x ∈ R2 ). R2

Proof. Before we start the proof let us introduce for w1 , w2 ∈ R the notations q¯(w1 , w2 ) := q(w2 ) − q(w1 ) − w1 (f (w2 ) − f (w1 )),

f¯(w1 , w2 ) := f (w1 ) − f (w2 ) − ∇f (w2 )(w1 − w2 ).

(24)

By assumption the inequality (22) holds for u1 and u2 where the smooth solution u2 satisfies (22) in fact with equality. Using the notations (24) this implies the following relations. 1 1 2 2 (u1 − u2 ) ωt + q¯(u1 , u2 ) · ∇ω dx dt + (u01 − u02 ) ω(., 0) dx ΩT 2 R2 2 ≥ − u2 (u1 − u2 )ωt + u2 (f (u1 ) − f (u2 ))∇ω dx dt Ω T , 2 − u1 T[u1 ] − u2 T[u2 ] ω dx dt u02 (u01 − u02 ) ω(., 0) dx − 2 ΩT R , , [ωu2 ]t − ωu2,t (u1 −u2 ) − ∇[ωu2 ] − ω∇u2 · (f (u1 ) −f (u2 )) dx dt = − ΩT 2 − u02 (u01 − u02 ) ω(., 0) dx 2 R , , − u1 T[u1 ] − u2 T[u2 ] ω + u2 T[u1 ] − u2 T[u1 ] ω dx dt. ΩT

We now use that ωu2 is a test function and that u1 , u2 are weak solutions of (15), (16). The estimate above can be simplified to 1 1 2 2 (u1 − u2 ) ωt + q¯(u1 , u2 ) · ∇ω dx dt + (u01 − u02 ) ω(., 0) dx 2 2 2 R ΩT , ¯ ≥ T[u1 ] − T[u2 ] (u1 − u2 )ω dx dt. ∇u2 · f (u1 , u2 )ω dx dt − ΩT

ΩT

We now choose ω to be the standard cutoff function of the characteristic cones associated with u1 and u2 (cf. [Daf00, Chap. 5.2]). Since u1 and u2

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are bounded in L∞ (ΩT ) by assumption, by enlarging the support of the test function for all t ∈ [0, T ] we get the estimate 1 2 (u1 (., t) − u2 (., t)) dx 2 2 R 1 2 ≤ (u01 − u02 ) dx + ∇u2 L∞ (ΩT ) |f¯(u1 , u2 )| dx ds (25) R2 2 Ωt t , + T[u1 ] − T[u2 ] (u1 − u2 ) dx ds. 0

R2

Note that the integrals in (25) are well defined due to the natural regularity of entropy solutions. In the next step we use the dissipative effect of the integral operator. Since φ is an even function we observe for w ∈ L2 (R2 ) H[w](x) dx = −2 φ(x − y)(w(x)w(y) − w2 (x)) dy dx 2 2 R2 R2 R R = −2 w(x)T[w](x) dx. R2

This identity and the L∞ -bounds on u1 , u2 imply that there is a constant C1 = C1 (u1 L∞ (Ωt ) , u2 L∞ (0,T ;W 1,∞ (R2 )) ) > 0 such that 1 2 (u1 (., t) − u2 (., t)) dx + H[u1 (., s) − u2 (., s)](x) dx ds R22 Ωt 1 1 2 2 (u01 − u02 ) dx + C1 (u1 (., t) − u2 (., t)) dx ds. ≤ Ωt 2 R2 2 Gronwall’s inequality leads to the statement (23). Remark: (i) The idea underlying the proof has been developed by Dafermos and DiPerna. They proved a stability estimate for solutions of systems of homogeneous conservation laws which are equipped with a strictly convex entropy. The reader is referred to [Daf00, Chap. 5]. (ii) It is interesting to compare the stability result (23) with the corresponding estimate for solutions of the parabolic equation ut + div(f (u)) = ∆u.

(26)

Similar arguments as in the proof of Thm. 2 lead to the result 2 2 u1 (., t) − u2 (., t)L2 (R2 ) + (∇u1 (x, s) − ∇u2 (x, s)) dx ds Ωt

2

≤ Cu01 − u02 L2 (R2 ) .

In fact the convolution operator shares many properties with the Laplaceoperator but is less regularizing (cf. [Fif96, Roh03]).

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(iii) Even though Thm. 1 might be of some interest in itself, it is not clear why it could be useful if compared with the statement in Thm. 2 for a more general class of solutions. In Thm. 4 below we show that it allows us to prove better convergence estimates for FV-schemes. 2.2 The Scalar Model Problem: FV-Schemes To introduce the FV-discretization for the Cauchy problem (15), (16) we introduce the auxiliary problem consisting of (16) and the truncated equation ∂t uS (x, t) + divf (uS (x, t)) = TS [u(., t)](x) := χS (x)T[u(., t)](x)

(27)

for a compact set S ⊂ R2 . Note that Thm. 1 and 2 hold also for this choice of the integral operator. For h > 0 we consider a family {Th } of unstructured polygonal grids. For the index set Ih of the grid, Ti (i ∈ Ih ) denotes a cell volume with center of gravity wi . The grid parameter h is supposed to satisfy h = supi∈Ih {diam(Ti )}. Moreover, let a family Fh = {Fij ∈ C 2 (R2 ) | i ∈ Ih , j ∈ N(i)} of monotone numerical fluxes for (f, Th ) be given ([Kr¨ o97]). The operator TS is approximated by an admissible operator TS,h which maps functions in the set of piecewise constant functions Vh to functions in Vh . We suppose that an admissible operator TS,h satisfies (17) and that there exists a continuous monotonically decreasing function γapp : R≥0 → R≥0 with γapp (0) = 0 such that for i ∈ Ih and wh ∈ Vh ∩ L1 (R2 ) ∩ L∞ (R2 ) # # #T[wh ] − TS,h [wh ]# dx ≤ |Ti |γapp (h). (28) Ti

γapp might depend on the L1 - or L∞ -norm of wh but not on S. Admissible operators with γapp (h) = O(h| ln(h)|). have been constructed in [DR04]. As the next step we define the finite volume scheme for the time step ∆t > 0. For n ∈ N and i ∈ Ih , we define iteratively = uni − un+1 i

∆t Fil (uni , unj ) + ∆tTS,h [uS,h (., tn )](wi ), |Ti |

(29)

j∈N(i)

; where u0i is given by u0i = |T1i | Ti u0 (x) dx. The approximate solution uS,h : R2 × [0, T ] → R is defined by uS,h (x, 0) = u0i for x ∈ R2 , i ∈ Ih and by for (x, t) ∈ Ti × (tn , tn+1 ]. In order to discuss the converuS,h (x, t) = un+1 i gence of the scheme (29) we suppose in the sequel u0 ∈ BV (R2 ) ∩ L∞ (R2 ),

supp (u0 ) bounded.

(30)

For either a Kruzkov solution or a smooth entropy solution of (15), (16) we assume that we have for t ∈ [0, T ] the decay estimate

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T[u(., t)](x) = O(exp(−|x|))

(|x| → ∞).

175

(31)

Then we can prove Theorem 3. Assume that there exist bounded Kruzkov solutions u, uS ∈ BV (ΩT ) for the Cauchy % problems (15), (16) and (27), (16) . Choose S = Sh to be a ball of radius r = | ln(h)| around the origin such that supp (u0 ) ⊂ Br (0). If the time step ∆t > 0 satisfies some CFL-condition then uS,h L∞ (ΩT ) is uniformly bounded and we obtain the error estimate u − uS,h L1 (ΩT ) ≤ C| ln(h)|h1/4 .

(32)

The constant C > 0 depends on the quantities CL , CT , T, u0 and the grid geometry but not on h. The proof can be found in [DR04]. It splits up into estimates for the errors u − uSh L1 (R2 ) and uSh − uSh ,h L1 (R2 ) . Both errors can be estimated by more general versions of Thm. 1. We now turn to the case of smooth solutions for (15), (16). A general error estimate for hyperbolic balance laws with local source terms has been developed in [JR03] using the above-mentioned stability theorem of DiPerna and Dafermos. Using the corresponding Thm. 2 for nonlocal terms and following [JR03] closely one gets Theorem 4. We suppose again the simplified relation B(θ) = θ. Assume that for the Cauchy problems (15), (16) and (27), (16) there exist smooth entropy solutions u, uS ∈ W 1,∞ (ΩT ). With ∆t and Sh as in Thm. 3 the estimate u − uS,h L1 (ΩT ) ≤ C| ln(h)|h1/2

(33)

holds with a constant C > 0 depending on the quantities CL , CT , T, u0 and the grid geometry but not on h. We observe that in the case of a smooth solution we can use the better regularity to increase the order of the error estimate from 1/4 to 1/2 (up to logarithmic terms). Let us note that numerical experiments suggest that the optimal orders in the homogeneous case are 1/2 and 1, respectively.

3 Design of Efficient Approximate Riemann Solvers The fluxes of the MHD system are invariant with respect to rotations, i.e., for all unit vectors n and all U ∈ U we have , F(U) · n = Fx (U)nx + Fy (U)ny + Fz (U)nz = R−1 (n)Fx R(n)U (34)

for a block-diagonal matrix R(n) which consists of standard rotation matrices for the vectorial components of U and unit matrices elsewhere. Thus numerical fluxes gij for arbitrary directions according to (11) are given by

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, gij (t) := |Sij |R−1 (nij )g R(nij )Li (zij , t), R(nij )Lj (zij , t)

(35)

for t ∈ (tn−1 , tn ] provided that g(UL , UR ) is such an approximation to the MHD system in one space dimension for any pair of admissible states UL , UR ∈ U. Here we assume that for i ∈ I we have linear functions Li (x, t) which are constant with respect to t ∈ (tn−1 , tn ] and linear with respect to x ∈ Ω with Li (wi , tn ) = Uni ; with wi and zij we denote the barycenter of Ti and the midpoints of Sij , respectively (see also Fig. 4 top left). The most simple choice is given by Li (x, t) :≡ Uni , which in combination with (13) and (35) results in a first-order accurate scheme. Clearly, there are three major factors that can improve the overall efficiency of the scheme: the optimal selection of the base scheme g, which is addressed in Sect. 3.1, a more elaborate choice of the functions Li (cf. Sect. 3.4), and an economic computational grid. The latter issue is considered in Sect. 3.5, and in Sect. 3.6 we try to give a ranking of these three approaches. 3.1 The Base Scheme The key ingredient for any approximate one-dimensional Riemann solver is the eigensystem of the underlying system of conservation equations. In the MHD case we have seven eigenvalues λ1 (U) ≤ · · · ≤ λ7 (U) and an additional constant eigenvalue λ8 ≡ 0. Depending on the value of the state vector U these eigenvalues may be either pairwise distinct, or some of them may coincide. In addition, for some values of U even some important properties of the eigenvectors can degenerate. Thus our first concern was to ensure that the base schemes do not degenerate (e.g., lose certain symmetries with respect to their arguments) in the vicinity of such degenerate points. This was achieved by a careful selection of the eigensystem and by switching off “critical” parts if necessary. Furthermore, we checked for a series of one- and two-dimensional test problems whether these schemes indeed converge to the correct solutions, and we enhanced their efficiency in terms of achieved error versus computational time. For a perfect gas (3) our newly-introduced solver MHD-HLLEM which is described below turned out to be the most efficient choice. In particular, MHD-HLLEM clearly outperforms other solvers like DW (see also Sect. 3.2) and HLLE (which is based on [EMRS91] and provides the basis for MHD-HLLEM, see below) for certain advection problems as in Fig. 4 (cf. the first-order results on bottom right); for more details we refer to [Wes02, Wes03]. For any pair of admissible states UL , UR ∈ U and UM := 21 (UL + UR ) the numerical flux of the MHD-HLLEM scheme reads g(UL , UR ) := gHLLE (UL , UR ) − Φ(UM )a(UL , UR ). With gHLLE we denote the numerical flux of the standard HLLE scheme gHLLE (UL , UR ) :=

cR cL , cR Fx (UL ) − cL Fx (UR ) UR − UL + cR − cL cR − cL

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for cL := min{λ1 (UL ), λ1 (UM ), 0} and cR := max{λ7 (UM ), λ7 (UR ), 0}. The anti-diffusion term a(UL , UR ) :=

6 cR cL δi αi ri (UM ) cR − cL i=2

is switched off by means of Φ(UM ) :=

0 if λ1 (UM ) = λ2 (UM ) 1 if λ1 (UM ) < λ2 (UM )

if and only if it causes problems due to degeneracies of the eigensystem. The coefficients αi result from the decomposition of UR L into the contributions −U 8 of the right eigenvectors ri , i.e., UR − UL = i=1 αi ri (UM ). (Note that α8 ≡ 0 due to the divergence constraint; therefore there is no contribution of r8 to the anti-diffusion term.) The anti-diffusion coefficients satisfy δi = δi (UM ), δ2 = δ6 , and δ3 = δ5 ; their precise definition as well as the Lipschitz-continuous version of the switch Φ which we use in our implementation can be found in [Wes02, Wes03]. 3.2 Extensions for an Arbitrary EOS For an arbitrary EOS of the form (2) (instead of (3)) the formal structure of the MHD eigensystem does not change. Thus any solver for a perfect gas could be extended directly to the general case. However, for the sake of computational efficiency the solvers should be reformulated in primitive variables wherever possible thus avoiding unnecessary but expensive conversions between conservative and primitive variables. For example, in [Wes02, Wes03] we consider the DW solver for perfect gases, which is based on an approximate Riemann solver introduced by Dai and Woodward in [DW95]. Its fundamental idea is to exploit the fact that Riemann invariants of hyperbolic systems are constant along characteristics (see [Kr¨o97]). Thus it is possible to obtain appropriate values of the conservative variables at the cell interfaces. The numerical flux is then computed by evaluating the exact flux Fx for these interface values and by adding a quadratic viscosity term for stabilization of the scheme. The main part of the RGDW algorithm, which is an extension of DW to the case of an arbitrary EOS, reads 1. Convert the left-hand and right-hand states UL and UR to WL and WR , respectively, for the primitive variables W := (τ, u, B, p)T with τ := ρ1 . 2. Set WM := 21 (WL + WR ), Udiff := UL − UR . 3. Compute the eigenvalues λ1 , . . . , λ8 , the left eigenvectors l1 , . . . , l8 , and the right eigenvectors r1 , . . . , r8 directly for the “primitive” state WM . 4. For k ∈ {1, . . . , 8} set WR if λk < 0 k W0 := and bk := lk W0k . WL if λk > 0

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5. Set Wref := (r1 , . . . , r8 )b for b := (b1 , . . .,, b8 )T% . 6. Set gRGDW (UL , UR ) := Fx (Wref ) + min 0.05 |Udiff |2 , 1 Udiff .

Our full implementation contains some additional fixes and switches. For details see [Wes03]; applications of RGDW can also be found in [DW01, DRW03a]. In a similar manner we extended the MHD-HLLEM scheme, which was discussed in the previous section, to the real gas version RGMHDHLLEM. However, within RGMHD-HLLEM we need an additional conversion between conservative and primitive variables. Thus RGMHD-HLLEM loses efficiency with respect to RGDW, but its application can still be advisable in certain situations. A different ansatz is the energy relaxation method, which allows to apply arbitrary solvers for perfect gases (3) also in the general case. We have extended this approach, which was suggested in [CP98] for the Euler equations of gas dynamics, to the MHD system. The idea is based on a splitting of the internal energy into two parts ε = ε1 + ε2 . The first part ε1 corresponds to a given simple pressure law p1 = p1 (ρ, ε) (for example, the polytropic gas law (3)); the nonlinear disturbances of the original pressure law p are simply advected with the fluid using ε2 . In the resulting relaxation system the energy conservation law (1d) is replaced by , ∂t (ρλ eλ1 ) + ∇ · (ρλ eλ1 uλ + P1λ uλ ) = qρe + Qrad + λρλ ελ2 − Ψ (ρλ , ελ1 ) , , ∂t (ρλ ελ2 ) + ∇ · (ρλ ελ2 uλ ) = −λρλ ελ2 − Ψ (ρλ , ελ1 ) , , 1 1 |Bλ |2 I − 4π Bλ (Bλ )T . for λ ∈ R+ and the pressure tensor P1λ = pλ1 + 8π The energy function Ψ = Ψ (ρ, ε) is chosen so that in the equilibrium limit λ → ∞ the original MHD system is recovered. The ER scheme is best understood as an operator splitting scheme for the relaxation system in the equilibrium limit. In the splitting approach the evolution of the conserved quantities is performed in two steps. The first step is the relaxation step. Here we neglect all spatial derivatives and solve the remaining system of ODEs for λ → ∞. The solution computed in the first step is then used in the second step as initial condition for the homogeneous system, which does not depend on λ. Since the homogeneous system coincides with the standard MHD system with pressure law p1 , we can reuse a numerical flux function for this system in the second step. The solution to the first step can be derived analytically. For a theoretical justification of the energy relaxation method we observe that if the limit function U(x, t) := lim (ρλ , ρλ uλ , Bλ , ρλ eλ1 + ρλ ελ2 )T (x, t) λ→∞

of a uniformly bounded family of classical solutions exists then it is a solution of the MHD equations (1) if we choose Ψ (ρ, ε1 ) = ε(ρ, p1 (ρ, ε1 )) − ε1 . Here ε(ρ, ·) is the inverse of p(ρ, ·). The proof and further aspects of this method can be found in [Ded03].

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While we observed no differences in the quality of the approximations obtained with the two methods described above, the energy relaxation technique requires less recoding and seems to have advantages concerning the overall efficiency [DW01, DRW03a, Ded03]. The more expensive the evaluation of an EOS becomes, the more important it is to restrict its calls within a solver to an absolute minimum. In addition, for an expensive EOS the use of interpolated values from a locallyadapted table rather than direct evaluations can pay off significantly. For example, in the case of a partially-ionized plasma as it occurs in the solar photosphere we were able to save 90% of the computational time by this approach [DRW03a, Wes03]. 3.3 Including the Divergence Constraint As we discussed in Sect. 1, on the analytical level the divergence constraint (1e) is not an additional elliptic condition but in fact only a condition on the initial data B0 for the magnetic field. This is due to the induction equation (1c) which leads to ∂t ∇ · B = 0. In our finite volume approach we cannot directly take this relation into account. Therefore it is possible that a simulation yields an approximate magnetic field which is not divergence free, which often leads to wrong magnetic field topologies and to the breakdown of the simulations. In this section we describe a method for reducing the problems which are due to such “divergence errors” in MHD simulations. This approach was introduced in [MOS+ 00] for the Maxwell equations, and we extended it to the MHD equations in [DKK+ 02, DRW03b]. The method is based on an extension of the MHD system that we call the Generalized Lagrange Multplier (GLM-)MHD system. The divergence constraint is coupled with the induction equation by means of an auxiliary function ψ, and the structure of the modified divergence constraint can be varied, thereby leading to different correction mechanisms. The GLM-MHD system consists of equations (1a), (1b), and (1d). The divergence constraint (1e) is coupled with the induction equation (1c) by introducing a new scalar function ψ and a linear differential operator D (36a) ∂t B + ∇ · (uBT − BuT )+∇ψ = 0 (modified induction equation), D (ψ) +∇ · B = 0 (modified divergence constraint). (36b) The correction mechanism is strongly influenced by the choice of the operator D. We have tested many different approaches: the elliptic correction where the divergence of the magnetic field satisfies △∇ · B = 0, the parabolic correction where we find ∂t ∇ · B − c2p △∇ · B = 0, and the hyperbolic correction with ∂tt ∇ · B − c2h △∇ · B = 0 (where ch , cp are suitable constants). For the latter case we take D (ψ) = c12 ∂t ψ. This approach is especially well suited for h the discretization using a finite volume scheme since the resulting GLM-MHD

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100

180

mixed GLM-MHD source term fix base scheme

mixed GLM-MHD source term fix base scheme

160 140

10

120 100 80

1

60 40 20

0.1

0 0

0.1

0.2

0.3

0.4

0.5 time

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0.9

1

0

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0.6

0.7

0.8

0.9

1

Fig. 2. Evolution of divergence errors (in L1 (left) and L∞ (right)) for the exploding magnetic ﬂuxtube. We show results for the base scheme, the mixed GLM-MHD scheme, and the source term ﬁx from [Pow94]. In the case of the unmodiﬁed scheme the simulation breaks down due to the distortion of the magnetic ﬁeld caused by divergence errors.

system is a hyperbolic system in divergence form. We can combine the transport effect of the hyperbolic correction with the damping introduced by the parabolic approach by choosing D (ψ) = c12 ∂t ψ + c12 ψ. Note that the addip h tional term only leads to a linear term of order zero in the modified divergence constraint and can thus be easily included in any numerical scheme. In fact the discretization of the resulting mixed GLM-MHD system only requires minor modifications of the original scheme (cf. [DKK+ 02]). In [DKK+ 02, Ded03, Wes03] we compared the different correction mechanism with each other and with other methods found in the literature. We found that the mixed correction leads to the best results by far — with respect to both the size of the divergence errors and the stability of the scheme. At the same time this correction leads to hardly any additional computational cost since the modifications required in each time step are negligible and no additional stability restrictions have to be imposed; for a suitable choice of the parameters ch , cp which allows us to maintain the time step of the base scheme we refer to [DKK+ 02, Ded03]. In Fig. 2 we demonstrate the effectiveness of our approach using the 3d magnetic fluxtube setting described in Sect. 6. 3.4 Higher Order Extensions In one space dimension it is quite straightforward to improve the convergence rate of a solver of the form (13) and thus its efficiency by defining functions Li (x, t) whose components are linear with respect to x. The standard approach is to compute linear reconstruction(s) of the constant cell data and to apply a suitable limiter. In [DEO92] Durlofsky, Engquist, and Osher suggest the following generalization of this approach to unstructured conforming triangular grids in 2d according to Fig. 4 top left. Assume that p1 , p2 , p3 ∈ R2 are chosen such that p2 − p1 and p3 − p1 are linearly independent and let L := Lp1 ,p2 ,p3 ,v1 ,v2 ,v3 be the uniquely defined linear function with L(pj ) = vj

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for vj ∈ R and 1 ≤ j ≤ 3. Then for each time level n and each component Ui of Uni 1. set Li :≡ Ui if Ui is a local maximum or minimum; otherwise 2. set Di1 := Lwi ,w2 ,w3 ,Ui ,U2 ,U3 , i.e., Di1 connects the constant values in all barycenters but w1 , and define Di2 and Di3 analogously. 9 i1 , D 9 i2 , D 9 i3 . 3. Apply a limiter to Di1 , Di2 , Di3 for obtaining D 9 9 9 4. Select Li ∈ {Di1 , Di2 , Di3 }.

An additional check on oscillations in derived physical quantities like the pressure can be advantageous. Here we assumed without loss of generality that the neighbors of Ti are given by T1 , T2 , and T3 . In [DEO92] two pairs of limiters and selection principles are suggested, which in our notation yield the approaches “DEOmin” and “DEOmax.” However, we were not able to simulate a simple hydrodynamic Rayleigh-Taylor instability with either of these choices. As reasons for these problems we identified an insufficient limitation and the disregard of the reconstructed values within the neighboring triangles. At the same time, sometimes the limitation proved to be unnecessarily strict since the limiters were applied in the wrong points [DRW03a, Wes03]. Our new approach “DEOmod” overcomes all these difficulties: For 1 ≤ j ≤ 3 1. we compare discrete directional derivatives gij := ∇Dij · (wj − wi ) ,

dij := Uj − Ui and

2. we limit the linear functions by reducing the gradients

for

9 ij (x) := Ui + mij ∇Dij · (x − wi ) D

⎧ dij gij ≤ 0 ⎨0 mij := dij /gij dij gij > 0, |gij | > |dij | . ⎩ 1 otherwise

3. Finally, we choose the steepest function by means of 6 5 9 ij | = max |∇D 9 ij with |∇D 9 ij | . Li :≡ D 0

0

1≤j≤3

For this approach we can prove that initially linear data is exactly reproduced and that the reconstruction is oscillation-free in the sense # # # # # # #Ui − Li (cij )# + #Lj (cij ) − Uj # ≤ #Ui − Uj #

for 1 ≤ j ≤ 3 where cij is the intersection point of the barycenter connection with the corresponding side of Ti . In addition, for problems to those also the DEOmin and DEOmax approaches are applicable, DEOmod yields the most efficient scheme (cf. Fig. 4 top right).

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Table 1. Initial data for the 2d Riemann Problem WaalsC. For the construction and the full precision initial data see [Wes03]. 2d Riemann Problem WaalsC EOS: van der Waals (37); computational domain: [−1, 1] × [−1, 1] boundaries: inﬂow condition (numerical solution to 1d Riemann problems) quadrants: 1 : x > 0, y > 0 / 2 : x < 0, y > 0 / 3 : x < 0, y < 0 / 4 : x > 0, y < 0 3 → 4: 1d Riemann problem with exact solution (Waals1mhd); 3 → 2: λ1 –shock; 2 → 1: λ1 –rarefaction uy uz Bx By Bz p qdr. ρ ux 1 176.735 245.018 −75.3096 5.05254 6749.27 3439.68 −2103.41 16173057.0 2 272.850 1.26850 −66.0187 −0.62898 6749.27 5472.65 −3346.61 48609863.0 3 250.000 0.00000 0.00000 0.00000 6175.23 5472.65 −3061.97 35966800.0 4 176.697 1.34771 −27.7180 15.5083 6175.23 3439.68 −1924.51 28064061.6

Fig. 3. Isolines of the DEOmod RGDW reference solution on 1048576 triangles at time t = 0.0011 for the WaalsC problem. From left to right: ρ, ux , uy , Bx , By , Bz . advection of Bz: dw, quartering

pi3

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Fig. 4. Top: Notation used for DEOmod and eﬃciency of diﬀerent DEO approaches for the advection of a smooth impulse in Bz . Bottom: Eﬃciency of diﬀerent schemes for the WaalsC problem and the advection of Bz for a perfect gas (3).

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3.5 Local Grid Adaption The basic idea behind local grid adaption or adaptive mesh refinement is the creation and use of economic grids for the problem under consideration. This means that the grid is refined in those regions whose contribution to the overall error is large, whereas in other parts a coarse grid is used. Up to now there are no analytically-justified error estimators for nonlinear timedependent hyperbolic systems like MHD, which would allow us to obtain a numerical solution that complies with an arbitrarily-chosen error tolerance. Thus our locally-adaptive finite volume scheme for the MHD equations has to rely on heuristic error indicators. By applying an heuristic indicator we want to obtain solutions of comparable quality as on globally-refined grids, but in significantly less computational time. To this end, we apply the following design principles. A suitable indicator should • yield locally-adapted grids on which we obtain the same errors as on globally-refined grids with cell diameters corresponding to the finest adaptive level, • be cheap to evaluate, and • facilitate a straightforward extension to the multidimensional case. The last point reflects the fact that usually the construction starts in 1d. For the derivation of our new indicator we assume that in linear parts of the numerical solution a coarse grid is sufficient, whereas fine grids are needed in the vicinity of discontinuities, maxima, minima, and in nonlinear parts. In essence, the indicator is expected to generate a “two-level” grid: in uncritical parts the coarse macrogrid is preserved, whereas in critical parts the grid is refined to the finest level admitted. In addition, we want to obtain an indicator which can easily be applied to an arbitrary set of quantities since for MHD problems no single quantity suffices for detecting all waves. The 1d version of our “simple but versatile” grid indicator svindV is based on the comparison of discrete one-sided and central differences (see [Wes03]). Within this section we focus on its generalization to triangular grids in 2d using directional derivatives along the barycenter connections. In the notation of the previous section our approach reads: Assume that Ti and Tj are neighboring triangles. For the Euclidean distance |. − .| of two vectors in R2 we set , vj − vi . svindV Ti , Tj , vi , vj ) := |wj − wi | Then for a given set V ⊂ {ρ, ρux , ρuy , ρuz , Bx , By , Bz , ρe, ux , uy , uz , e, p, s} where s is the specific entropy we define , svindV Ti , T1 , T2 , T3 , Ui , U1 , U2 , U3 # # # , wj − wi ## V # . := max max #svind Ti , Tj , vi , vj − ∇Lw1 ,w2 ,w3 ,v1 ,v2 ,v3 · v∈V 1≤j≤3 |wj − wi | #

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Usually, we set V := {ρ, Bx , By , Bz }. Within our algorithm a triangle is refined, if the value of the indicator exceeds a certain threshold value and if the refinement complies with the selected minimal cell diameter. On the other hand, a cell is marked for coarsening if the value of the indicator is sufficiently small. However, the actual coarsening is only performed if it is compatible with the grid conformity constraints, if it reverses a previous grid refinement, and if all other involved cells are also marked for coarsening. In [Wes03] we demonstrate for a broad range of examples that the indicator svindV meets the requirements which we stated above. An enhancement of the asymptotical efficiency can also be observed in Fig. 4 bottom left. 3.6 An Economic Way Towards Efficiency Any comparison of the accuracy and the efficiency of different solvers requires • a suitable set of test problems with fully- or partially-known exact solutions which cover a broad range of physical regimes and • reliable techniques for computing the errors of a given numerical solution. Both issues are addressed in detail in [Wes03]. For the van der Waals EOS p(ρ, ε) =

R (ε − ε0 + ρa) − ρ2 a cV ( ρ1 − b)

(37)

with the definitions a := 1684.54, b := 0.001692, R := 461.5, cV := 1401.88, and ε0 := 0 we suggested several two-dimensional Riemann problems. For example, the problem “WaalsC” according to Tab. 1 and Fig. 1 comprises the constructed 1d Riemann problem “Waals1mhd” with known exact solution plus two simple waves. Thus only for one of the four arising 1d problems the exact solution is unknown. Test problems of this type allow comparisons of the one-dimensional structures and convergence tests for the full solutions by means of a highly-resolved reference solution (see Fig. 4 bottom left). Further examples are one- and multidimensional advection problems which can be formulated for an arbitrary EOS (cf. the right column of Fig. 4). Our recommended solvers converged for all test problems that we considered. For predicting the efficiency of one-dimensional Riemann solvers in multidimensional applications it is not sufficient to compare their efficiency in 1d. Rather we also have to consider the achieved errors on fixed grids since the errors are proportional to some positive powers of the maximal cell diameter h, whereas the number of cells and thus the number of flux evaluations scales with h−2 and h−3 in 2d and 3d, respectively, instead of h−1 in 1d. This explains our observation that if a scheme is more efficient than other solvers in 1d but needs significantly finer grids to achieve the same errors, it can be clearly less efficient than its competitors in multiple space dimensions. Similarly, if we compare the runtime of different solvers on fixed grids, the fraction of EOS-dependent operations within faster solvers tends to be larger than in

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slower ones. Thus it can happen that by switching to a more complex EOS the ranking of the solvers according to their efficiency is reversed. For more details concerning both phenomena we refer to [Wes03, Sect. 3.4.8]. Altogether we recommend using MHD-HLLEM for perfect gases and RGDW or the energy relaxation extension of MHD-HLLEM for a general EOS. From our experience and the numerical tests we draw the following conclusions: • The multidimensional finite volume code directly benefits from any improvement of the underlying one-dimensional Riemann solvers. • In general, both local grid adaption and the second-order extension by means of the DEOmod-reconstruction enhance the efficiency of the twodimensional base schemes. • The actual gain in efficiency by the DEOmod-extension depends strongly on the structure of the underlying grid: If the grid gets more irregular, the observed convergence rate and thus the asymptotical efficiency decreases. • The generalization of local grid adaption from one to multiple space dimensions seems to be more straightforward than the extension of limited linear reconstruction. In particular, the properties of a higher-order scheme turned out to be sensitively dependent on the limitation procedure chosen, and finding an appropriate one requires many — necessarily multidimensional — tests. We expect that in the case of non-conforming grids the construction of a higher-order scheme gets even more involved. Thus we favour the following order of enhancements of multidimensional finite volume codes: 1. Selection of an “optimal” one-dimensional approximate Riemann solver. 2. Implementation of local grid adaption. 3. Extension to formal second-order accuracy.

4 Coping with the Needs of Atmospheric Simulations In atmospheric flows the solution U can often be represented as a compactly9 of a stratified background atmosphere U. ˚ This supported perturbation U background solution is given by balancing the pressure gradient with the force of gravity and is a stationary solution of the MHD system. Even if the convergence of a numerical scheme has been shown for a grid size h → 0, simulations have to be performed for finite values of h. On these grids approximation errors, although small, can still be of the same magnitude or even larger than the perturbations that contribute to the solution. Consequently, the physical characteristics of the problem cannot be captured without a modification of the numerical scheme. A suitable strategy is discussed in Sect. 4.1. Initially, the perturbation is restricted to a small domain, but fast moving ˚ waves can quickly lead to a significant enlargement of the region where U − U

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does not vanish. Quite often these fast waves have a very small amplitude and their evolution is not of great interest. Therefore it is desirable to reduce the size of the computational domain as far as possible. This leads to artificial boundaries which require the formulation of boundary conditions which allow these waves to move through the boundary without being reflected. The derivation of suitable boundary conditions is described in Sect. 4.2. 4.1 Balancing the Force of Gravity The Bgfix modification of the base scheme presented in Sect. 3.1 is based on a suitable equation for the perturbations themselves. Its main feature is that the background solution is captured without any approximation error. As we have shown in [Ded03], this allows for a very efficient solver for this setting. We derive the scheme for a general system of balance laws of the form (7) ∂t U(x, t) + ∇ · F(U(x, t)) = q(U(x, t)), ˚ 0) + U 9 0 (·) U(·, 0) = U(·,

(38b)

˚ ˚ ˚ ∂t U(x, t) + ∇ · F(U(x, t)) = q(U(x, t)).

(39)

(38a)

˚ is a known smooth function satisfying where U

By subtracting (39) from (38a) we arrive at an equation for the perturbation 9 ˚ U(x, t) := U(x, t) − U(x, t): 9 9 U(x, 9 9 9(U(x, t) + ∇ · F( t), x, t) = q t), x, t), ∂t U(x, 9 0) = U 9 0 (·) U(·,

9 and the source q 9 via if we define the flux F

9 ˚ ˚ F(u, x, t) := F(u + U(x, t)) − F(U(x, t)), ˚ ˚ 9(u, x, t) := q(u + U(x, t)) − q(U(x, t)). q

(40a)

(40b)

(40c) (40d)

9 also satisfies a system of hyperbolic balance laws Therefore the function U with a flux function and a source term, which depend explicitly on x and t. If 9 ˚ 9 is a weak solution to (40), it we define V(x, t) := U(x, t) + U(x, t) where U is easy to see that V is a weak solution of (38). For the scalar case, we show in [Ded03] that not only does (40) have the same weak solutions, but also that 9 is the entropy solution of (40) then V is the entropy solution of (38). if U 9 ≡ 0 is the classical solution to (40) and 9 0 ≡ 0 then U Note that if U ˚ Since even first order finite volV is equal to the background solution U. ume schemes reproduce constant solutions without any approximation error, a scheme based on (40) can fulfill the requirement that the background solution be reproduced exactly in the case of a vanishing initial perturbation 9 0. U

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In the case of the finite volume scheme presented in Sect. 1.2 we use the 9 numerical flux function gij for F to define a numerical flux for F: n ˚ ij , tn ), W + U(z ˚ ij , tn )) + |Sij |F(U(z ˚ ij , tn )) · nij . 9ij g (V, W) := gij (V + U(z

9 We can now derive a finite volume scheme for approximating U:

n 9 n+1 = U 9 n − ∆t 9 n, U 9 n ) + ∆tn q 9 n , wi , tn ), 9(U 9n (U U g i i j i |Ti | j ij i

(41)

90 = U 9 0 (wi ). Since we are interested in an approximation of U and not with U i 9 we define the approximation on Ti × (tn , tn+1 ] by of U, 9 n + U(x, ˚ Vh (x, t) = U t). i

9 n = 0 for all i, n and therefore we have Vh (x, t) = 9 0 ≡ 0 then U Note that if U i ˚ U(x, t) as required. In [Ded03] we prove the convergence of the scheme in the scalar case and demonstrate its efficiency for a wide range of test cases. Our method is especially efficient when used in combination with local ˚ is reproduced without loss of grid adaption. Since the background solution U accuracy independent of the size of the grid elements, we can use a very coarse 9 grid in those parts of the computational domain where the perturbation U 9 vanishes. Therefore we can use an adaption indicator based only on U. In the original scheme the refinement criterion has to be based on U itself and thus ˚ varies strongly. a fine grid is required in every part of the domain where U In Fig. 9 we demonstrate this effect considering as example a 2d calculation of a magnetic flux tube. The grid produced without the Bgfix modification is larger by approximately a factor of 2.5 than the grid with the modification. Note also the loss of symmetry in the calculation without our modification. 4.2 Transparent Boundary Conditions For our simulations in the lower solar convection zone the computational domain can only be a small section of the full convection zone. It has to be chosen with two aims in mind. On the one hand, small structures have to be resolved and their evolution has to be tracked over a long time period; on the other hand, the computational domain has to be as small as possible to minimize computational costs. Covering only a small portion of the full domain in a simulation requires the specification of suitable boundary conditions on the artificial vertical and horizontal boundaries to close the MHD system. It would be desirable if all boundaries — the vertical as well as the horizontal — were transparent for outgoing waves so that the solution in the computational domain is independent of the chosen position of the boundaries. Definition (Transparent Boundary Conditions): Let Ω1 be a compact subset of Rd with supp(U0 ) ⊂ Ω1 and let U1 be the solution to the MHD

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Fig. 5. Left: evolution of L1 -error for Dirichlet, Neumann, local, and non-local transparent boundary conditions. Up to the time t = 30, where the large-scale perturbations reach the upper boundary, the transparent boundary conditions lead to almost no error compared to the reference solution. Right: percentage of total time required for computing the transparent boundary conditions.

system on Ω1 using boundary conditions BC. These are said to be transparent if for all compact subsets Ω2 ⊂ Rd with Ω1 ⊂ Ω2 we have U1 ≡ U2 in Ω1 where U2 is the solution to the MHD system on Ω2 using boundary conditions BC. In most solar physical simulations the important structures move upwards through the atmosphere and, therefore, the top boundary is the most critical one; the influence of the vertical boundaries is much smaller. Therefore we focus on both horizontal boundaries, while we assume periodic vertical boundaries in accordance with [CMIS95, FZL98]. In [DKSW01a, DKSW01b] we derived boundary conditions that fulfill the requirement stated in the above definition at least for small perturbations. The boundary condition necessarily includes a non-local convolution term with respect to time at the artificial boundaries. However, by using a special approximation of the convolution kernel, this non-local term can be evaluated in a time-stepping manner so that the numerical method stays local with respect to time. Again the starting point for the derivation are the equations for the pertur9 We assume that the perturbations at the boundary are sufficiently bations U. small and smooth. Furthermore, we study the special case of an exponentially˚ that permits a sufficiently far-reaching decaying background atmosphere U analytical study. At the same time, the application of our boundary conditions to other models for the background atmosphere inside the computational domain seems to pose no problems. To derive the boundary conditions the MHD equations are linearized. We can prove that for the linearized system these boundary conditions are transparent for background solutions with exponential decay in the sense of our definition. The proof can be found in [DKSW01b], where we also discuss implementational aspects and compare our boundary conditions with other more direct approaches. Additional implementational details can be found in [Wes03]. Our numerical examples

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illustrate that the structure of the solution is considerably influenced by the choice of the boundary conditions. Moreover, using our boundary conditions we find that even large perturbations are hardly reflected at the artificial boundaries. The examples indicate that the proposed transparent boundary conditions yield good results (cf. Fig. 10) and are very cheap with respect to their computational costs. In fact, the costs for the numerical evaluation of the boundary conditions are almost negligible: in a 2d test calculation it took less than 6% of the overall CPU time, see Fig. 5.

5 Approximating the Radiation Source Term 5.1 Discrete Ordinate Method To include the energy transport through radiation in our finite volume scheme, the average radiation source term Qrad has to be approximated. Using the average radiation intensity 1 J(x, t) = I(x, t, µ)dµ 4π S 2 we have to find an approximation of 1 1 J Qrad (x, t)dx = 4πχ(x, t) (J(x, t) − B(x, t)) dx (42) Qrad (t) := |T | T |T | T for each element T of a given grid T. By means of a quadrature rule for the integral over the unit sphere in the definition of J, we reduce the computation of the radiation source term to a summation involving the radiation intensities Im (x, t) := I(x, t, µm ) for a fixed set of directions {µm }M m=1 . In [DV02, Ded03] we studied different schemes for computing an approximation Im,h of the solution Im to the radiation transport equation (1f) in a fixed direction µm . We focus here on the Discontinuous Galerkin (DG) method first studied in [LR74, JP83] and on the extended short characteristic (ESC) method which we proposed in [DV02]. These steps lead to the following approximation for QJrad T ∈ T: QJrad h (t) :=

1 |T |

T

χ(x, t)

M

m=1

ωm (Im,h (x, t) − B(x, t)) dx .

(43)

M The constants ωm are the weights of the quadrature with m=1 ωm = 4π. A second approximation can be derived by using the radiation flux µ · ∇I(x, t, µ)dµ F (x, t) = S2

since we have (using the RT equation (1f) and Gauß’ theorem)

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QJrad = −

1 |T |

T

F (x, t)dx = −

1 |T |

∂T

S2

I(x, t, µ)µ · n =: QF rad .

The approximation steps outlined above lead to the following expression QF rad h (t)

1 := |T |

M

∂T m=1

ωm Im,h (x, t)µm · n .

(44)

Depending on the scheme used for computing Im,h , the approximations (43) and (44) might not lead to the same result for a fixed grid T. Since the approximate intensity functions Im,h can be discontinuous over element boundaries, we have to clarify how we compute the boundary integrals in (44). Let Sij be the faces of the element T = Ti and denote with Im,i the discrete intensity defined on the element Ti and with Im,j the intensity; defined on the neighboring elements Tj . We define the value of the integral Sij Im,h required to compute (44) by using the intensity in upwind direction, i.e. 0; Im,j if nij · µ < 0 , Im,h = ;Sij I otherwise . Sij Sij m,i

In the following all boundary integrals involving the discrete intensity function are to be understood in this sense. For the element integrals in (43) we use only values defined on the element itself. Remark: The two terms QJrad h and QF rad h differ only in the case of a non-conservative scheme (cf. [Ded03]). In the case of the DG-method both expressions are identical and can differ only after the integrals have been approximated by quadrature rules. However, this difference is negligible so that we only have to compare the approximations QJradh and QF radh in the case of the non-conservative ESC-methods. 5.2 The Average Intensity versus the Radiation Flux For a quantitative comparison of the approximation to Qrad we use the simple setting I(x, y, µ) := (cos(2πx) + 2) sin(πy)2 + 1, χ(x, y) := (1000 tanh(−15y) + 1000.001)(sin(πx) + 1.25)

1 to compute B(x, y, µ) = χ(x,y) µ · ∇I(x, y) + I(x, y). Note that I is independent of µ so that we easily compute Qrad ≡ 0; as a consequence we now have a source term B which depends on µ. We measure the quality of the approximation to Qrad using the L1 -norm. In Fig. 6 we plot the approximations to QJrad and QF rad using the second order ESC2-method. We can clearly distinguish two regions. For y > 0 the approximation of QJrad is very close to zero,

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whereas the error in QF rad is quite large. The situation is reversed for y < 0, where the error in QJrad is very large. Between these two regions the absorption coefficient χ varies considerably in magnitude: for y < 0 the absorption is dominant (χ large) and for y > 0 the transport of radiation is dominant (χ small). A closer look at the RT equation (1f) can help to explain the influence of χ on the radiation source term. When χ is large the radiation intensity I is close to B. To approximate QJrad in (43) the difference of I and B is taken and then multiplied by χ. Thus small errors in I − B are amplified and this leads to the observed oscillations. On the other hand when χ is small then µ · ∇I is also small. This causes oscillations when discrete derivatives in QF rad are computed. These observations were already made in [BVS99] for the standard short-characteristics method. With the following example we quantify the different quality of the approximation to Qrad by using either the average radiation intensity J or the radiation flux F . Example: We study the solution to a model problem that corresponds to the radiation transport equation (1f) in 1d with only two directions µ and constant data: u′ (s) = χ(B − u(s)) , − v ′ (s) = χ(B − v(s)) for s ∈ [0, h] , u(0) = u0 , v(h) = v1 .

(45a) (45b)

We assume that χ, B, u0 , and v0 are given constants. We are interested in ;h , approximating the average radiation source term QJrad = h1 0 χ u(s) + v(s) − 2B ds. It is straightforward to compute the solutions to the initial value problems in (45) so that we obtain J QF rad = Qrad = (u0 + v1 − 2B)

1 − e−χh . h

(46)

As in the ESC1-method, we use the one step Radau IIa method to approximate u(h) and v(0) by values u1 and v0 , respectively, and use these values to define linear functions on [0, h]: QJradh = (u0 + v1 − 2B)

χ + 12 hχ2 , 1 + hχ

QF radh = (u0 + v1 − 2B)

χ . 1 + hχ

For χ → 0 we see that both QJrad h and QF rad h tend to zero which is the correct limit. On the other hand, for χ → ∞ we have QJradh → ∞, but 1 QF radh → h (u0 + v1 − 2B). Thus the approximations differ greatly for χ large. Using (46) we see that h1 (u0 + v1 − 2B) is the correct value for χ → ∞. Therefore only QF radh can lead to a good approximation for χ large. In Fig. 7 J we plot both approximations and the errors |QJradh − QJrad | and |QF radh − Qrad | as functions of χ. We see that the approximation using QJradh is closer to the correct value for hχ < 1 whereas only QF radh is close to the correct value

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for hχ > 1; the errors are equal for hχ ≈ 1.1. Note that for h → 0 both approximations converge to the correct value limh→0 QJrad = χ(u0 + v1 − 2B). To compute Qrad the authors of [BVS99] propose using QF radh on those elements T where χ is greater than some threshold value χ0 and to use QJrad h otherwise. They suggest a threshold value that is either a constant or depends on the element size hT . Our example indicates that the threshold should not be chosen independent of the element size, but rather that hT χ, which is a dimensionless quantity, is the relevant parameter. Consequently we define 0 M 1 J hT χ(wT ) < 1 , m=1 QradT,µm Qrad h := |T | 1 (47) M F otherwise . − |T | m=1 QradT,µ m

In Fig. 8 (left) we plot the L1 -error for the approximations QJradh , QF radh , and also for Qrad h as defined in (47). The advantage of our mixed definition of Qrad can be clearly seen: on the finest grid level the use of our indicator leads to a gain in runtime of more than a factor of 50 compared to using only QJrad J and to a gain of a factor of four compared to using QF rad . The curves for Qradh show a higher convergence rate than the curves for QF radh , which have about the same slope as the corresponding curves for Qrad h . Consequently, these curves will intersect for h small enough. Our indicator takes this behavior into account since for small h we use QJrad to compute Qrad . In Fig. 8 (right) we compare the first and second order ESC- and DGmethods and the conservative extensions of the ESC-methods (CESC-methods). The advantage of using the ESC2-method together with the indicator for defining Qrad is very clear. Due to strong oscillations in those parts of the domain where χ is large, the DG1-method is far less efficient than the ESC2method (even less efficient than the ESC1-method for the grid resolutions shown here).

6 Dynamic Load Balancing for Complex 3d Simulations The ultimate numerical goal of this ANumE-project was the development of an efficient code for solving the equations of radiation magnetohydrodynamics in 3d. Parallelization and local grid adaption are important tools for increasing the efficiency. In this section we discuss the design of our software package for use on distributed memory computers. Due to limitations in space, we cannot present all the features and restrict ourselves to the construction of one key feature: the load balancing algorithm which is a central aspect if local grid adaption and parallelization are to be combined. A complete presentation of the software can be found in [DRSW04]. The object-oriented grid data structure goes back to [Sch99].

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6.1 Some Notes on the Load Balancing Algorithm We achieve parallelism by decomposing the spatial domain Ω into subdomains Ωk , k ∈ {1, . . . , K} and by distributing it on the available K ∈ N processors. Initially the mesh is equally distributed to all processors. Due to our adaptive algorithm the mesh is refined and coarsened during the computation depending on the evolution of the approximate solution. In consequence, after some time the numerical effort of some processors might deviate substantially from the mean effort of all processors. This destroys the performance of the parallel algorithm. Dynamical load balancing cures this problem. We describe now the basic load balancing algorithm as implemented in our code. At first we need some details about the administration of the mesh. The mesh is constructed from a macro grid consisting of a set V 0 of cell volumes and a set S 0 of surfaces of the cell volumes, i.e., V 0 := {Vi }i∈I ,

S 0 := {Sij | joint surface of Vi , Vj ∈ V 0 (if it exists)}.

Here I is some index set. We also need E 0 = {(i, j) ∈ I 2 | Sij ∈ S 0 }. The macro grid cannot be coarsened further. Refinement is done by refining elements of V 0 . Each new cell volume and each inner surface that occurs during refinement (or coarsening up to the level of the macro grid) is administrated by the corresponding cell volume of the macro grid. The surfaces that belong to the boundary of a macro grid cell volume are kept as the leafs of a tree with root being the macro grid surface they belong to. The partitions Ωk , k ∈ {1, . . . , K} are supposed to be unions of cell volumes of V 0 . The load balancing algorithm is only executed on the level of the macrogrid pair (V 0 , S 0 ). To (re)distribute the cell volumes of the macro grid to the processors we have to compute the total computational effort for each volume V ∈ V 0 and for each surface in S 0 . For the volumes we have to determine the costs for methods that act on cell volumes that are leafs of the tree with root V and those acting on all surfaces within V . The total computational effort for a surface of the macro grid S 0 is given by the overall costs of numerical methods that act on all surfaces administrated by S 0 . These include in particular all surfaces of the physical boundary ∂Ω. The number of surfaces in a cell volume V is proportional to the number of cell volumes in the tree belonging to V : the computational effort associated with V is then α(V ) ∼ #volumes administrated by V .

(48)

The computational effort associated with a surface S of S 0 is computed from β(S) ∼ #surfaces administrated by S.

(49)

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Costs of global operations are neglected in our estimate of the effort. The load balancing algorithm is now based on a representation of the macro grid as a graph. Consider the weighted graph G = (V 0 , E 0 , α, β). The functions α : V 0 → N and β : E 0 → N are the weight functions for the nodes V 0 and the edges E 0 of the graph, respectively. Define the set A as the set of partition functions Π : V 0 → {1, . . . , K} such that for all k ∈ {1, . . . , K} we have # # # # 1 # α(V )## ≤ max0 α(V ). α(V ) − # K V ∈V 0 0 V ∈V

V ∈V , Π(V )=k

All the partitions in A lead to an (almost) equal distribution of the computational costs on volumes. To find the optimal distribution we look for a ¯ such that the functional F : A → [0, ∞) given by partition function Π F [Π] = β(i, j) (i,j)∈E 0 , Π(Vi ) =Π(Vj )

¯ A solution of the minimization problem optimizes the is minimized by Π. computational costs due to the inner boundary surfaces under the side condition of (almost) the same load for all partitions. A solution to this finite dimensional problem exists but might not be unique. Moreover computing the exact solution might require extreme computational effort which is not compensated by the gain due to a better partitioning. We use standard graph partitioning software provided by the METIS-software package [KK] for the approximation of the exact solution. Load balancing is not performed in each time step of the parallel algorithm. We fix a number τ > 1 and balance the load if and only if there is a 9 k ∈ {1, . . . , K} such that τ α(V ) ≥ α(V ). (50) K 0 V ∈V 0 , Π(V )=9 k

V ∈V

Remark: The code is designed for runs on machines with distributed memory architecture. Since the equations of magnetohydrodynamics involve only local (differential) operators, parallelization using spatial decomposition is the natural choice. Inside a partition we need only access to the locally stored data. If we want to solve the equations of radiative magnetohydrodynamics we are confronted with the problem that the evaluation of the nonlocal operator requires the access to the complete data on all processors. A possible solution of this problem which avoids costly data exchange would be an iterative algorithm as sketched in [Ded03]. The latter is under construction. 6.2 Performance of the Algorithm for a Realistic Problem in 3d We present the results of a three-dimensional magnetohydrodynamic simulation with our code. The underlying physical problem is the evolution of

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an initially-kinked magnetic flux tube of high magnetic field strength in a gravitationally-layered atmosphere. Since the density of the flux tube at initial time is lower than the density of the surrounding plasma it starts to rise and, when reaching a certain height with less dense material, explodes in a mushroom-like form. For this simulation the efficient Riemann solver, the GLM divergence correction, and the Bgfix scheme for stabilizing the background atmosphere were essential tools for performing this simulation. The left-hand pictures in the series displayed in Fig. 11(a) show the evolution of the physical entropy at times t = 0.0, t = 6.0, t = 9.0, and t = 12.0 in two different kinds of visualization. The corresponding right pictures show the refinement level of the mesh (lower graph) and the mesh together with the partitioning at the given times. We observe that the refinement is concentrated in the spatial domain of the explosion. Furthermore, in this region most of the partitions are located. This underlines the qualitatively good performance of the overall algorithm. To get more insight into the functioning of the algorithm we present Fig. 11(b). It shows the evolution of different characteristic numbers versus the time steps. The peaks in the execution time correspond to load balancing in the time step. From the difference between maximum/minimum runtime and average runtime we observe that condition (50) is true for those time steps. It remains to validate quantitatively that the combination of local refinement/coarsening and load balancing leads to an efficient algorithm. This means one has to compare the results with a computation on a globallyrefined mesh with comparable resolution, which requires no load balancing. Furthermore, the parallel performance (speedup, efficiency, scaling) has to be measured. We refer to [DRSW04] where this has been done and has led to very satisfying results.

7 Conclusions In our project we studied an explicit finite volume scheme for solving the coupled system (1) of radiation magnetohydrodynamics. We implemented the scheme in one, two, and three space dimensions, in a first and second order version on both structured and unstructured grids. The starting point of our discussion was a standard finite volume scheme for solving the equation of ideal magnetohydrodynamics based on approximate Riemann solvers. In the design of every necessary extension to this base scheme, our main concern was its efficiency measured by the error to runtime ratio. The computational costs required for simulating the evolution of a plasma — including the energy transport by radiation — is very high; therefore, the efficiency of the scheme is a central aspect. Since our scheme falls into two parts, evolving the fluid quantities and computing the radiation field, we studied these two parts separately. We took care not to introduce any new stability restriction into the base scheme and not to increase its computational costs either. For

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the solution of the MHD equations, the flux computation is by far the most expensive part. Therefore our first goal was to derive an efficient Riemann solver for a perfect gas. The resulting MHD-HLLEM flux function was the basis for all further extensions of the MHD part of our scheme. We modified this method to include more general pressure laws, leading to the direct extension RGMHD-HLLEM and the ER-HLLEM scheme. The latter one reuses the flux for the perfect gas law within the energy relaxation framework. The remaining modifications were aimed at improving the robustness and accuracy of the scheme. We introduced the GLM–MHD schemes that take the divergence constraint on the magnetic field into account. Furthermore, the Bgfix method improves the approximation of solutions near an equilibrium state. These methods require only small modifications of the flux computation and were, therefore, easy to add to our base scheme. In fact, none of the modifications presented in this study are restricted to the finite volume framework; they can be used in other schemes as well. For the approximation of the radiation source term we presented the (C)ESC schemes. We demonstrated that the ESC framework leads to very efficient schemes that can be easily coupled to any scheme solving the MHD equations. For all the above-mentioned schemes we derived both first and second order versions optimizing the additional costs caused by the higher order extensions compared to their reduction of the approximation error. For the radiation transport solver this led to the higher-order extensions of the shortcharacteristics approach; for the flux computation we introduced the DEOmod reconstruction method. To further reduce the computational costs of the scheme we used local grid adaption to concentrate the grid elements in those regions of the domain where a high resolution is necessary. Moreover, the use of suitable transparent boundary conditions can lead to a significant reduction in the size of the computational domain without a significant increase in the approximation error. As a last step we used parallel computing strategies which allowed us to perform the simulations on a large number of grid elements. The resulting software package is now ready-to-use for complex solar physical applications. To study the quality of our methods, we had to rely primarily on numerical tests since analytical results for complex systems of balance laws are scarce. For an analytical justification of the schemes, we studied simplified settings. For a scalar model problem which we derived from the coupled system (1) we were able to establish the convergence of a first-order finite volume scheme.

References [BVS99]

J.H.M.J. Bruls, P. Vollm¨ oller, and M. Sch¨ ussler. Computing radiative heating on unstructured spatial grids. Astron. Astrophys., 348:233, 1999.

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Fig. 9. Results using the mixed GLM-MHD scheme for a 2d magnetic ﬂux tube simulation, on the left with the Bgﬁx modiﬁcation, on the right using only the base scheme. The Bz component is shown in a small region of the domain (top) and a color representation of the grid reﬁnement (bottom), red indicating a very high grid resolution and blue a very coarse grid. With the Bgﬁx correction the grid consists of about 12000, without the correction of about 32000 elements. Note also the loss of symmetry caused by the base scheme. (See also color ﬁgure, Plate 14.)

Fig. 10. Density for a 2d atmospheric test case for t = 35. The main structure is starting to move through the upper boundary. The result using our transparent boundary conditions (left) is still very close to the reference solutions (middle); for a comparison we included results using Dirichlet conditions (right).

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(a) Entropy isosurface and levelsurface in the central part of the domain (left) and the grid partitioning and reﬁnement (right) at times t = 0.0, 6.0, 9.0, and t = 12.0.

simulation time 70

execution time

60 50 40 30 20 10 0 0

1

2

3

no. of elements, maximum: 6687440 minimum runtime for numerics average runtime for numerics

4

5

6

7

8

maximum runtime for numerics runtime, total: 233928 s

(b) Eﬀect of the load balancing on the runtime. Top: grid partitioning, reﬁnement, and relative load of partitions at t = 6.2, 6.6, 6.65. Bottom: graphs of runtime. Load balancing takes place about every 200 steps (peaks in runtime graph) and requires approximately the runtime of three normal time steps. For example, load balancing occurs shortly before t = 6.2. Due to grid reﬁnement we observe an increase of the load, e.g., for the light green and the brown partition between t = 6.2 and t = 6.6. Thus the maximal and minimal runtime diverge. Consequently, the total runtime increases more than it would be justiﬁed by the growth of the number of elements. Following our load balancing strategy the grid is therefore repartitioned between t = 6.6 and t = 6.65. On average the total runtime increases by the same amount as the number of elements in the grid. Note that this is the optimal behaviour for an explicit ﬁnite volume scheme. Fig. 11. Simulation of an exploding ﬂux tube in 3d on 8 processors of an IBM-SP2. (See also color ﬁgure, Plate 15.)

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[CMIS95]

P. Caligari, F. Moreno-Insertis, and M. Sch¨ ussler. Emerging ﬂux tubes in the solar convection zone. I: Asymmetry, tilt, and emergence latitude. Astrophys. J., 441(2):886–902, 1995. [CP98] F. Coquel and B. Perthame. Relaxation of energy and approximate Riemann solvers for general pressure laws in ﬂuid dynamics. SIAM J. Numer. Anal., 35(6):2223–2249, 1998. [Daf00] C.M. Dafermos. Hyperbolic conservation laws in continuum physics. Springer-Verlag, New York, ﬁrst edition, 2000. [Ded03] A. Dedner. Solving the system of Radiation Magnetohydrodynamics : for solar physical simulations in 3d. PhD thesis, Albert-Ludwigs-Universit¨ at, Mathematische Fakult¨ at, Freiburg, http:/ /www.freidok.uni-freiburg.de/volltexte/1098, 2003. [DEO92] L.J. Durlofsky, B. Engquist, and S. Osher. Triangle based adaptive stencils for the solution of hyperbolic conservation laws. J. Comput. Phys., 98(1):64–73, 1992. oner, C.-D. Munz, T. Schnitzer, and M. We[DKK+ 02] A. Dedner, F. Kemm, D. Kr¨ senberg. Hyperbolic divergence cleaning for the MHD equations. J. Comput. Phys., 175(2):645–673, 2002. [DKRW01a] A. Dedner, D. Kr¨ oner, C. Rohde, and M. Wesenberg. Godunovtype schemes for the MHD equations. In E.F. Toro, editor, Godunov Methods: Theory and Applications, pages 209–216. Kluwer Academic/Plenum Publishers, November 2001. [DKRW01b] A. Dedner, D. Kr¨ oner, C. Rohde, and M. Wesenberg. MHD instabilities arising in solar physics: A numerical approach. In H. Freist¨ uhler and G. Warnecke, editors, Hyperbolic Problems: Theory, Numerics, Applications, volume 140 of International Series of Numerical Mathematics, pages 277–286, Basel, 2001. Birkh¨ auser. [DKRW03] A. Dedner, D. Kr¨ oner, C. Rohde, and M. Wesenberg. Eﬃcient divergence cleaning in three-dimensional MHD simulations. In E. Krause and W. J¨ ager, editors, High Performance Computing in Science and Engineering ’02, pages 323–334. Springer, Berlin, 2003. [DKSW01a] A. Dedner, D. Kr¨ oner, I.L. Sofronov, and M. Wesenberg. Absorbing boundary conditions for astrophysical MHD simulations. In E.F. Toro, editor, Godunov Methods: Theory and Applications, pages 217– 224. Kluwer Academic/Plenum Publishers, November 2001. [DKSW01b] A. Dedner, D. Kr¨ oner, I.L. Sofronov, and M. Wesenberg. Transparent boundary conditions for MHD simulations in stratiﬁed atmospheres. J. Comput. Phys., 171(2):448–478, 2001. [DR] A. Dedner and C. Rohde. Existence, uniqueness, and regularity of weak solutions for a model problem in radiative gas dynamics. In preparation. [DR02] A. Dedner and C. Rohde. FV-schemes for a scalar model problem of radiation magnetohydrodynamics. In R. Herbin and D. Kr¨ oner, editors, Finite Volumes for Complex Applications III: Problems and Perspectives, pages 179–186, Paris, 2002. Herm`es Science Publications. [DR04] A. Dedner and C. Rohde. Numerical approximation of entropy solutions for hyperbolic integro-diﬀerential equations. Numer. Math., 97(3), 2004. [DRSW04] A. Dedner, C. Rohde, B. Schupp, and M. Wesenberg. A parallel, loadbalanced MHD code on locally adapted, unstructured grids in 3d. Comput. Visual. Sci., 7:79–96, 2004.

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Kinetic Schemes for Selected Initial and Boundary Value Problems Wolfgang Dreyer1 , Michael Herrmann1 , Matthias Kunik2 , and Shamsul Qamar1,2 1

2

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, D - 10117 Berlin, Germany dreyer|[email protected] Institute for Analysis and Numerics, Faculty of Mathematics, Otto-von-Guericke University Magdeburg, Postfach 4120, D - 39016 Magdeburg, Germany matthias.kunik|[email protected]

Summary. The hyperbolic system that describes heat conduction at low temperatures and the relativistic Euler equations belong to a class of hyperbolic conservation laws that result from an underlying kinetic equation. The focus of this study is the establishment of an kinetic approach in order to solve initial and boundary value problems for the two examples. The ingredients of the kinetic approach are: (i) Representation of macroscopic ﬁelds by moment integrals of the kinetic phase density. (ii) Decomposition of the evolution into periods of free ﬂight, which are interrupted by update times. (iii) At the update times the data are refreshed by the Maximum Entropy Principle. Key words: Boltzmann-Peierls equation, hyperbolic moment systems, Bose-gas, phonons, relativistic Euler equations, kinetic schemes, maximum entropy principle, shock waves

1 Introduction In this article we study (i) initial value problems for kinetic equations and (ii) initial and boundary value problems for the corresponding hyperbolic moment systems. We consider two different physical phenomena that, however, lead to similar equations which can be solved by kinetic schemes. 1. The evolution of heat in crystalline solids at low temperature is driven by the transport of phonons, which form a gas like structure in the solid. The phonons behave as Bose particles and their evolution may be described by the Boltzmann-Peierls equation (BPE), which is an integro-differential equation for the phase density of the phonon gas. The entropy of a Bose gas and

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the Maximum Entropy Principle (MEP) are used to derive a hierarchy of hyperbolic moment systems. 2. The evolution of transport processes in a gas, whose particles have velocities that are comparable with the speed of light, is described by the relativistic Boltzmann equation. In this study we consider the framework of special relativity and the limiting case of small free flight times of the gas particles. Furthermore we restrict the gas particles to obey Boltzmann statistics, so that local equilibrium is described by the J¨ uttner phase density. The Maximum Entropy Principle (MEP) serves to derive the relativistic Euler equations for the first five moments of the phase density. Regarding their mathematical structure, the two examples have many similarities so that we can apply the same numerical method to solve the two described problems. There numerical method is a kinetic scheme which consists of periods of free flight and update times. In both examples, the periods of free flight is described by the same collision free kinetic transport equation. The macroscopic fields appear as moments of the phase density which are formed by integrals over the kinetic variable. In both cases, the moment integrals may be reduced to integrals over the unit sphere. The update procedure relies in its essential part on the MEP. Thus, we are confronted with the problem whether the MEP exists at all. It was Junk who has pointed out, that the MEP for the Boltzmann equation does not exist, because the moment integrals have an infinite domain. Guided by Junk’s seminal contribution, Dreyer, Junk and Kunik [11] studied the Fokker-Planck equation and proved nonexistence also in that case. However, we could prove the existence of the MEP, for the BPE as well as for the ultra-relativistic Euler equations, because both cases lead to moment integrals over the unit sphere. The described kinetic approach leads to numerical schemes that are first order in time. However, we will describe suitable correction terms that lead to second order schemes. The first part of this report deals with the BPE. At first we introduce a reduced kinetic equation which has a simpler structure than the BPE. Moreover, if we restrict to the macroscopic 1D case, a further simplification of the kinetic equation is possible. Secondly we give a positive existence result for the MEP. Finally we establish kinetic schemes for the kinetic equation as well as for the hierarchy of hyperbolic moment systems. In the second part of this paper we apply the kinetic approach to the ultrarelativistic Euler equations. We write these in terms of the particle density n, the spatial part of the four-velocity u and the pressure p.

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2 The Boltzmann-Peierls Equation In this section we use kinetic schemes in order to solve the Boltzmann-Peierls Equations (BPE) as well as the moments systems that are derived by means of the Maximum Entropy Principle. Here we present a survey of results that are explained in more detail in [9, 30, 17, 16]. Further results concerning the BPE and its moment systems may be found in [14, 18] and the references therein. First we give a brief summary on the kinetic theory of heat conduction in 2.1. In 2.2 we introduce a reduced model with a simplified kinetic variable. However, the reduced equation contains all physically relevant information. Afterwards in 2.3 we discuss the strategy of Extended Thermodynamics and Maximum Entropy Principle. In particular, we derive the moment systems of hyperbolic PDEs that approximate the kinetic equation. Finally, in 2.4 we present the kinetic schemes mentioned above. We conclude with some illustrating numerical examples in 2.5. 2.1 The kinetic theory of heat conduction in solids In 1929, Peierls [35] proposed his celebrated theoretical model to describe transport processes of heat in solids. According to the model the lattice vibrations responsible for the heat transport can be described as an interacting gas of phonons. An overview on phonon theory and its applications is given by Dreyer and Struchtrup in [18]. The BPE is a kinetic equation that describes the evolution of the phase density f (t, x, k) of a phonon gas. The microscopically three dimensional BPE reads ∂f ∂f + cki = Sf, ∂t ∂ xi

(1)

where t, x = (x1 , x2 , x3 ), k = (k1 , k2 , k3 ) denote the time, the space and the wave vector, respectively. The positive constant c is the Debye speed and S abbreviates the collision operator that will be defined below. The moments of the phase density f reflect the kinetic processes on the scale of continuum physics. The most important moments are the energy density e and the heat flux Q = (Q1 , Q2 , Q3 ) which are defined as e(f ) = c |k| f (k) dk, Q(f ) = c2 kf (k) dk. (2) R3

R3

Since f depends on time and space the moments e(f ) and Q(f ) depend on t and x, too. Phonons are identified as Bose particles, see [35, 18]. Thus, the kinetic entropy density-entropy flux pair (h, Φ) is given by

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f f f f 1+ ln 1 + − ln dk, y y y y R3 f ki f f f 1+ ln 1 + − ln dk, ϕi (f ) = y c |k| y y y y h(f ) = y

(3)

(4)

R3

where y = inequality

3 8π 3 .

The kinetic equation (1) implies the following entropy ∂ h(f ) ∂ Φj (f ) + ≥ 0. ∂t ∂ xj

(5)

In contrast to ordinary gas atoms, phonons may interact by two different collision processes, called R- and N-processes. N -processes describe phononphonon interactions, while R-processes take care of interactions of phonons and lattice impurities. The N -processes conserve energy as well as momentum, while the R-processes conserve only the energy. The Callaway approximation of the collision operator is a suitable simplification of the actual interaction processes (cf. [2, 18]). The Callaway collision operator is written as the sum of two relaxation operators modeling the R- and N -processes separately. There holds ' 5 6 1& Pα f − f , α ∈ R, N . (6) Sf = SR f + SN f, Sα f = τα

The positive constants τR and τN are the relaxation times, PR and PN are two nonlinear projectors. The phase densities PR f and PN f are defined as solutions of the two optimization problems 5 6 h(f ′ ) : e(f ′ ) = e(f ) , h(PR f ) = max (7) ′ f 5 6 h(f ′ ) : e(f ′ ) = e(f ), Q(f ′ ) = Q(f ) . (8) h(PN f ) = max ′ f

These maximization problems may be solved explicitly. The resulting expressions for PR and PN in terms of e and Q may be found in [18, 30]. 2.2 The reduced Boltzmann-Peierls Equation

In this section we recall results from [9, 30] in order to derive a reduced kinetic equation for a reduced phase density. This procedure relies on the observation that for any solution f of (1) there exists a corresponding solution of a reduced equation that determines all physically important moments of f . For any phase density f depending on the wave vector k ∈ R3 we define the reduced phase density ϕf of f depending on a normal vector n ∈ S 2 by ϕf (n) = c

∞ 0

|k|3 f (|k| n) d|k|,

(9)

Kinetic Schemes for Selected Initial and Boundary Value Problems

207

where n = (n1 , n2 , n3 ) = k/|k|. Let m be a homogeneous moment weight of degree 1, i.e. m(λk) = λm(k) for all λ ≥ 0, and let u be the corresponding moment function. Note that all physically important moments are homogeneous of degree 1. A straight forward calculation yields (10) u(f ) = c m(n)ϕf (n) dS(n), S2

whereas dS(n) denotes the usual measure on the unit sphere S 2 . We conclude that the moment of f is given by a respective moment of it’s reduced phase density ϕf . In particular we find e(f ) = e(ϕf ) and Q(f ) = Q(ϕf ) with e(ϕ) = ϕ(n) dS(n), Q(ϕ) = c nϕ(n) dS(n). (11) S2

S2

Furthermore, we introduce an entropy density-entropy flux pair (h, Φ) for reduced phase densities by 3 3 (12) h(ϕ) = µ ϕ 4 (n) dS(n), Φi (ϕ) = µ c ni ϕ 4 (n) dS(n), S2

S2

where µ is a given constant. We summarize the main results in the following theorem. Theorem 2.1 1. There exist two operators ΘR and ΘN such that ϕ(Pα f ) = Θα (ϕf )

for

α ∈ {R, N }.

(13)

2. If f is a solution of the BPE, then its reduced phase density ϕf is a solution of the following reduced BPE ∂ϕ ∂ϕ + cni = Ψ ϕ, ∂t ∂ xi where Ψ = ΨR + ΨN and Ψα ϕ = τ1α (Θα ϕ − ϕ). 3. ΘR and ΘN have similar properties as PR and PN , i.e. 5 6 ′ ′ h(ϕ h(ΘR ϕ) = max ) : e(ϕ ) = e(ϕ) , ϕ′ 5 6 ′ ′ ′ h(ϕ ) : e(ϕ ) = e(ϕ), Q(ϕ ) = Q(ϕ) . h(ΘN ϕ) = max ′ ϕ

(14)

(15) (16)

4. There holds ΘR ϕ =

e , ΘN ϕ = 4π

3e(4 − F )3 6 ( '4 , F = & & '2 . F n·Q 4πF 1 − 4 c e 1 + 1 − 43 |Q| ce

(17)

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5. The reduced BPE implies the entropy inequality ∂ h(ϕ) ∂ Φi (ϕ) + ≥ 0. ∂t ∂ xi

(18)

6. The reduced BPE leads to an hierarchy of balance laws. For any vector of moment weights m(n) ! we obtain ∂ !u(ϕ) ∂ F!i (ϕ) ! + = Π(ϕ), ∂t ∂ xi

(19)

where !u(ϕ) =

m(n)ϕ(n) ! dS(n),

(20)

cni m(n)ϕ(n) ! dS(n),

(21)

S2

F!i (ϕ) =

S2

! Π(ϕ) =

S2

, m(n) ! Ψ ϕ (n) dS(n),

(22)

denote the densities, the fluxes and the productions, respectively. One-dimensional Reduced Kinetic Equation To conclude this section we summarize results from [30] that allow a further simplification of the reduced BPE. In the macroscopically one dimensional case we have x = (x, 0, 0) and Q = (Q, 0, 0). We introduce the new variables −1 ≤ ξ ≤ 1, 0 ≤ ϑ ≤ 2π by % % n1 = ξ , n2 = 1 − ξ 2 sin ϑ , n3 = 1 − ξ 2 cos ϑ , (23)

with the surface element dS(n) = dξdϑ. Furthermore we eliminate the angle ϑ by setting 2π ϕ(t, x, 0, 0, n) dϑ. (24) ϕ(t, x, ξ) = 0

The reduced BPE (14) then further reduces to ∂ϕ ∂ϕ 1 1 + cξ = (ΘR ϕ − ϕ) + (ΘN ϕ − ϕ) , ∂t ∂x τR τN where ΘR ϕ and ΘN ϕ are given by expressions similar to (17).

(25)

Kinetic Schemes for Selected Initial and Boundary Value Problems

209

2.3 The Maximum Entropy Principle The strategy of Extended Thermodynamics The objective of Extended Thermodynamics is to solve initial and boundary value problems for truncated moment systems instead of solving the kinetic equation. To this end only the first N equations of the infinite hierarchy of moment equation are used, and the Maximum Entropy Principle (MEP) serves to close the truncated system. For the formulation of the MEP we start with a fixed N -dimensional vector m ! = m(n) ! of moment weights. The vector m ! induces a vector !u of densities, cf. (20). In the following we call the pair (m, ! !u) a moment pair of dimension N . The MEP corresponding to (m, ! !u) can be formulated as follows. For any given phase density ϕ we seek a phase density ϕM that maximizes the entropy, i.e. h (ϕM ) = max { h (ϕ′ ) : !u (ϕ′ ) = !u (ϕ) } . ′ ϕ

(26)

In order to indicate that ϕM obviously depends on ϕ, we write ϕM = ΘM ϕ. The MEP assumes, that for any reasonable phase density ϕ there always exists a phase density ϕM = ΘM ϕ that maximizes the entropy according to (26). Thus, the MEP ends up with an operator ΘM with the following properties 2 1. ΘM is a nonlinear projector, i.e. ΘM = ΘM . 2. ΘM f depends only on the moments !u (ϕ), i.e. !u (ϕ1 ) = !u (ϕ2 ) implies ΘM ϕ1 = ΘM ϕ2 .

We call the operator ΘM the MEP projector corresponding to the moment pair (m, ! !u). We mention that, according to (15) and (16), the operators ΘR and ΘN appearing in the reduced collision operation Ψ are also MEP projectors. Next we consider the closure problem of Extended Thermodynamics. We start with a finite number of balance equations derived from the kinetic equation, cf. (19). As before we denote the corresponding vectors of densities and fluxes by !u and F!j , respectively. The densities are now considered as the independent variables. Since in general the fluxes F!j do not depend on the densities !u, there arises the so called closure problem. The closure problem is solved by a reasonable ansatz that provides the fluxes and the productions as functions of the densities. A very popular closure ansatz in Extended Thermodynamics is the MEP leading to the so called MEP moment system, which is achieved from (14) by a formal replacement of the phase density ϕ by the MEP density ΘM ϕ: ∂!u (ΘM ϕ) ∂ F!j (ΘM ϕ) + = !u (Ψ ΘM ϕ) . ∂t ∂xj

(27)

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W. Dreyer, M. Herrmann, M. Kunik, S. Qamar

Since ΘM ϕ depends on ϕ via the densities !u, the system (27) is in fact a closed system with respect to the variables !u. The resulting system of PDEs is symmetric hyperbolic. For further details we refer to the standard textbook on Rational Extended Thermodynamics by M¨ uller/Ruggeri ([33]) and to [9]. The existence of the MEP projector is a nontrivial and subtle problem, because there are counterexamples in which the MEP fails. Junk has observed, that for the Boltzmann Equation the corresponding MEP density does not exist in general. A detailed discussion of this problem may be found in [24, 25, 11]. However, in the case of the reduced BPE these problems do not arise. This topic will be discussed in the next subsection. The MEP and the reduced equation We apply the MEP to the reduced kinetic equation and to the entropy (12). In particular, we give a positive existence result for MEP projectors ΘM . Let (m, ! !u) be a moment pair of dimension N . We call the pair (m, ! !u) admissible, if (i) the energy density e is among the components of !u and if (ii) the components of m ! are smooth (at least C 3 ). In the following, we consider exclusively admissible pairs (m, ! !u). For r ∈ {1, ∞} we define 5 6 Lr+ (S 2 ) = ϕ ∈ Lr (S 2 ) : ∃ δ = δ(ϕ) > 0 with ϕ ≥ δ a.e. . (28) For given ϕ ∈ L1+ (S 2 ) the MEP leads to the following optimization problem with constraints. Problem 2.1

5 6 ′ ′ 1 2 ′ h(ϕ h(ϕM ) = max ) : ϕ ∈ L (S ), ! u (ϕ ) = ! u (ϕ) . + ′ ϕ

(29)

Next we introduce the conjugate functional h⋆ of the entropy h, that reads 4 1 3 ⋆ h (ψ) = − µ ψ −3 (n) dn. (30) 3 4 S2

2 ⋆ Note that h⋆ is well defined for all ψ ∈ L∞ + (S ). Using this functional h we formulate the following dual problem of 2.1, namely

Problem 2.2

6 5 ˜ Λ ˜ Λ) !M ) = min h( ! : Λ ! ∈ DM , h( Λ 6 5 2 ! ∈ Rn : Λ !·m DM := Λ ! ∈ L∞ (S ) , + ' & ˜ Λ) ! = −h⋆ Λ !·m ! h( ! + !u(ϕ0 ) · Λ,

(31) (32) (33)

Kinetic Schemes for Selected Initial and Boundary Value Problems

211

which is an optimization problem without constraints. There is a close relation !M of Problem between the Problems 2.1 and 2.2. In particular, the solution Λ 2.2 are the Lagrange multipliers corresponding to the solution ϕM of Problem 2.1. The main results concerning the MEP are summarized in the following theorem. Theorem 2.2 For any ϕ ∈ L1+ (S 2 ) there holds 1. There exists a unique solution ϕM of problem 2.1. !M of problem 2.2. 2. There exists a unique solution Λ 3. There holds the identity 4 3 µ ϕM = . !M · m 4Λ !

(34)

The proof of a similar result for two dimensions is contained in [9]. 2.4 Kinetic schemes Kinetic solutions of the kinetic equation In this section we derive kinetic schemes that allow the construction of approximate solutions of (14) in the time interval [0, ∞). The solution of the Cauchy problem of the collisionless kinetic equation ∂ϕ ∂ϕ + cni = 0, ∂t ∂ xi

(35)

is given by the free transport group T (t) acting on phase densities ϕ depending on x and n according to & ' T (t)ϕ (x, n) := ϕ(x − ctn, n). (36)

In particular, T (t)ϕ0 is a solution of (35) with initial data ϕ0 . The solution of the corresponding Cauchy problem for the reduced BPE (14) can be represented by means of Duhamel’s principle as 0

ϕ(t) = T (t)ϕ +

t 0

, T (t − s) ΨR ϕ(s) + ΨN ϕ(s) ds.

(37)

Note that for any t the function ϕ(t) is a phase density depending on x and n. Obviously, the formula (37) is not explicit in ϕ(t). In order to find approximate solutions, we shall replace the integrals in (37) by Riemann sums. If we introduce a small parameter τ˜ > 0, we find , τ˜ T (t − k˜ τ ) ΨR ϕ(k˜ τ ) + ΨN ϕ(k˜ τ) . (38) ϕ(t) ≃ T (t)ϕ0 + k : 0 ≤ k τ˜ < t

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This approximate representation of solutions of (14) immediately gives rises to an explicit semi discrete kinetic scheme. Using the abbreviations ϕk± = lim ϕ(k˜ τ ± t) for t↓0

k ≥ 0,

(39)

and ϕ0− = ϕ0 , we find by a straight forward calculation, that (38) with equality sign is equivalent to ϕ(k˜ τ + t) = T (t)ϕk+ , 0 < t < τ˜, & τ˜ τ˜ τ˜ τ˜ ' ΘR ϕk− + ΘN ϕk− + 1 − − ϕk+ = ϕk− . τR τN τR τN

(40) (41)

The time intervals (k˜ τ , k˜ τ + τ˜) are called transport intervals, whereas the multiples of τ˜ are called update times. For any strictly positive initial data ϕ0 and sufficiently small parameter τ˜, the kinetic scheme (40)-(41) defines an approximate solution ϕ of (14) with the following properties. Lemma 2.1 1. ϕ(t) is strictly positive for all t and there exist the left-hand and right-hand limits at the update times. 2. ϕ satisfies exactly the conservation of energy, that is ∂ e(ϕ) ∂ Qi (ϕ) + = 0. ∂t ∂ xi

(42)

3. The entropy production is nonnegative, i.e. ∂ h(ϕ) ∂ Φi (ϕ) + ≥ 0. ∂t ∂ xi

(43)

The equation (42) and the inequality (43) are satisfied in the sense of distributions. For further details see [9]. Kinetic solution of the MEP moment systems In this section we shall briefly describe how kinetic schemes can be used in order to solve moment systems of the reduced kinetic equation that are derived by means of the MEP. It will turn out, that there is a close relationship between kinetic schemes for the kinetic equation and kinetic schemes for its moment systems. A more detailed discussion is contained in [9]. In the following we consider an admissible moment pair (!u, m) ! together with the resulting MEP projector ΘM (cf. Subsection 2.3). The moment system corresponding to !u is given by

Kinetic Schemes for Selected Initial and Boundary Value Problems

213

∂ !u(ΘM ϕ) ∂ F!j (ΘM ϕ) + = !u(Ψ ΘM ϕ). ∂t ∂ xj

(44)

The standard kinetic approach of the Cauchy problem for this moment system can be summarized as follows. 1. We start with initial data of the form ΘM ϕ0 that correspond to the given macroscopic initial data !u0 , i.e. !u0 = !u(ΘM ϕ0 ). 2. For a small but fixed time τM we solve the kinetic equation (14) for in the time interval [0, τM ], at least approximately. 3. The resulting phase density will be used to calculate the moments !u. 4. At the time τM the phase density ϕ(τM ) will be replaced by the MEP phase density ΘM ϕ(τM ) and we restart the scheme. Kinetic schemes of this kind are well known and studied by many authors for moment systems relying on various kinetic equations ( see [10, 12, 13, 24, 36] for moment systems of the Boltzmann Equation, [14, 15] for a moment system of the BPE). In view of this standard approach we consider the following kinetic equation ∂ϕ ∂ϕ + cni = ΨR ϕ + ΨN ϕ + ΨM ϕ (45) ∂t ∂ xi The newly introduced quantity is ' 1 & ΨM ϕ = ΘM ϕ − ϕ (46) τM

that is again a relaxation operator with an artificial relaxation time τM . If we apply the moment maps !u to (45), we formally obtain for the limiting case τM →0 the system (44). We can thus interpret equation (45) as a kinetic approximation of the moment system (44). Next we apply the approach from above to the kinetic equation (45). There result the following kinetic scheme ϕ(k˜ τ + t) = T (t)ϕk+ , 0 < t < τ˜, τ˜ τ˜ τ˜ ΘR ϕk− + ΘN ϕk− + ΘM ϕk− + ϕk+ = τR τN τM & τ˜ τ˜ τ˜ ' ϕk− . 1− − − τR τN τM

(47)

(48)

This scheme differs from (40)-(41) just in the update rule (48). However, all assertions of Lemma 2.1 remain valid. Fully Discretized First Order Scheme In order to get a fully discretized piecewise constant solution of the reduced BPE (25), we first define a grid in the reduced phase-space consisting of cells Ci,j = Ii × Jj centered around (xi = i∆x, ξj = j∆ξ),

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# # ∆x , (x, ξ) ∈ R2 ##|x − xi | ≤ 2

Ci,j =

|ξ − ξj | ≤

∆ξ 2

,

where ∆x = xi+ 21 − xi− 21 and ∆ξ = ξj+ 21 − ξj− 12 . The cell-average of ϕ at time t = tn over the cell Ci,j is given by xi+ 1 ξj+ 1

ϕni,j

1 = ∆x∆ξ

2

2

ϕ(t, x, ξ) dξdx .

(49)

xi− 1 ξj− 1 2

2

With the characteristic function χi,j (x, ξ) of the cellCi,j we can write the desired piecewise constant phase density in the form ϕni,j χi,j (x, ξ). 2 3 2 3 Integrating (40)-(41) over xi− 12 , xi+ 12 × ξj− 21 , ξj+ 21 and dividing by ∆x∆ξ, we get for a time step τ˜ = ∆t ' & n 2 n n n ϕn+1 (50) − F = ϕ − λ F 1 1 i,j i,j i− ,j + ∆t Si,j + O(∆t) , i+ ,j 2

where λ =

∆t ∆x ,

2

and for the CFL condition ∆t ≤ n Si,j = n Fi+ 1 ,j = 2

α∈R,N

∆x 2

we have

1 , Θα ϕni,j − ϕni,j , τα

(51)

c, ξj ϕni,j + ξj ϕni+1,j − |ξj |∆ϕni,j , 2

(52)

where ∆ϕni,j = ϕni+1,j − ϕni,j . In order to get the average values of the moments from this discrete phase density at any time tn in each cell Ii we use the Riemann sums as eni

= ∆ξ

Nξ j=1

ϕni,j ,

Qni

= c∆ξ

Nξ

ξj ϕni,j ,

Nin

j=1

= ∆ξ

Nξ

ξj2 ϕni,j ,

(53)

j=1

where Nξ is the number of elements in the interval −1 ≤ ξ ≤ 1. Second Order Extension of the Scheme For the second order accuracy in space and time we have the following three steps. (I) Data Reconstruction: nStarting with a piecewise-constant solution in time and phase-space, ϕi,j χi (x), one reconstruct a piecewise linear (MUSCL-type) approximation in space, namely n x (x − xi ) n (54) χi,j (x, ξ) . ϕj (x) = ϕi,j + ϕi,j ∆x Here, ϕxi,j abbreviates a first order discrete slope.

Kinetic Schemes for Selected Initial and Boundary Value Problems

215

The extreme points x = 0 and x = ∆x, in local coordinates correspond to the intercell boundaries in general coordinates xi− 21 and xi+ 12 , respectively, see Figure 1. The values of ϕi,j at the extreme points are 1 x 1 x n R n ϕL i,j = ϕi,j − ϕi,j , ϕi,j = ϕi,j + ϕi,j , 2 2

(55)

and are usually called boundary extrapolated values. A possible computation 1 ∆ϕi+1,j ∆x

x

xi+1

ϕR i,j

ϕn i+1,j

xi+ 1

2

xi

ϕR i−1,j

ϕn+1 i,j

ϕn i,j

xi− 1 2

xi−1

ϕn i−1,j

ϕL i,j

ϕL i−1,j

t

Fig. 1. Second order reconstruction.

of these slopes, which results in an overall non-oscillatory schemes (consult [39]), is given by family of discrete derivatives parameterized with 1 ≤ θ ≤ 2, i.e., for any grid function ϕi,j we set θ ϕxi,j = M M θ∆ϕi+ 21 ,j , (∆ϕi− 21 ,j + ∆ϕi+ 12 ,j ), θ∆ϕi− 21 , j . 2 Here, ∆ denotes the forward differencing, ∆ϕi+ 12 ,j = ϕi+1,j − ϕi,j , and M M denotes the min-mod nonlinear limiter ⎧ ⎨ mini {xi } if xi > 0 ∀i , M M {x1 , x2 , ...} = maxi {xi } if xi < 0 ∀i , (56) ⎩ 0 otherwise .

The interpolant (54), is then evolved exactly in time and projected on the cell-averages at the next time step. R (II) Evolution: For each cell Ii , the boundary extrapolated values ϕL i,j , ϕi,j 1 in (55) are evolved for a time 2 ∆t by

216

W. Dreyer, M. Herrmann, M. Kunik, S. Qamar L ϕˆL i,j = ϕi,j −

! ∆t n λ R L + Fi,j − Fi,j S , 2 2 i,j

R ϕˆR i,j = ϕi,j −

! L

(57)

∆t n λ S , F R − Fi,j + 2 i,j 2 i,j

L R R where Fi,j = cξj ϕL i,j and Fi,j = cξj ϕi,j . In order to calculate the source term at the half time step we use

ϕˆi,j = ϕni,j − n = cξj ϕni,j and where Fi,j

eˆi = ∆ξ

Nξ j=1

! ∆t n λ n n + Fi+1,j S , − Fi,j 2 2 i,j ϕˆi,j ,

ˆ i = c∆ξ Q

Nξ

ξj ϕˆi,j .

(58)

(59)

j=1

(III): Finally we use the conservative formula (50) in order to get the discrete phase density at the next time step ' & ∆t n+ 12 n+ 12 n + ϕn+1 = ϕ − λ F (Θα ϕˆi,j − ϕˆi,j ) , (60) − F i,j i,j i+ 12 ,j i− 21 ,j τα α∈R,N

where the numerical fluxes are defined by ! c n+ 1 Fi+ 12,j = ξj ϕˆR ˆL ˆL ˆR (61) i,j + ξj ϕ i+1,j − |ξj |(ϕ i+1,j − ϕ i,j ) . 2 2 The above scheme can be extend to the two-dimensional case in a dimensionby-dimension manner. We have used the same algorithm in order to extend the scheme to the two-dimensional case. In the two-dimensional case the scheme use again the MUSCL-type initial reconstruction and min-mod non-linear limiters for the calculation of discrete slopes. Our two-dimensional examples presented in this article use the same approach. 2.5 Numerical Examples The results of the preceding section shall be illustrated by some numerical examples. Example 1: The phenomenon of second sound The first two examples we have taken from [9] although there we rely on the microscopic two dimensional version of the BPE. However, the qualitative behavior does not depend on the number of microscopic dimensions. For both examples we assume that τR = ∞. Further we assume that the phase density only depends on x1 . In order to simulate interesting phenomena, we consider the following macroscopic initial data for energy density e and the momentum density Q.

Kinetic Schemes for Selected Initial and Boundary Value Problems

0 1.5 e (x1 ) = 1.0 0

if |x1 | ≤ 0.01 , if |x1 | > 0.01

217

Q1 (x1 ) = 0 .

For the details of the discretization we refer to [9]. 1.02

1.00 -1.01

Energy density, t 1.6

Energy density, t 1.2

0

Energy density, t 2.

1.02

1.01

1.00 -1.01

1.01

0

1.01

1.00 -1.01

0

1.01

Fig. 2. Example 1. Evolution of the energy pulse for τN = 2.0 Energy density, t 1.2

1.02

1.00 -1.01

1.01

0

1.01

1.00 -1.01

Energy density, t 2.

Energy density, t 1.6 1.01

0

1.01

1.00 -1.01

0

1.01

Fig. 3. Example 1. Evolution of the energy pulse for τN = 1.0 Energy density, t 1.2

1.02

1.00 -1.01

1.01

0

1.01

1.00 -1.01

Energy density, t 1.6

Energy density, t 2. 1.01

0

1.01

1.00 -1.01

0

1.01

Fig. 4. Example 1. Evolution of the energy pulse for τN = 0.5

We study the evolution of the initial energy pulse according to different values of τN (τN = 2., τN = 1. and τN = 0.5). The Figures 2-4 show the spatial dependence of the energy density at different times (t = 1.2, t = 1.6 and t = 2.0). According to [18] we can interpret the results as follows. For large values of τN , as in Figure 2, the pulse is ballistic and its fronts move with the Debye speed c to the left and to the right. Figure 4 illustrates the case of small τN . Here, the shape of the pulse reflects the characteristic behavior of the so called second sound, that propagates with a speed less than c. In Figure 3 we observe a transition regime. The pulse starts as a ballistic pulse. After about 1.6 time units it changes its shape and becomes second sound. Example 2: Kinetic equation versus MEP moment systems This example illustrates the relationship between solutions of the kinetic equation and solutions of the moment systems. The initial data are the same as in the first example, the relaxation time τN is set to 0.7. The energy density

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W. Dreyer, M. Herrmann, M. Kunik, S. Qamar Kinetic Equation, t 0.6

1.04

Kinetic Equation, t 1.3 1.01

1.00 1.01

0

1.01

1.01

1.00 1.01

0

1.01

Kinetic Equation, t 2.

1.00 1.01

0

1.01

Fig. 5. Example 2. Evolution of the energy pulse according to the kinetic equation.

corresponding to the reduced BPE is depicted in Figure 5, whereas Figure 6 show the evolution of the initial energy pulse according to various moment systems. We mention, that the moment system of order n consists of 2n + 1 independent balance equations. For the details we refer to [9]. The Figures 5 and 6 reveal, that moment systems with a small number of moments produce quite bad approximations. However, the results become better if the number of moments is increased. Finally we have a good correspondence of the kinetic equation and of the moment system of order 41 in Figure 6. The rows of Figure 6 correspond to different hyperbolic systems with 3, 4, 21, and 41 moments as variables, respectively. Thus the appearing pulses indicate the various waves of the different systems, and are not induced by numerical deficiencies. In particular, the number of pulses depend on the number of moments but not on the numerical discretization. The Table 1 shows that the numerical effort for calculating the MEP projectors ΘM increases tremendously with the number of moments. Example 3: Two Interacting Heat Pulses This test problem demonstrates the interaction of two heat pulses, which leads to a large increase of the energy density at the collision point during a short time interval. The initial data are ⎧ ⎧ 1, x ≤ 0.3 0, x ≤ 0.3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.3 ≤ x ≤ 0.4 ⎨ 2 , 0.3 ≤ x ≤ 0.4 ⎨1 , 0.4 ≤ x ≤ 0.6 . (62) e(0, x) = 1 , 0.4 ≤ x ≤ 0.6 , Q(0, x) = 0 , ⎪ ⎪ ⎪ ⎪ 2 , 0.6 ≤ x ≤ 0.7 −1 , 0.6 ≤ x ≤ 0.7 ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ 1, x ≤ 1.0 0, x ≤ 1.0 Table 1. Comparison of Computational Time. Equation Kinetic System System System System

equation with 3 moments with 4 moments with 21 moments with 41 moments

Figure 5 6 6 6 6

independent CPU time moments 100% = 169 sec 5 1686% 7 2334% 41 10563% 81 19951%

Kinetic Schemes for Selected Initial and Boundary Value Problems 1.11

System with 3 moments, t 0.6

System with 3 moments, t 1.3 1.07

1.05

219

System with 3 moments, t 2.

1.00 1.00 1.00 1.01 0 1.01 1.01 0 1.01 1.01 0 1.01 System with 4 moments, t 0.6 System with 4 moments, t 1.3 System with 4 moments, t 2. 1.09 1.05 1.03

1.00 1.00 1.00 1.01 1.01 0 0 1.01 1.01 1.01 0 1.01 System with 21 moments, t 0.6 System with 21 moments, t 1.3 System with 21 moments, t 2. 1.02 1.01

1.04

1.00 1.00 1.00 0 1.01 1.01 1.01 1.01 0 1.01 0 1.01 System with 41 moments, t 0.6 System with 41 moments, t 2. System with 41 moments, t 1.3 1.04 1.01 1.01

1.00 1.01

0

1.01

1.00 1.01

0

1.01

1.00 1.01

0

1.01

Fig. 6. Example 2. Evolution of the energy pulse according to various moment systems

We solve the BPE for the above problem at time t = 0.2 for two values of τN , i.e., τN = 1 and τN = 0.1, while τR = 1.0. Figure 7 shows the results. From the comparison of the initial and final curves of the energy density, we observe a large increase of the energy density e at the collision point x = 0.5. Figures 7 give the comparison between the kinetic upwind and the kinetic central schemes for both, first order and second order. The first order kinetic central scheme is exactly the same as the well known first order LaxFriedrichs scheme. The second order central scheme is a staggered extension of the first order Lax-Friedrichs scheme, which exactly following the approach of Nessyahu and Tadmor [34, 23]. The first order and second order upwind schemes are the usual kinetic flux vector splitting (KFVS) schemes, see Xu [42, 41]. Both second order central and upwind schemes utilize the MUSCLtype approach for the linear reconstruction. Example 4: Heat Pulse in 2D In this example we solve a two-dimensional hyperbolic moment system. We consider a two-dimensional energy pulse inside a square box of sides length 0.02 with out-flow boundaries. Initially the heat fluxes are zero. The energy

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W. Dreyer, M. Herrmann, M. Kunik, S. Qamar energy density at t = 0.0

energy density, τ = 0.1, τ = 1.0 , t = 0.2

energy density, τ = 0.5 , τ = 1.0 , t = 0.2

2.5

N

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first order upwind scheme first order central scheme second order upwind scheme second order central scheme

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first order upwind scheme first order central scheme second order upwind scheme second order central scheme

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heat flux at t =0.0

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first order upwind scheme first order central scheme second order upwind scheme second order central scheme

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Fig. 7. Example 3: Evolution of energy and heat ﬂux. energy density , t = 1.5

energy density at y = 1.0

1.002

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second order KFVS scheme second order central scheme

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heat flux at y = 1.0 second order KFVS scheme second order central scheme

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Fig. 8. Example 4: Evolution of energy density and heat ﬂux in 2D.

density is 1.5 inside a small square box of sides length 0.02 in the center of the large box, while energy density is unity elsewhere. The results are shown at t = 1.2 in Figure 8. In all the results we have used 200 × 200 mesh points. We take τR = ∞. In Figure 8, KFVS is an abbreviation for the kinetic flux vector splitting (KFVS) schemes, see Xu [42, 41].

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Example 5: Explosion in a Box. Here we also solve a two-dimensional hyperbolic moment system. We consider a two-dimensional energy pulse inside a square box of sides length 2.0, with periodic boundaries. Initially the heat fluxes are zero. The energy density is energy density at t = 0.5

2

1

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Fig. 9. Example 5: Explosion in a box problem.

2.0 inside a small square box of sides length 0.5 in the center of the large box, while energy density is unity elsewhere. The results are shown in Figures 9 at t = 0.5, t = 1.5 and t = 2.0. In all the results we have used 300 × 300 mesh points. We take τR = ∞.

3 Relativistic Euler Equations 3.1 Introduction We consider gas flows with thermal and macroscopic velocities that both are comparable with the speed of light. In this case, space and time are coupled and the relativistic Euler equations of gas dynamics become more complicate than the classical ones. However, in some fixed reference frame it is still possible to write the relativistic Euler equations as a first order hyperbolic system. Relativistic gas dynamics plays an important role in areas of astrophysics, high energy particle beams, high energy nuclear collisions and free-electron

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laser technology. Here we consider exclusively the ultra-relativistic limit within the framework of special relativity. Kinetic approaches to solve the classical Euler equations of gas dynamics were successfully applied to several initial- and boundary value problems, see for example Reitz [38], Deshpande [7, 8], Xu [42, 41], Dreyer and Kunik [12], Dreyer, Herrmann, Kunik [10], and Qamar [37]. Some interesting links between the Euler system and the so called kinetic BGK-model, which was introduced by Bhatnagar, Gross and Krook [1], are discussed in the textbooks by Cercignani [3] as well as by Godlewski and Raviart [22]. J¨ uttner [26] extended the non-relativistic kinetic theory of gases, which was developed by D. Bernoulli, Clausius, Maxwell and Boltzmann, to the domain of relativity. He succeeded in deriving the relativistic generalization of the Maxwellian equilibrium phase density. Later on this phase density and the whole relativistic kinetic theory was structured in a well organized Lorentzinvariant form, see Chernikov [4], [5], M¨ uller [32] and the textbook of deGroot, van Leeuven and van Weert [6]. In the textbook of Weinberg [40] one can find a short introduction to special relativity and relativistic hydrodynamics with further literature also on the imperfect fluid (gas), see for example Eckart’s seminal papers [19, 20, 21]. In [29, 27, 28, 31, 37] Kunik, Qamar and Warnecke have formulated two different kinetic schemes in order to solve the initial and boundary value problems for the ultra-relativistic Euler equations as well as in the general case. The first kind of kinetic schemes are discrete in time but continuous in space. These schemes are explicit and unconditionally stable. Furthermore, the schemes are multi-dimensional and satisfy the weak form of conservation laws for mass, momentum, and energy, as well as an entropy inequality. The schemes preserve the positivity of particle density and pressure for all times and hence they are L1 −stable. Moreover, these schemes may be extended to account for boundary conditions, see [29, 27, 31, 37]. The second kind of kinetic schemes are discrete both in time and space, see [28, 37] and have an upwind conservative form. We use flux vector splitting in order to calculate the free flight moment integrals. The structure of the light cone implies a natural CFL condition. These schemes are called kinetic flux vector splitting (KFVS) schemes which we have extended to the two-dimensional case by dimension splitting. We use a MUSCL-type data reconstruction to obtain second order accuracy. In the following we restrict to the ultra-relativistic limit, where we meet a simpler mathematical structure. In particular, all moments are completely determined by surface integrals with respect to the unit sphere. Due to this fact, the ultra-relativistic Euler equations may be treated similar to the moment systems of the Boltzmann-Peierls equation.

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223

3.2 The ultra-relativistic Euler equations The coordinates with respect to a fixed reference frame are given by the 4vector xµ , µ ∈ {0, 1, 2, 3}, where x0 = t is the observer time. The three vector x = xi , i ∈ {1, 2, 3}, denotes the spatial coordinates of any event xµ . For simplicity we set c = = kB = 1. Furthermore we assume that the metric tensor gµν is given by a diagonal matrix gµν = g µν = diag(1, −1, −1, −1). The kinetic variable is the four-momentum of the individual gas particles q µ = (q 0 , q) with the spatial part q = (q 1 , q 2 , q 3 )T . However, not all components of the 4-momentum are independent, because q µ q µ = m2 ,

(63)

where m is the rest mass of the particles. The invariant volume element dω of the momentum space is given by dω =

1 1 1 2 3 dq dq dq = d3 q. q0 q0

(64)

The phase density f (xµ , q m ) ≡ f (t, x, q) gives the number density of particles in the element dω at xµ . From now on we consider exclusively particles without rest mass, i.e. m = 0, so that q0 ≡ |q|

and dω =

d3 q |q|

(65)

This is the ultra-relativistic limit, and the macroscopic quantities that appear in the relativistic Euler equations can be calculated from the following moments of the phase density d3 q and (66) N µ = N µ (t, x) = q µ f (t, x, q) |q| 3 d3 q , (67) T µν = T µν (t, x) = q µ q ν f (t, x, q) |q| 3 which give the particle 4-vector and the energy-momentum tensor, respectively. Furthermore we consider exclusively non-degenerate gas particles so that the entropy four vector is given by d3 q µ µ . (68) S = S (t, x) = − q µ f (t, x, q) ln (f (t, x, q)) |q| 3 There are conservation laws for N µ , T µν and an inequality in conservative form for S µ , viz. ∂N µ = 0, ∂xµ

∂T µν = 0, ∂xµ

∂S µ ≥ 0. ∂xµ

(69)

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We read off from (66)-(69) the interpretations: N 0 - particle density, N i particle flux vector, T 0j - momentum density, T ij - momentum flux, T 00 energy density, T i0 - energy flux, S 0 - entropy density, and S i - entropy flux, where i, j ∈ {1, 2, 3}. We conclude from the symmetry T µν = T νµ that the momentum flux is equal to the energy flux. Note that the particle flux vector is not equal to the momentum density, as it is the case in the non-relativistic limit. Next we introduce the macroscopic 4-velocity uµ by % 1 u µ = N µ , n = N ν Nν , (70) n so that uµ uµ = 1. We define the local rest frame of the gas by uµ = (1, 0, 0, 0). We can use uµ and the combination hµν = (uµ uν − gµν ) to define further macroscopic fields that have a suggestive meaning in the local rest frame. These are e = uµ uν T µν - internal energy density, p = 1/3hµν T µν - pressure, Qµ = −hµλ uν T λν - heat flux, and p<µν> = (hµλ hνκ − 1/3hµν hλκ )T λκ pressure deviator, where p<µν> denotes the trace free part of pµν . There follows N µ = nuµ

and T µν = euµ uν + phµν + Qµ uν + Qν uµ + p<µν> .

(71)

In the ultra-relativistic limit we have gµν q µ q ν = 0 and (67)2 and (71)2 imply e = 3p. In the ultra-relativistic case, the phase density that maximizes the entropy density (68) in the local rest frame under the constraints of given values for n and e is called the ultra-relativistic J¨ uttner phase density, cf. [26, 27]. It reads uµ q µ n exp − fJ (n, T, u, q) = 8πT 3 T n |q| % q 2−u· = exp − 1 + u . (72) 8πT 3 T |q|

Herein T denotes the temperature, which is defined by T = p/n. Next we calculate the particle 4-vector and the energy-momentum tensor from the J¨ uttner phase density. We obtain Qµ = 0 and p<µν> = 0 and the conservation laws (69) formally transform into the ultra-relativistic Euler equations 3

% ∂ ∂ (n 1 + u2 ) + (n uk ) = 0, ∂t ∂xk k=1

∂ (4pui ∂t

%

3 ∂ (p δ ik + 4pui uk ) = 0, 1 + u2 ) + ∂xk k=1 3

% ∂ ∂ (3p + 4pu2 ) + (4puk 1 + u2 ) = 0. k ∂t ∂x k=1

(73)

Kinetic Schemes for Selected Initial and Boundary Value Problems

225

3.3 Kinetic Schemes As mentioned in the introduction, the kinetic approach for the ultra-relativistic Euler equations consists of periods of free flight and update times. In particular, we prescribe a time step τM > 0 and define the update times tm = m τM for m = 0, 1, 2, 3... The evolution during the periods of free flight is given by the collision transport equation which reads in the ultra-relativistic case 3

∂f q k ∂f + = 0. ∂t |q| ∂xk

(74)

k=1

Since we cannot expect the phase densities to be continuous at the update times, we have to distinguish between the left-hand and right-hand limits w.r.t. time. We thus define ± (x, q) := lim f (tm ±τ , x, q). fm

(75)

τ ց0+

Within the m-th period of free flight, i.e tm−1 < t ≤ tm , the moments of f are given by d3 q q + (x − τ , q) , (76) N µ (tm−1 + τ , x) = q µ fm−1 |q| |q| 3 d3 q q + T µν (tm−1 + τ , x) = q µ q ν fm−1 , q) , (77) (x − τ |q| |q| 3 whereas the fields n, u, T , and p are determined by the algebraic equations n=

%

N µ Nµ ,

uµ =

1 µ N , n

T =

1 uµ uν T µν , 3n

p = nT.

(78)

− At the update time tm we use the free flight density fm in order to calculate + uttner phase density fm as a J¨ & ' + ˜ m (x), q . ˜ m (x), T˜m (x), u (79) fm (x, q) = fJ n

˜ m so that the densities N 0 and T 0ν are We choose the fields n ˜ m , T˜m , and u conserved across the update times. In particular, for all tm and all x we have to ensure the continuity conditions d3 q d3 q 0 + − = q 0 fm , (80) q fm (x, q) (x, q) |q| |q| 3 3 d3 q d3 q − + = q 0 q ν fm (x, q) (81) q 0 q ν fm (x, q) |q| |q| 3 3

It is important to note that the conditions (80) and (81) guaranty the continuity of the densities N 0 and T 0ν at the update times, but they do not imply

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the continuity of the fields n, T , p, and u at the update times. We mention ˜ m turn out be the right-hand limits of n, T , that the fields n ˜ m , T˜m p˜m , and u p, and u, respectively. The update procedure maximizes the entropy in any point (tm , x) under the constraints of prescribed densities. For this reason we call the update times maximization times. From (79), (80), and (81) we may derive the following explicit expressions ˜ m, n ˜ m , and T˜m for u u ˜km = %

0k Tm 00 ) 4˜ pm (˜ pm + T m

n ˜m = %

,

0 Nm , ˜ 2m 1+u

p˜m T˜m = . n ˜m

(82)

0 0ν Here Nm (x) = N 0 (tm , x) and Tm (x) = T 0ν (tm , x) are the densities at the update time tm and p˜m is given by ) ⎞ ⎛ * 3 * 1 00 0k )2 ⎠ . 00 )2 − 3 (83) (Tm + +4(Tm p˜m = ⎝−Tm 3 k=1

Reduction to surface integrals The moment integrals (76) and (77) may be simplified as follows. We split the microscopic variable q into its length |q| and its direction q w = (w1 , w2 , w3 )T = ∈ S2, (84) |q|

where S 2 denotes the unit sphere. Due to the ultra-relativistic structure of the moment integrals in (76) and (77), we may carry out the integration with respect to |q|. There result the following expressions µ (85) N (tm + τ , x) = wµ Φm (x − τ w, w) dS(w), S2

T µν (tm + τ, x) =

wµ wν Ψm (x − τ w, w) dS(w),

(86)

S2 0

where w = 1 and µ, ν ∈ {0, 1, 2, 3} and Φm (x, w) =

Ψm (x, w) =

1 & 4π % 3 & 4π %

n ˜ m (x)

'3 ˜ m (x) ˜ 2m (x) − w · u 1+u p˜m (x)

'4 . ˜ 2m (x) − w · u ˜ m (x) 1+u

(87)

(88)

The functions Φm and Ψm are the counterparts to the reduced phase densities for the Boltzmann-Peierls equation, cf. Section 2.2. The surface integrals in (85) and (86) reflect the fact that in the ultrarelativistic case the particles are moving on the surface of the light cone.

Kinetic Schemes for Selected Initial and Boundary Value Problems

227

Kinetic scheme in one space dimension Here we consider phase densities f that do not depend on x2 and x3 , and we will show that this restriction gives rise to a further simplification of the kinetic scheme. In the following we write x = x1 , n = n(t, x),

u = (u(t, x), 0, 0),

p = p(t, x),

T = T (t, x),

(89)

and so on. Next we introduce new variables −1 ≤ ξ ≤ 1 and 0 ≦ ϕ ≦ 2π by % % w2 = 1 − ξ 2 sin ϕ , w3 = 1 − ξ 2 cos ϕ. (90) w1 = ξ ,

The surface element then becomes dS(w) = dξdϕ. Now we can carry out the integration with respect to the angular ϕ in (85) and (86) and we obtain µ

N (tm + τ , x) =

1

wµ Φm (x − τ ξ, ξ) dξ,

(91)

1

wµ wν Ψm (x − τ ξ, ξ) dξ,

(92)

−1

T

µν

(tm + τ , x) =

−1

where Φm (x, ξ) =

Ψm (x, ξ) =

1 & 2 % 3 & 2 %

n ˜ m (x)

'3 , 1+u ˜2m (x) − ξ u ˜m (x) p˜m (x)

'4 . 1+u ˜2m (x) − ξ u ˜m (x)

(93)

(94)

3.4 Numerical Examples Problem 3.1: Relativistic shock tube The initial data are (n, u, p) =

(5.0, 0.0, 10.0) (1.0, 0.0, 0.5)

if x < 0.5 , if x ≥ 0.5 .

The spatial domain is taken as [0, 1] with 400 mesh elements and the final time is t = 0.5. For the kinetic scheme we consider 100 maximization times. This problem involves the formation of an intermediate state bounded by a shock wave propagating to the right and a transonic rarefaction wave propagating to the left. The fluid in the intermediate state moves at a mildly relativistic speed (v = 0.58c) to the right. Flow particles accumulate in a dense shell

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particle density Exact Riemann Solution First Order Godunov Scheme First Order Central Scheme First Order Kinetic Scheme

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velocity

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10

Exact Riemann Solution First Order Godunov Scheme First Order Central Scheme First Order Kinetic Scheme

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Exact Riemann Solution 2nd Order Godunov Scheme 2nd Order Central Scheme 2nd Order Kinetic Scheme

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0.9 position

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Fig. 10. Problem 3.1: Comparison of the results at time t = 0.5.

behind the shock wave compressing the fluid and heating it. Figures 10 show u and pressure p. Both central the particle density n, fluid velocity v = √1+u 2 and KFVS scheme give a comparable accuracy in both first order as well as second order approach. However, we found that our first and second order continuous kinetic scheme give better resolution. The advantage of the central and KFVS schemes over the continuous kinetic scheme is the simplicity, compactness and less computational time. KFVS scheme also utilize all the upwind properties of the schemes and it is very easy to apply the boundary conditions exactly in the same way as in the Godunov upwind schemes, see

Kinetic Schemes for Selected Initial and Boundary Value Problems

229

Toro [39]. One can extend the one-dimensional KFVS scheme to the two-dimensional case in a dimension-by-dimension manner. We have used the same technique in order to extend the scheme to the two-dimensional KFVS scheme. We have used again the MUSCL-type initial reconstruction and min-mod non-linear limiters for the calculation of discrete slopes. In the following we solve a twodimensional numerical test problem by using first order and second order central and KFVS schemes. Problem 3.2: Implosion in a box

reﬂecting

n=1 u1 = 0 u2 = 0 p=1

1.25

0.75

0

reﬂecting

In this example we consider a twodimensional Riemann problem inside a square box of sides length 2, with reflecting walls. Initially the velocities are zero. The pressure is 10 and density is 4 inside a small square box of sides length 0.5 in the center of the large box, while pressure and density are unity elsewhere. The results are shown at t = 3.0 in Figure 12. We have used 400 × 400 mesh points.

reﬂecting

2

n=4 u1 = 0 u2 = 0 p = 10

0.75

1.25

2

reﬂecting

Fig. 11. Initial Data.

4 Conclusion We have derived a kinetic approach that can be used for hyperbolic systems with an underling kinetic equation. The approach treats the kinetic equation and the hyperbolic moment system in a unified manner, which allows an easy comparison of solutions of the kinetic equation with the solutions of the corresponding hyperbolic systems. The kinetic method consists of periods of free flights and update times, where we apply update rules. Two import examples illustrate the numerical approach: 1. Heat conduction at low temperature where the physics is embodied in the kinetic Boltzmann-Peierls equation. The kinetic equation may be approximated by various hyperbolic moment systems. 2. The second example concerns the evolution of the ultra-relativistic ideal gas. In this report we studied exclusively the ultra-relativistic Euler equations, and we did not compare the solutions with those that result in a certain limit of the relativistic Boltzmann equation.

230

W. Dreyer, M. Herrmann, M. Kunik, S. Qamar particle density

particle density

2.3 2.2

2.8

First Order Central Scheme First Order KFVS Scheme First Order BGK−type KFVS Scheme

2.6

2nd Order Central Scheme 2nd Order KFVS Scheme 2nd Order BGK−type KFVS Scheme

2.1 2.4

2 2.2

1.9 1.8

2

1.7

1.8

1.6 1.6

1.5 1.4

1.4

1.3 0

0.5

1 position

1.5

2

1.2 0

0.5

4.5

3.8 3.6

1

1.5

2

position

pressure

pressure First Order Central Scheme First Order KFVS Scheme First Order BGK−type KFVS Scheme

4

3.4 3.2

2nd Order Central Scheme 2nd Order KFVS Scheme 2nd Order BGK−type KFVS Scheme

3.5

3

3

2.8 2.6

2.5

2.4

2.2

2

2

1.8 0

0.5

1 position

1.5

2

1.5 0

0.5

1 position

1.5

2

Fig. 12. Problem 3.2: Implosion in a box at t = 3.0.

Both examples describe physically total different phenomena, however their mathematical structure exhibit a common behavior. For the numerical implementation, we use two versions of the kinetic approaches: The first one is continuous in space but discrete in time. The second one is discrete in both space and time. The second approach is usually called a kinetic flux vector splitting (KFVS) scheme. We have also extended the second version to second order which utilizes the MUSCL-type initial reconstruction. We have implemented this scheme in both, one and two space dimensions. Acknowledgement. This project was founded by the DFG Priority Research Program ”Analysis and Numerics for Conservation Laws”

References 1. P.L. Bhatnagar, E.P. Gross, and M. Krook. A model for collision processes in gases. i. small amplitude processes in charged and neutral one-component systems. Phys. Rev., 94:511–525, 1954. 2. J. Callaway. Quantum theory of the solid state. Academic Press, San Diego, 1991. 3. C. Cercignani. The Boltzmann equation and its applications, volume 67 of Applied Mathematical Sciences. Springer, New York, 1988. 4. N.A. Chernikov. Equilibrium distribution of the relativistic gas. Acta Phys. Pol., 26:1069–1092, 1964.

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5. N.A. Chernikov. Microscopic foundation of relativistic hydrodynamics. Acta Phys. Pol., 27:465–489, 1964. 6. S.R. deGroot, W.A. van Leeuven, and Ch.G. van Weert. Relativistic kinetic theory: Principles and applications. North Holland, Amsterdam, 1980. 7. S.M. Deshpande. A second order accurate, kinetic-theory based, method for inviscid compressible ﬂows. NASA Langley Tech. Paper, (2613):1–39, 1986. 8. S.M. Deshpande. Kinetic ﬂux splitting schemes. In M. Hafez and K. Oshima, editors, Computational Fluid Dynamics Review. John Wiley & sons, 1995. 9. W. Dreyer, M. Herrmann, and M. Kunik. Kinetic solutions of the BoltzmannPeierls equation and its moment systems. In Preprint, number 709. Weierstrass Institute Berlin (WIAS), 2001. to appear in Cont. Mech. Thermdyn. 10. W. Dreyer, M. Herrmann, and M. Kunik. Kinetic schemes and initial boundary value problems. Transport Theory in Statistical Physics, 31:1–33, 2002. 11. W. Dreyer, M. Junk, and M. Kunik. On the approximation of kinetic equations by moment systems. Nonlinearity, 14:881–906, 2001. 12. W. Dreyer and M. Kunik. The maximum entropy principle revisited. Cont. Mech. Thermodyn., 10:331–347, 1998. 13. W. Dreyer and M. Kunik. Reﬂections of Eulerian shock waves at moving adiabatic boundaries. Monte Carlo Methods and Applications, 4.3:231–252, 1998. 14. W. Dreyer and M. Kunik. Initial and boundary value problems of hyperbolic heat conduction. Cont. Mech. Thermodyn., 11.4:227–245, 1999. 15. W. Dreyer and M. Kunik. Hyperbolic heat conduction. In Trends in applications of Mathematics to Mechanics, volume 109 of CRC Monographs and Surveys in Pure and Applied Mathematics. Chapman & Hall, 2000. 16. W. Dreyer and S. Qamar. Kinetic ﬂux-vector splitting schemes for the hyperbolic heat conduction. In Preprint, number 861. Weierstrass Institute Berlin (WIAS), 2003. 17. W. Dreyer and S. Qamar. Second order accurate explicit ﬁnite volume schemes for the solution of Boltzmann-Peierls equation. In Preprint, number 860. Weierstrass Institute Berlin (WIAS), 2003. 18. W. Dreyer and H. Struchtrup. Heat pulse experiments revisited. Cont. Mech. Thermodyn., 5:1–50, 1993. 19. C. Eckart. The thermodynamics of irreversible process I: The simple ﬂuid. Phys. Rev., 58:267–269, 1940. 20. C. Eckart. The thermodynamics of irreversible process II: Fluid mixtures. Phys. Rev., 58:269–275, 1940. 21. C. Eckart. The thermodynamics of irreversible process III: Relativistic theory of the simple ﬂuid. Phys. Rev., 58:919–928, 1940. 22. E. Godlewski and P.A. Raviart. Numerical approximation of hyperbolic systems of conservation laws, volume 118 of Applied Mathematical Sciences. Springer, New York, 1996. 23. G.-S. Jaing and E. Tadmor. Non-oscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput., 19:1892–1917, 1998. 24. M. Junk. Kinetic Schemes: A New Approach and Applications. Shaker-Verlag, 1997. 25. M. Junk. Domain of deﬁnition of Levermore’s ﬁve-moment system. J. Stat. Phys., 93:1143–1167, 1998. 26. F. J¨ uttner. Das Maxwellsche Gesetz der Geschwindigkeitsverteilung in der Relativtheorie. Ann. Phys., 34:856–882, 1911.

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27. M. Kunik, S. Qamar, and G. Warnecke. Kinetic schemes for the relativistic gas dynamics. In Preprint, number 21. Otto-von-Guericke University Magdeburg, 2002. 28. M. Kunik, S. Qamar, and G. Warnecke. A BGK-type kinetic ﬂux- vector splitting schemes for the ultra-relativistic Euler equations. In Preprint, number 4. Otto-von-Guericke University Magdeburg, 2003. 29. M. Kunik, S. Qamar, and G. Warnecke. Kinetic schemes for the ultra-relativistic Euler equations. J. Comput. Phys., 187:572–596, 2003. 30. M. Kunik, S. Qamar, and G. Warnecke. A reduction of the Boltzmann-Peierls equation. In Preprint, number 6. Otto-von-Guericke University Magdeburg, 2003. 31. M. Kunik, S. Qamar, and G. Warnecke. Second order accurate kinetic schemes for the ultra-relativistic Euler equations. In Preprint, number 18. Otto-vonGuericke University Magdeburg, 2003. 32. I. M¨ uller. Speeds of propagation in classical and relativistic extended thermodynamics. Max Planck Institute for Gravitational Physics, 1999. 33. I. M¨ uller and T. Ruggeri. Rational Extended Thermodynamics. Springer, New York, 1998. 34. H. Nessyahu and E. Tadmor. Non-oscillatory central diﬀerencing fo hyperbolic conservation laws. SIAM J. Comput. Phys., 87:408–448, 1990. 35. R.E. Peierls. Quantum theory of solids. Oxford University Press, London, 1995. 36. B. Perthame. Boltzmann type schemes for gas dynamics and the entropy property. SIAM J. Numer. Anal., 27.6:1405–1421, 1990. 37. S. Qamar. Kinetic Schemes for the Relativistic Hydrodynamics. PhD thesis, Faculty of Mathematics, Otto-von-Guericke University Magdeburg, 2003. 38. R.D. Reitz. One-dimensional compressible gas dynamics calculations using the Boltzmann equation. J. Comput. Phys., 42:108–123, 1981. 39. E.F. Toro. Riemann solvers and numerical method for fluid dynamics. Springer, New York, 1999. 40. S. Weinberg. Gravitation and cosmology. Wiley, New York, 1972. 41. K. Xu. Gas evolution dynamics in Godunov-type schemes and analysis of numerical shock instability. ICASE Report, (99-6):1–17, 1999. 42. K. Xu. Gas-kinetic theory based ﬂux splitting method for ideal magnetohydrodynamics. J. Comp. Phys., 153:334–352, 1999.

A Local Level-Set Method under Involvement of Topological Aspects F. V¨ olker, R. Vilsmeier, and D. H¨ anel Institute of Combustion and Gasdynamics, University Duisburg-Essen [email protected] [email protected] Summary. This paper proposes a general multi-dimensional front tracking concept for various physical problems involving specially discontinuous solution features. The tracking method is based on the level-set approach with a restricted dynamic deﬁnition range in the vicinity of the fronts on ﬁxed grids of arbitrary cell structure. To combine the front tracking procedure with the continuous part of the ﬁeld to be simulated, a double sided ﬂux discretization called ﬂux-separation and a set of inner boundary conditions over the discontinuities are used. The methods developed are not restricted to ﬂuid dynamics, however all examples relate to this class of simulation problems. Special attention is drawn to the restrictions of the classical level-set method, i.e. accuracy issues and topological restrictions. In this concern, an improved time integration method for the front motion is introduced and the problem of interacting discontinuities is addressed. The methods are integrated in the object oriented Finite-Volume solution package MOUSE [1] for systems of conservation laws on arbitrary grids.

1 Introduction Discontinuous flow features, as material interfaces, shocks or detonations, occur in many flow problems. In most cases, such flow features are resolved numerically by capturing methods. Accordingly, these captured discontinuities become smeared out by artificial intermediate states. In many cases such a non-physical representation is tolerable in conjunction with conservative discretization methods. Besides, grid adaption with high resolution near fronts can be used to restrict the physical space of the intermediate states. In some cases however, the non-physical intermediate states may cause errors in the solution, that are unacceptable. As example, consider the travel speed of the front shock of a detonation wave, which is severely influenced by the onset of exothermal reaction source terms in the smeared zone. An alternative to capturing methods are front tracking methods, which enable an exact representation of discontinuities as discrete jumps in the set

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of variables. This is of advantage because consequences of intermediate states are avoided. However, a special treatment of discontinuous solution parts is then required, introducing additional algorithmic difficulties. Many pioneering developments have to be mentioned in this context, such as the work of Richtmyer and Morton [2] or Moretti [3]. One of the most promising approaches however, is the newer level-set method [4, 5, 6, 7], also used in the present paper. In this approach, the front position is defined as an iso-value of a smooth scalar function. Besides its strong assets, the level-set method in its classical form shows some severe drawbacks. First of all, topological restrictions apply, as only closed, differentiable interfaces can be described. Also, most implementations show accuracy problems concerning the preservation of the exact front position, when preparing the level-set function for further physical motion steps. The present paper attempts to overcome the drawbacks and restrictions of the classical method, however introducing further algorithmic diversification and thus difficulties. To provide an overview, the basic level-set implementation as previously published in [9] is repeated in a very brief manner, while algorithmic enhancements are presented in more detail. Special attention is given to the topological restrictions and the ongoing work to overcome these. The explanations are supplemented by sample calculations in two and three space dimensions. All methods developed are implemented into an open source object oriented framework for the simulation of partial differential equations [1], in continuous development at the site of the authors. However, the specific methods presented in this paper are not yet commonly available.

2 Basic Numerical Concept A general, time dependent system of conservation laws in integral form as used in the MOUSE [1] package reads: d Q dV + H · n dA = S dV , (1) dt V

A

V

where Q, H and S represent the variables, fluxes and source terms respectively. In the frame of fluid dynamics, being the physical basis of all examples within this paper, the conserved quantities are mass, momentum and energy. In the case of reacting fluids, also species mass conservation is considered. The system (1) is solved on arbitrary grids employing a nodal finite volume method, Fig. 1. For a discrete control volume Vd the above system without source term reads:

A Local Level-Set Method under Involvement of Topological Aspects

∆Q |V d + Res∆,V d = 0 with: ∆t Res∆,V d =

nv 1

VV d

235

(2) P

Hi ni ∆Ai ,

(3)

i=1

where V d is a control volume with the size VV d and the sum is carried out over all bounding segments. For the developments in the present paper, a fully explicit time integration method is used throughout.

K2

P

n∆A

K1

Fig. 1. Control volume.

3 Basics of Level-Set Methods 3.1 Discontinuous solutions Neglecting effects of diffusion, then discontinuous solutions can appear as weak solutions of hyperbolic partial differential equations. A typical set of such nonlinear equations are the Euler equations for compressible, inviscid flows, which may be described in conservative form by (1). The corresponding weak solution of an embedded discontinuity is given then by R

[H − cd Q ]L · nd = 0 ,

(4)

where [X]R L = XR − XL means the jump of variables left and right the discontinuity and cd is the velocity and nd the unit normal vector at the front of the discontinuity. Analyzing (4) one finds two types of discontinuities: • Passively transported fronts are e.g. material interfaces, contact discontinuities or shear layers. These discontinuities float along with a carrier (flow) speed v, i.e. it is cd · nd = v · nd . From the point of view of front tracking the treatment is easier, since the transport velocity v is available everywhere, where needed. • Actively transported fronts are e.g. shocks or detonation waves. The speed of the front cd is usually larger than the velocity of the underlying fluid flow, i.e. it is cd · nd = v · nd . Its value is ruled by additional physical relations, i.e. by exploration of (4), known as the Rankine Hugoniot equations for the Euler equations. Thus the front velocity cd is known only on the front itself, which leads to a more complex formulation of tracking algorithms. To calculate both, passive and active discontinuities, a locally narrowbanded level-set formulation is chosen in the present paper. A narrow band means here a restricted number of grid cells and nodes in the neighborhood

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of the front of the discontinuity on which the level-set function is evaluated. As the front tracking method is intended to qualify for arbitrary physical problems, the more difficult class of active discontinuities must be considered throughout. Therefore, the succeeding descriptions are related to this more complex case of actively transported discontinuities, essentially for examples of moving shock waves. 3.2 Level-Set Function The level-set function is used to describe the geometry and the transport of a front. This function is represented by a scalar function G(x, t). The front itself is defined by a constant value of G, typically chosen to G(x, t) = 0. In this case both sides of the level-set function are easily distinguishable by the sign of the corresponding values of the level-set function G. The rate of change of a constant level G = G0 is described in the Lagrangian way by ddGt |G0 = 0, but is usually used in an Eulerian frame. It results in the transport equation of a level-set: ∂G + c · ∇G = 0 . (5) ∂t In principle, only the level G(x, t) = 0 is of interest for detecting the front position. However, values of neighboring levels of G are required to evaluate the metric coefficients containing gradients of G. For example, the normal vector nd at the front of a discontinuity is given by nd =

∇G . |∇G|

(6)

The level-set function is usually normalized away from G = 0 to satisfy a slope condition expressing it as a distance function to the front. The function is then a measure of the distance normal to the front, which makes the geometrical exploration of the level-set much more accurate. A distance function requires that the modulus of the local gradients of G remains constant, usually set equal one: | ∇G | = 1 . (7) Thus the solution procedure for the level-set function consists of two steps, the transport step, solving (5), and the normalization step, satisfying (7) away from G = 0. In addition, the local propagation speed of front c has to be determined for actively transported discontinuities by exploring the weak solution Equation (4), usually in form of the Rankine-Hugoniot conditions. Finally the left and right states of the discontinuity have to be updated according to the Rankine-Hugoniot conditions. The numerical realization for a finite-volume method on arbitrary unstructured grids is described in following sections.

A Local Level-Set Method under Involvement of Topological Aspects

237

4 Numerical Level-Set Approach A narrow band level-set algorithm has been designed for arbitrary, unstructured grids and integrated into a node-centered finite-volume method. The level-set function is required in the neighborhood of the front only, therefore its definition range is locally restricted to a few number of grid levels around the front (narrow band). The grid used for the level-set computation in this range is identically with the grid for computing the fluid flow. The generalization to arbitrary grids and the local treatment of the level-set function requires some special algorithmic developments for the normalization and transport step and for the implementation of the jump conditions at the front. The algorithmic realization is described in this section for the more complicated cases of active discontinuities, as e.g. shocks. The algorithmic descriptions reveal in addition that all level-set procedures are interacting and require therefore careful treatment to track correctly a discontinuous front. 4.1 Grid arrangement for Local Level-Set Method Discrete values for the function G are stored at the nodes of the mesh, while the definition range is restricted to a given number of neighborhood levels apart the front. In Fig. 2 a zoom to a hybrid mesh with two neighborhood levels is sketched. Points and edges are marked with P and K respectively and the number index refers to the neighborhood level. Edges of type K0 refer to those crossing the front and the nodes appended to these edges are of first neighborhood level P1 . Other edges connected to nodes of type P1 are themselves of type K1 , etc ... Accordingly, nodes and edges farther apart are of increasing neighborhood level. The exact front position is P1 K1 P2 described by the value of G = 0 K0 on the edges of type K0 upon linK1 P1 K1 K0 ear reconstruction. In the present K1 P1 P2 K 1 P2 method, the front motion is reK0 K1 K 1 stricted to overrun at most the P1 K1 nodes P1 of the first level in a sinK0 P2 P2 P K1 1 K1 P K0 gle time step. Therefore, at least P1 2 K1 two neighborhood levels are reK1 K1 K0 P1 P quired to ensure that the inforP2 K1 K1 2 P K0 P2 1 K1 mation on the front position reK0 mains available after the motion P1 K1 step. If nodes are overrun, the K1 P2 K0 P1 K P2 1 restricted definition range of the scalar function G is adjusted acFig. 2. Control volume. cordingly. The minimal narrow-band formulation consists then of 2 plus 2 equal 4 neighborhood node levels in total. In the present paper a higher order for-

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mulation of the transport equation (5) is employed, which uses an extended discretization stencil. In the case the band width is enlarged to 4 plus 4 neighborhood levels. Further, multiple fronts are considered in the course of this work. The narrow band formulation of one single applies to each of the fronts independently and their discrete definition ranges may overlap in arbitrary ways without restriction. 4.2 Level-Set normalization Assuming an arbitrary distribution of the level-set function G is given. For reasons of accuracy it is desired that these values satisfy the condition for a distance function (7). This normalization procedure for solving (7) performs iteratively (iteration index k) in two steps starting from an initial field of level-set values G. The gradients and the new, normalized level-set function are determined by pointwise least square approaches. The direction of the gradient ∇Gk at a node i is computed in the first step. The least square approach for a node i, see Fig. 3, with a restricted number of appending neighbor nodes ˜j ∈ Epar (i), reads: Φpar =

˜ j∈Epar (i)

− Gik−1 ) ∇Gki · mi˜j − (G˜k−1 j |mi˜j |q/2

or equivalent

2

→ minimum

(8)

d Φpar = 0. d(∇Gki )

The vector connecting the nodes i and j is mi˜j = X˜j − Xi and q ≥ 0 is a distance weighting exponent. The result is the gradient ∇Gki . The functional value of Gk is set to satisfy the slope condition in the second step. The corresponding approach is:

Φhyp =

j∈Ehyp (i)

or equivalent

⎛ ⎝

∇Gk i |∇Gk i|

· mij − (Gjk−1 − Gki ) |mij |q/2

⎞2

⎠ → minimum

(9)

d Φhyp = 0. dGki

The result of the second step is a new functional value for Gk . The iteration procedure is continued until the slope condition (7) is sufficiently satisfied. In above equations, the indices par and hyp refer to restricted ranges of influence. Further details can be found in [9].

A Local Level-Set Method under Involvement of Topological Aspects

239

Unfortunately, the normalization procedure alters slightly the front position, determined before by linear interpolation for G = 0. Additional corrections are necessary to preserve the exact front position. This can be done by storing the front position and correcting the level-set function after the normalization such that the original front position is preserved. Details are given e.g. in [9]. Such a correction influences the level-set function and thus the exact slope condiFig. 3. Discontinuity and edge with tion is violated again, which is howthe vertices i and j and gradients of ever more tolerable than a falsified the scalar level-set function G. front position itself. Due to its iterative nature, the normalization procedure is by far the most CPU-consuming task in the present level-set formulation. 4.3 Transport step of level-set function The motion of the front during a time step ∆t is carried out by an explicit scheme of (5). Starting from a given (normalized) distribution Gn , the new front position is defined by Gn+1 = 0, from which the front location and its direction (normal vector) can be derived from. A discretization of first order accuracy in time of (5) reads: Gn+1 = Gn − c · ∇Gn ∆t .

(10)

This transport equation for the level-set function G can be solved under the assumption of frozen front velocity c during a time step and for given gradients of G updated after the normalization procedure. Experiments with the scheme (10), e.g. in [9], have shown that remarkable errors in the front position can occur, if the slope condition (7) is not sufficiently fulfilled. Therefore a higher order extension of scheme (10) is introduced, which takes into account second derivatives of the level-set function G and thus is less sensible to deviations of the slope condition (7). With a Taylor expansion to formal second accuracy in time Gn+1 = Gn +

∂ 2 G ∆t2 ∂G ∆t + ∂t ∂t2 2

the extended scheme of the transport equation (5) reads: 1 Gn+1 = Gn − c · ∇Gn ∆t + ∆t2 cT (∇ ⊗ (∇G)) · c , 2

(11)

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where cT means the transposed vector c and ∇ ⊗ (∇G) a dyadic vector product. For example, this product results in two dimensions in: cT (∇ ⊗ (∇G)) · c = c2x

∂2G ∂2G ∂2G + 2 cx cy . + c2y 2 ∂x ∂x∂y ∂y 2

The higher order approach, (11) stabilizes the explicit transport step and corrects deviation from the slope condition (7). In particular, the number of iterations performed to approximately normalize the level-set function can be essentially lowered, improving the overall performance. On the other side, this approach requires the formulation of gradients of second order and thus a larger range of neighborhood levels, compared to the first order formulation in (10), which was explored in [9]. With the restriction that only one neighborhood level can be overrun per time step, it has been stated that at least two neighborhood point levels were required for updating the gradient ∇G. With the extended approach (11) however, a larger stencil is needed and thus an additional neighbor level, a third one, is required. Finally a fourth level is introduced to avoid an influence of new nodes onto the computations in the case the dynamic definition range is adjusted after a node has been overrun. Obviously, the performance enhancements due to a lower amount of iteration steps needed within the normalization process are partially lost, but the extended approach has become more flexible with respect to complex front behavior. The gradients ∇Gn in (10) or (11) can easily be computed on the nodes of an unstructured grid, employing a Green-Gauss integration on finite volumes or by using a least square method as shown in the normalization procedure. Since the level-set function Gn and the gradients are given by that, the (11) can be discretized within the definition range of the level-set around the front and integrated in the manner of a Lax-Wendroff scheme. Basic requirement is however that the transport velocity c is known on all connecting edges. For passive discontinuities, like material interfaces, the velocity c, equal the flow velocity v, is given in the whole definition range of the level-set. In this case, the tracking procedure simplifies, its utilization is topic of separate studies. In this paper, the so called active discontinuities, like shocks or detonation waves, are considered, which are characterized by the fact, that the transport velocity c is given only at the front (G = 0) itself and has to be determined from the left and right states at the front by exploring the weak solution (Rankine-Hugoniot conditions). It means that the discretization concept of (10) or (11) can only be employed to nodes of the first neighborhood (nodes P1 in Fig. 2), which have a connecting edge cut by the front. An edge-based discretization is developed for this case and described in the next subsection. For nodes P2,··· ,n , more far away from the front, which have to be moved consistently with front speed, special extrapolation techniques are used.

A Local Level-Set Method under Involvement of Topological Aspects

241

Edge based discretization of the transport equation The indices i and j refer to neighboring nodes in the following equations, while the double index k(i, j) refers to the edge connecting the nodes pi and pj . The set of edges appended to a node pi will be called k(i). Nodes P1 at edges k0 , cut by the front, are considered first. The nodal discretization of (10) or (11) can be written as: Gn+1 = Gni + △t · RES{Gni } . i

(12)

The residual RES{Gni } contains the propagation speed c, which is defined on the edge k(i, j) and contains gradients of G, evaluated on the nodes i and j. For a consistent approach respecting the exact position of c, the nodal residual RES{Gni } in (12) is expressed as a function of edge-wise residual contributions RES{Gnk(i,j) } RES{Gni } = f (RES{Gnk(i,j) })

∀ k(i, j) ∈ k(i) ,

(13)

where f is a weighting function to be detailed later. According to (10) and (11) the edge-wise residual for an edge k(i, j) reads: 1 RES{Gnk(i,j) } = −ck(i,j) · ∇Gnk(i,j) + cTk(i,j) (∇ ⊗ (∇Gnk(i,j) ))ck(i,j) 2

(14)

and thus the velocity ck(i,j) as well as ∇Gnk(i,j) and ∇ ⊗ (∇Gnk(i,j) ) must be reconstructed on the edge k(i, j). For all edges k(i, j) ∈ K0 hitting the discontinuity, the propagation speed ck(i,j) is directly available from the physics of the discontinuity and thus accessible. The terms ∇Gnk(i,j) and ∇ ⊗ (∇Gnk(i,j) ) are computed on the nodes and reconstructed by linear interpolation along the edge onto the position of the discontinuity. Exemplarily for ∇Gnk(i,j) in Fig. 3 the interpolation reads: ∇Gk(i,j) = ∇Gj

| mi,o | | mi,o | + ∇Gi 1 − , | mi,j | | mi,j |

(15)

with mi,0 = xi − x0 and mi,j = xi − xj , where xi , xj and x0 are the positions of the nodes i, j and the position of the discontinuity on the edge respectively. The residual RES{Gnk(i,j) } according to (14) is now available for edges k0 and can principally be redistributed to the connecting nodes i and j.

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However a node i can be connected with several edges of type k0 which contribute with their residuals to node i. Figure 4 sketches such a local situation. A node i of level li = 1 is connected to three edges k(i, j1), k(i, j2) and k(i, j3), all of them are of level 0, thus crossing the discontinuity. To get a representative residual for node i the an- Fig. 4. Discontinuity and discrete edges gle of the appended edges to the di- with the vertices i and jn . rection of ∇Gi , stored at node i, is calculated. The preferable residual is found by a linear interpolation between the edge with the smallest angle to the left (α1 ) and to the right side (α2 ) of the gradient direction ∇Gi . For the situation in Fig. 4 the weighting function would read: RES{Gni }

=

cos(α1 ) RES{Gnk(i,j1) } + cos(α2 ) RES{Gnk(i,j2) } cos(α1 ) + cos(α2 )

.

(16)

If no interpolation is possible, the contributing edge with the lowest angle against the normal direction by ∇Gi is chosen. The extension of the interpolation (16) to 3D is generalized to n ′ cos(mk ′ , ∇Gi ) RES{Gk ′ } ; k ′ ∈ k(i) ; lk′ < li , (17) RES{Gni } = k ′ , ∇Gi ) cos(m ′ k k

where mk′ = mi,j refers to the direction of the edge k ′ = k(i, j). The residuals RES{Gnj } of all points P1 next to the front are updated in the same way and the new values of the level-set function Gn+1 are computed according to (12). j Nodes Pp, p>1 on edges of higher level, k ∈ Kl ; l > 0, away from the front, are treated in a slightly different way, since direct information from the front are not available. The speed vector c for these edges is taken from the appended node of lowest level: ⎧ ⎨ ci if li < lj , cj if li > lj , (18) ck(i,j) = ⎩ void if li = lj . Note, that only edges connecting nodes of different levels li and lj are considered and information is only transported away from the discontinuity. In contrast to the reconstruction of the terms ∇Gnk(i,j) and ∇ ⊗ (∇Gnk(i,j) ) for the edges in the set K0 , a simple average is taken for all edges of higher level: 1 ∇Gk(i,j) = (∇Gi + ∇Gj ) ; lk(i,j) > 0 . (19) 2

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The residuals RES{Gnk(i,j) } on edges can be constructed now according to (14). The distribution of the edge residual to the nodes is analog to (17) for the case of multiple edges, connected with the node. The new values of the level-set function Gn+1 are computed according to (12). i Space marching algorithm The accuracy of the front position is primarily ruled by the discretization on all edges in the set K0 and the calculation of the transport equation, which starts there. Then it proceeds recursively to nodes and edges of increasing level. This is done by considering the special role of the propagation speed vectors. In general case these are accessible at the edges in the set K0 only and are thus extrapolated via the above reconstruction rules over nodes and edges of increasing neighborhood level. Speed vectors provided, the discretization is non-recurrent and is carried out explicitly.

5 Numerical Treatment of the Jump Conditions The present level-set method is a general, kinematic approach, independent from the physical type of the discontinuity. The knowledge of the propagation speed cd is the only parameter to be prescribed for propagating a front in time and space. A complete front tracking method consists of kinematic and dynamic descriptions. The dynamic part introduces the physics of the underlying problem. It has to take into account the jump conditions over the front, enables the determination of the propagation speed and couples the states ”left” and ”right” to the front with the surrounding solution. The physics can roughly be expressed by the two different types of discontinuities, as classified in a previous section in passive (e.g. material interfaces) and active discontinuities (e.g. shock waves). The conditions prescribed left and right to the front are called here inner boundary conditions. They are derived from the conservative jump conditions (4) and of additional constraints, which are given e.g. by the local Riemann problem for hyperbolic, gasdynamic problems. It is mentioned however that the present front tracking method is not restricted to gasdynamic problems. In [11], this method has been applied to interfaces in ground water flows [11], being ruled by Darcy type equations yielding parabolic Poisson equations for pressure. Further applications of the present concept deal with free surface flows of incompressible fluids for the prediction of flow around ships[12]. Finally, an algorithmic approach has been developed which enables the implementation of the inner boundary conditions consistent with the basic finite-volume method, which is called flux-separation method and described later in this section.

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5.1 Inner boundary conditions and propagation speed The derivation of inner boundary conditions and of the propagation speed is explained for the example of a gasdynamic discontinuity. Inviscid flow of a gas is described by the Euler equations, given e.g. by the integral form of (1). The corresponding jump conditions (4) in a relative frame fixed with the discontinuity describe the conservation of mass, of normal and tangential momentum and of energy over a discontinuity (ρ wn )L (ρ wn2 + p)L (ρ wn wt )L 1 2 ρ wn (e + (wt + wn2 )) 2 L

= (ρ wn )R , = (ρ wn2 + p)R , = (ρ wn wt )R , 1 = ρ wn (e + (wt2 + wn2 )) . 2 R

(20a) (20b) (20c) (20d)

The indices ”L” and ”R” refer to the left and right states at the discontinuity. The normal and tangential velocities in the absolute frame are vn and vt , the propagation speed normal to the front is cn . The velocities in the relative frame are designed as wn = vn − cn and wt = vt . The local Riemann problem in the relative frame in direction x normal to the front is required as additional constraint. The characteristics ddtx and the corresponding invariants read: dx |1,2 = wn , dt dx |3,4 = wn ± a , dt

dp − a2 dρ = 0 and dwt = 0 ,

(21)

dp ± ρa dwn = 0 .

(22)

The characteristics have to be applied independently on each side ”L” and ”R” of the discontinuity, where they identify the direction of influence and the corresponding invariants. The so called passive discontinuities are considered first. They describe the interface between immiscible fluids. Immiscible means, there is no throughflow through the interface, i.e. wn = vn − cn = 0. The jump conditions (20) yield the following inner boundary conditions at the interface, which couple the left and right states: vn,L = vn,R

and

pL = pR .

The condition of equal pressure pL = pR over the front can be substituted by the pressure jump pL − pR = σ/R in the presence of surface tension σ. The other quantities, as the tangential velocity wt = vt and inner energy e or density ρ are independent each from the other side of the front. The Riemann problem, applied to the states left and right, yields no additional information in this case, beside the fact that the pressure at the interface is a function of the left and right acoustic waves ( ddtx |3,4 = ±a). The propagation velocity c of the front, as required for the level-set formulation in the

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previous section is per definition equal to the fluid velocity v. The propagation velocity c on the edge for updating (10) or (11) is given node-wise: ci = ± v i .

(23)

The sign depends on the definition of the level-set, as the gradient vector of the level-set function G might be directed in positive or negative propagation direction. Active discontinuities, like shocks or detonation waves are the more difficult case to be considered next. The front is moving relative to the carrier fluid with speed cn , therefore a finite mass flux (ρ wn )L = (ρ wn )R exists and the propagation speed cn differs from the fluid speed vn . In general, the jump conditions (20) are reformulated in the form of the Rankine-Hugoniot conditions, which express the ratios of the left and right states as function of a shock Mach number M as . The Mach number M as is defined as the ratio of the relative, normal velocity wn = vn − cn to the corresponding speed of sound a on the ”cold”, supersonic side of the gasdynamic wave. Assuming the left side, index ”L”, of the front would be supersonic (in the frame fixed with the moving front), then the shock Mach number M as reads: M as =

wnL (vn − cn )L = . aL aL

The Rankine-Hugoniot conditions from (20) for an ideal gas read e.g. for the pressure ratio over the discontinuity: pR 2γ , M a2s − 1 . =1+ pL γ+1

(24)

The jump conditions (20) or as Rankine-Hugoniot conditions in form of (24) for each variable result in a sufficient number of equations, connecting the conservation variables QL and QR on both sides of the front. These conditions are interpreted here again as inner boundary conditions. The system of equations for inner boundary conditions is not yet closed, since it is not clear from which side variables have to be taken, and the propagation speed cn has to be defined as well. Similar to boundary conditions on the outer boundaries of a computational domain,the local Riemann problem (characteristics) are accounted concerning their influences onto the discontinuity. For the case of a moving shock wave, embedded in a variable flow field (known e.g. from a previous time level), the supersonic state (in the shockfixed frame) is found at the ”colder” side normal to the front or easier by the sign of the level-set function, let’s assume index ”L”. The other side ”R” has to be subsonic.

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Figure 5 sketches the situation for an one-dimensional shock wave in the relative frame. The characterL R istics at the left side L are directed towards the discontinuity. Accordingly all three incoming characteristics from the left influence the t+∆t shock, while no back influence from the shock onto the left side is present. The flow on the right side L R t R is subsonic and all characteristics beside of one are directed away from the discontinuity. This charFig. 5. Characteristics ruling the seacteristic, the backwards pressure lection of inner boundary conditions. wave, influences the discontinuity. It means that all but one values have to be overwritten on the subsonic (right) side and all values on the supersonic left side remain unchanged. It arises the question, which of the variables have to be considered. The most consequent method would be to use the Riemann invariants from (21). As for outer boundary conditions of a computational domain, this task is not always practical and instead reasonable primitive variables are chosen. Exemplarily for a shock wave the pressure at the subsonic side is preserved. All other variables on this side are determined via Rankine-Hugoniot equations (24). The propagation speed is now available from the pressure ratio pR /pL using (24): cn = vn − M as aL . (25) The propagation velocity c on the edge required in section 4.3 is: crel,k(i,j) = ±cn

∇Gk(i,j) ; |∇Gk(i,j) |

k(i, j) ∈ K0 .

(26)

Adding the local carrier speed yields the propagation speed in the inertial system as required for updating the residual (14): ck(i,j) = crel,k(i,j) + Vk(i,j) ;

k(i, j) ∈ K0 .

(27)

5.2 Finite Volume Implementation by Flux Separation The determination of the variables QL and QR in the nodes left and right the discontinuity by exploring the jump conditions and the characteristics was discussed in the previous subsection. The implementation of the inner boundary conditions into the Finite Volume framework is explained in the following. Referring back to Fig. 1, each edge of the mesh carries a segment of two adjacent control volumes. In smooth regions, a projection from the nodes

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storing the variables to the cell interface is performed and the flux is computed in central or upwind manner. The flux is used for both the adjacent control volumes. For all edges k ∈ K0 , cut by the discontinuity, this is no longer the case. Corresponding finite volume cells contain now the two states QL and QR , proportional their partial volumes ∆VL and ∆VR .The geometrical determination of these partial volumes, as done e.g. in [13] on Cartesian grids, is rather tedious for arbitrary grid structures. Therefore an alternative way is introduced here, called flux separation. It is assumed that a computational QL cell belongs completely to the state QR L defined by the node in the cell center, see Fig. 6. Then, fluxes on edges, HL HR R cut by the front, are double defined, belonging either to the left or the right cell. It means, the flux HL for the ”left” side is computed using the ”left” side projection HL (QL ), and Fig. 6. Flux Separation at an interface. accordingly for the right side. The flux difference on such an edge obeys the jump conditions (28), since the values QL and QR satisfy these conditions: HL − HR = cd (QL − QR ) nd = 0 .

(28)

If a node gets overrun by the discontinuity, then the corresponding cell is redefined to the new state. In this case the variable set stored at this node is overwritten by an extrapolated value on the corresponding side. In the present version, the time step is restricted to allow at most one neighborhood level to be overrun at once. So the state near the discontinuity jumps slightly, but the level-set and thus the front moves continuously. The flux separation approach is consistent with the finite volume concept and can be employed to any grid structure. The original control volumes of the mesh are preserved and the need for a sub-cell resolution, rather complex on grids of arbitrary structure, is avoided. In many problems, conservation plays an essential role. According to the formulation of the flux-separation and switching of states the code preserves conservation exactly in the asymptotic states (steady state or constant front speed). However, conservation even in capturing methods is mostly required to ensure correct propagation speeds and jump conditions on discontinuities, which can directly be insured here by the construction of jump conditions and the transport by the level-set method.

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6 Results and Discussion The present level-set method has shown to be a general, kinematic approach to track discontinuous solutions of different type. Nevertheless the classical formulation of the level-set method, the strict separation of regions as well as the numerical discretization causes some restrictions to be discussed in the following and demonstrated by some results. 6.1 Front tracking with single level-sets Depending on the problem under consideration, a discrete iso-value of the single, scalar level-set function G(x, t) may only describe closed curves (surfaces) or curves (surfaces) hitting the boundaries of bounded domain. As a matter of principle, this iso-value of a level-set separates a field in two, distinct regions coupled by corresponding jump conditions. These topological constraints for single level-sets are acceptable and useful for many passive and active discontinuous problems. Examples in fluid mechanics are: • two non-miscible fluids A and B • phase transitions with either evaporated or liquid (solid) material • flame fronts with either non-burned or burned gas However, in reactive or non-reactive gasdynamics, above restrictions are not always permissible, since discontinuities may interact causing e.g. triple points and thus non-differentiable front curves. The level-set function G(x, t), as the solution of a hyperbolic transport equation 5 enables in principle nondifferentiable front curves depending on the prescribed front velocity c. However such non-differentiable front curves cannot be represented by one single level-set, they are always smoothed. The reasons are that the additional discontinuity is usually captured (if not tracked again) and the level-set algorithm smoothes the solution due to truncation errors. This problem is demonstrated by a few examples. A relatively easy tracking problem for demonstration is the formation of a bow shock in front of blunt bodies. In Fig. 7 and Fig. 8 the bow shock of the inviscid flow past a sphere is computed on an unstructured tetrahedral grid as a single closed surface. The free stream Mach number is M a∞ = 2. Figure 7 shows the initialization of the shock front as a planar surface in front of the sphere. The converged solution is given in Fig. 8. Broken partitions of the discrete definition range of the level-set and iso-colors of pressure on a cut plane are shown in both figures. The bow shock around such simple body is a differentiable surface and can be represented very well by a single level-set. As an example for non-differentiable front curves, the inviscid supersonic flow at M a = 3 past a set of three cylinders is considered in Fig. 9 and in detail in Fig. 10. The front shock is tracked by a single level-set, starting from an initial straight shock as in previous example. The global Fig. 9 represents

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obviously the bow shocks and intersecting points well, but looking in more detail in Fig. 10, one can see it is not the case. Existing intersection points of the front curves are no more represented as sharp corners, but smeared out to a rounded radius according to the mesh density. As further example, the transient detonation wave, as given in chapter 7, Fig. 11, can be considered. Again the front shock was tracked by a single level-set. The front deformation is however ruled by transverse shocks in the appended reaction zone. These shocks are attached to the front shock, forming the well known detonation cells. Accordingly, in nature, the front shock is kinked and thus forms a non-differentiable curve. The classical level-set method however smears these out to a moderate curvature radius, depending strongly on the resolution of the grid. 6.2 Front tracking with two level-sets The smoothing of non-differentiable front curves can be avoided, if additional level-sets are used for additional discontinuities at intersecting points. First results are achieved for the simpler problem of a regular shock reflection. In this case it is known that at the location, where the incident shock hits the wall, the reflected shock appears. Thus two independent level-sets are prescribed for both shocks and treated independently, coupled only on the reflection point lying on the wall. Shock reflection is initialized in such a manner that the deflection angle of the incident shock does not correspond to the angle of the reflected shock. This causes a movement of the reflected shock to the exact angle given by the shock relations. The colors of constant pressure, the front line (G = 0) and the velocity vectors are presented in Fig. 15 for the initial state and in Fig. 16 for the final state. 6.3 Attempts for non-closed discontinuities However, in compressible gas-dynamics, above restrictions are not always permissible: Shock waves may end somewhere in the field (non-closed discontinuity) and degrade to compression waves and several shocks or shear layers may interact causing triple points. A classical closed line fails to represent such effects. A possible way to describe such an open discontinuity (an open level-set) is to limit its active range by intersection with other transverse level-sets. In the following, the active discontinuity will be called a master and its limiting level-sets slaves. The slaves separate the master in two distinct regions: valid as a discontinuity or not valid. Back to a topological understanding, the discrete value of a level-set will describe a geometrical object, one dimension below the actual space dimensions considered. Accordingly, the intersection with a transverse level-set will create a geometrical object two dimensions below the actual domain. In 2D, a slave thus creates a point and this point can be used to mark the ending point of a discontinuity in the field.

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Master and slaves are moved independently and a common speed vector c is formulated. The component of c normal to the discontinuity rules its displacement as in the classical method, while the movement of the slaves is responsible for alterations of its extension in time. For simplicity, level-set gradients of master and slaves are assumed orthogonal at the intersection, which is ensured by a reconstruction step of all slaves after motion. 6.4 Bifurcating discontinuities and intersection points A possible way to create interacting discontinuities is to decompose these in several open sections, whose ending points are attached to a common point of intersection. As several discontinuities are attached to a single intersection point and as this point may move according to the dynamics of the problem, it has to be ensured that the ending points of the appended discontinuities follow synchronously. In fact, this requirement reverses the original level-set idea to a certain extent: In the classical theory, a location on the discontinuity is given by a discrete value of a function, but to follow a given point, the function has to be reconstructed accordingly. In first tests, it was found to be useful to formulate a correction speed at all ending points to follow an intersection. With a given intersection point location PInt : (xInt , [t]) and the ending point location Pi : (xi , [t]) of a discontinuity i, the coupling speed ci is defined as: xInt − xi , (29) ci = ∆τ where ∆τ is an iteration time step, actually chosen as the true time step in explicit calculations. Alternatively, additional post-iterations to satisfy the matching condition with better accuracy are possible, if needed. As the values for the level-set functions involved are stored at the fixed nodes of a given mesh, it is useful to relax the above coupling speed over a certain range of influence. For first tests a linear approach in a relaxation radius Ri was chosen. For a node Pi within a level-set i the correction speed reads: i| ci · (1 − |xPR−x ) for |xP − xi | < Ri , i (30) cP,i = 0 for |xP − xi | ≥ Ri . Not all of the above mentioned is yet available as a reliable program code. In fact many questions remain, such as priority rules for intersection point motion or precision limits. In consequence however, the formulation allows the extension of the level-set method to a large number of further physical problems.

7 Computational Examples 7.1 2D/3D bow shock computations In this test case, the 3D bow shock of the inviscid flow past a sphere is computed as a single closed surface. The complete, unstructured grid consists of

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216940 tetrahedral elements. The free stream Mach number is M a∞ = 2. Figure 7 shows the initialization of the shock front as a planar surface in front of the sphere, Fig. 8 shows a converged solution. Broken partitions of the discrete definition range and iso-colors of pressure on a cut plane are shown in both figures below. The 2D computational example is calculated in the same manner as described in chapter 6.

Fig. 7. Initialization of a 3D bow shock with shock, subset grid, inner sphere and constant pressure values. (See also color ﬁgure, Plate 6.)

Fig. 8. Final state of the 3D bow shock calculation with front, subset mesh and pressure values in the cut plane. (See also color ﬁgure, Plate 6.)

Fig. 9. Snapshot of an inviscid bow shock with Mach=3 and three cylinders. Pressure values are plotted (See also color ﬁgure, Plate 6.)

Fig. 10. Zoom of area between both lower cylinders; pressure values, front, subset mesh and velocity vectors. (See also color ﬁgure, Plate 6.)

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7.2 Unstable inviscid 2D Chapman-Jouget detonation The second example is a snapshot of an unstable inviscid 2D ChapmanJouget(CJ) detonation on a triangulated grid, Fig. 11. Chemical source terms are calculated with a one step reaction model. The detonation is initialized as a steady CJ-detonation with an incoming flow of Ma=8.3 from the right side. The onset of detonation cells can be observed. In theory, the front shock has kinks at the position where transverse shocks in the reaction zone (captured here) are attached. The classical level-set theory fails to describe the effect with sharp corners. 7.3 Inviscid, transonic flow past a NACA 0012 airfoil The transonic, inviscid flow past a NACA 0012 airfoil is considered at M a∞ = 0.75 and αattack = 2o . The level-set propagation was combined with an explicit multi-grid method for the flow field [10]. In this test case, a single, limited recompression shock wave on top of the airfoil appears. It is simulated here by a single ending front, described by a master-slave combination as can be seen in Fig. 12. At present, no shock detection and transverse shock growth algorithms are available yet. Therefore, the master-slave combination has been initialized near the leading edge with an estimated altitude above the surface. While the shock moves towards the final steady state position, the limiting slave level-set has been frozen. Up to the ending point, inner boundary conditions on the shock apply, while degrading to the capturing method beyond.

Fig. 11. Snapshot of an inviscid unstable 2D CJ-detonation with pressure values calculated on an unstructured grid. (See also color ﬁgure, Plate 6.)

Fig. 12. Master-slave combination for a restricted re-compression shock wave on a transonic airfoil: Colors of constant Mach number and front position. (See also color ﬁgure, Plate 6.)

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7.4 Passive transport of a master-slave combination The next two figures (Fig. 13 and Fig. 14) show an example for a master-slave combination on a triangulated grid in a phantasy domain. The flow from the left is inviscid at M a∞ = 0.5. Both level-sets are transported with the local fluid velocity. In this case the fluid velocity causes the slave to extend the master. The examples can be interpreted to a moving twine in a spinnery.

Fig. 13. Initialization of a passively transported master-slave combination.

Fig. 14. Passive master-slave combination after several time steps.

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7.5 Regular shock reflection The last example is a regular shock reflection. Figure 15 and Fig. 16 show two independent master discontinuities, which are only coupled on the reflection point lying on the boundary. In this case a regular shock reflection is initialized in a such a manner that the absolute value of the deflection angle of the incident shock doesn’t correspond to the deflection angle of the reflected shock. This causes a movement of the reflected shock in the field depending on the boundary condition.

Fig. 15. Zoom of initialized regular shock reﬂection; pressure values, front and velocity vectors

Fig. 16. Zoom of ﬁnal regular shock reﬂection; pressure values, velocity vectors, front and stream lines

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7.6 Level-sets attached to moving points Another test example is a calculation of a master-slave combination guided by three moving points. The master level-sets have zero speed, while the three guiding nodes march with constant velocity to the right side. This is a purely academic case, testing the ability of restricted fronts to follow intersection points. The computation is performed on a structured grid. A sequence in time is given in Fig. 17.

Fig. 17. Two clipped fronts with ends following three moving points in the ﬁeld. Sequence, showing all level-sets involved (two masters, four slaves) left: initial condition middle: after 10 time steps right: after 100 time steps

8 Conclusion A tracking method for moving fronts based on level-set formulations is presented. The tracking algorithm is integrated in an object-oriented finitevolume method for conservation laws on unstructured grids. The level-set concept is formulated locally in a restricted range around the front, which enables the treatment of several or even crossing discontinuities. The tracking algorithm is described in detail and demonstrated by means of 2-D and 3-D shock wave problems. Further discussion is presented about issues of accuracy and topological restrictions.

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Acknowledgement. This project was supported by the ”Deutsche Forschungsgemeinschaft” (DFG) in the frame of the special research program ANUME.

References 1. Gloth O., Vilsmeier R. H¨anel D. (2004) Homepage of Project MOUSE, Online: http://www.vug.uni-duisburg.de/MOUSE (periodically updated) 2. Richtmyer R., Morton K. (1967) Diﬀerence Methods for Initial value Problems. Interscience, New York 3. Moretti G. (1987) A Technique for Integrating Two-Dimensional Euler Equations. Computers & Fluids 15: 59-75 4. Sethian AJ. (1999) Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics. Computer Vision and Materials Science Cambridge. Cambridge University Press, Cambridge 5. Kerstein A., Ashurst W., Williams F. (1988) Field Equation for Interface Propagating in an Unsteady Homogeneous Flow Field. Phys. Rev. A 37: 2728-2731 6. Mulder K., Osher S., Sethian AJ. (1992) Computing Interface Motion in Compressible Gas Dynamics. J. Comp. Phys. 100: 209-228 7. Sussman M., Smereka P., Osher S. (1994) A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow. J. of Comp. Phys. 114:146-159 8. Tran L., Vilsmeier R., H¨anel D. (1999) A local level set method for the treatment of discontinuities on unstructured grids. In: Vilsmeier R. Benkhaldoun F., H¨anel D. (Eds.) Finite Volumes for Complex Applications II, Hermes Science Publ., Paris, 623-630, 849-856 9. Vilsmeier R., H¨ anel D., Tran L. (2003) A Front Tracking Method on Unstructured Grids. Computer & Fluids 32: 547-570 10. Fournier L., Gloth O. (1999) An Attempt to Develop a Multi Purpose FAS Multigrid Algorithm. In: Vilsmeier R. Benkhaldoun F., H¨anel D. (Eds.) Finite Volumes for Complex Applications II, Hermes Science Publ., Paris, 623-630 11. Bouzouf B., Gloth O., H¨ anel D.,Vilsmeier R. (2001) Simulation of discontinuous ﬂows in porous media. Proc. of 1st Int. Conf. on Saltwater Intrusion and Costal Aquifers (SWICA), Essaouira, Morocco 12. Bet F., H¨ anel D., Sharma S. (1999) Numerical Simulation of Ship Flow by a Method of Artiﬁcial Compressibility. Twenty-Second-Symposium on Naval Hydrodynamics, National Academy Press, Washington, 522-531 13. Smiljanovski V., Moser V., Klein R. (1997) A capturing-tracking hybrid scheme for deﬂagration discontinuities. Comb Theory Modell 1:183-215

Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and K.P. Hadeler2 1

2

University of Alberta, Mathematical and Statistical Sciences, Edmonton T6G 2G1, Canada [email protected] Biomathematik, University of T¨ ubingen, Auf der Morgenstelle 10, D-72076 T¨ ubingen [email protected]

Summary. The standard models for groups of interacting and moving individuals (from cell biology to vertebrate population dynamics) are reaction-diﬀusion models. They base on Brownian motion, which is characterized by one single parameter (diﬀusion coeﬃcient). In particular for moving bacteria and (slime mold) amoebae, detailed information on individual movement behavior is available (speed, run times, turn angle distributions). If such information is entered into models for populations, then reaction-transport equations or hyperbolic equations (telegraph equations, damped wave equations) result. The goal of this review is to present some basic applications of transport equations and hyperbolic systems and to illustrate the connections between transport equations, hyperbolic models, and reaction-diﬀusion equations. Applied to chemosensitive movement (chemotaxis) functional estimates for the nonlinearities in the classical chemotaxis model (Patlak-Keller-Segel) can be derived, based on the individual behavior of cells and attractants. A detailed review is given on two methods of reduction for transport equations. First the construction of parabolic limits (diﬀusion limits) for linear and non-linear transport equations and then a moment closure method based on energy minimization principles. We illustrate the moment closure method on the lowest non-trivial case (two-moment closure), which leads to Cattaneo systems. Moreover we study coupled dynamical systems and models with quiescent states. These occur naturally if it is assumed that diﬀerent processes, like movement and reproduction, do not occur simultaneously. We report on travelling front problems, stability, epidemic modeling, and transport equations with resting phases. Key words: Chemotaxis, transport equation, hyperbolic system, Keller-Segel model, telegraph equation, parabolic limit, moment closure, Cattaneo system

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1 Introduction Hyperbolic models and transport equations are used in Mathematical Biology to model movement and growth of populations. For example, certain bacteria (like Escherichia coli or Salmonella typhimurium) show a very characteristic movement pattern. Periods of straight runs alternate with periods of random rotations which lead to reorientation of the cells ([BB72]). This behavior can be modeled by a velocity jump process, which in a continuum formulation leads to a transport equation ([Str74]). Transport models in one space dimension can be seen as hyperbolic systems. Moreover, in any space dimension, moment closure methods lead from transport models to hyperbolic systems. In contrast to diffusion based models, transport models and hyperbolic systems do not show the unwanted effect of infinitely fast propagation. Transport equations are based on detailed information on turning rate, turning distribution and mean speed. The relevant parameters can be extracted from the measurements of individual paths. Transport models for biological applications are closely related to transport models in physics, like semiconductor [MRS90], radiation [MWM83], and neutron transport [J¨ or58], as well as to thermodynamics [MR98] and the Boltzmann equation [Bel95, CIP94]. In typical physical applications the directional changes of the individual particles are driven by collisions. These collisions usually conserve mass, momentum and energy, and hence the collision operator has a five-dimensional null space. In a biological context only the total particle mass is conserved and directional changes have to be treated as spontaneous (without collisions), and they do not necessarily preserve energy or momentum. Hence the turning operator has a one-dimensional null space. This difference becomes important if one uses functional analytic properties of the operators involved (like spectral properties, stability, parabolic scaling limits, asymptotic behavior etc.). All moving species orient their movement on external information. For bacteria spatial information comes via their cell surface receptors which collect information mainly on chemical cues. In chemotaxis, for example, cells move towards high concentrations of a chemical attractant. Experiments on chemotaxis measure either the behavior of the population as a whole, e.g. in terms of densities (e.g. Woodward et al. [WTM+ 95]) or the paths of individuals are followed with a video apparatus (Frymier et al. [FFBC95]). Of course the behavior of the population results from the movement of its members. Mathematical modeling provides a way to relate the individual and collective movement parameters, e.g. by forming parabolic limits [HO00]. The first mathematical model for chemotaxis is the Patlak-Keller-Segel model (PKS), which is based on Brownian motion. It is known that the PKS model shows a great variety of different patterns, e.g. standing waves, aggregation, finite time blow-up, or spinodal decomposition patterns ([MPW98, HP01, Hor03]).

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We use the remainder of this introduction to recall basic facts about correlated random walk models and transport equations. In Section 2 we discuss three model classes for chemotaxis; the classical PKS models, a onedimensional hyperbolic model, and transport equations in arbitrary dimensions. In particular for transport equations we develop a spectral theory which ensures the existence of a parabolic limit. In Section 3 we discuss coupled dynamical systems and systems with quiescent states. In Section 4 we present the moment closure method and we apply it to chemotaxis models. 1.1 Correlated Random Walk in One Dimension Movement in one space dimension with constant speed γ and constant turning rate µ can be described by a correlated random walk. The total population density u(t, x) is split as u = u+ + u− into densities for right/left moving parts of the population, u+ , u− , respectively. These are the variables of the Goldstein-Kac model for correlated random walk ([Gol51, Kac56]), µ + + + − u+ t + γux = −µu + (u + u ), 2

µ + − − − u− t − γux = −µu + (u + u ). (1) 2

This system can be transformed to an equivalent system for the total population density u and the population flux v = γ(u+ − u− ): ut + vx = 0,

vt + γ 2 ux = −µv.

(2)

By eliminating the variable v (Kac’ trick) one obtains a telegraph equation or damped wave equation. 1 γ2 utt + ut = uxx . µ µ

(3)

Note that the transition from (1) to (3) (which can be generalized to systems with several dependent variables in any space dimension) is not completely invertible ([Had99b]). If we let the parameters µ and γ go to infinity such that the quotient γ 2 /µ converges to a number D > 0 then we formally obtain the parabolic limit ut = Duxx . Notice that in this linear case the transition is equivalent to a scaling of space and time τ = ε2 t and ξ = εx with µ and γ held constant. 1.2 The Linear Transport Equation We denote the population density at time t ≥ 0, position x ∈ IRn , and velocity v ∈ V ⊂ IRn with p(t, x, v). We assume that the set of possible velocities V ⊂ IRn is bounded and symmetric (i.e. v ∈ V ⇒ −v ∈ V ). Then the linear transport model, which is based on a velocity jump process (see e.g. Stroock [Str74] or Othmer et al. [ODA88]) reads

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∂ p(t, x, v) + v · ∇p(t, x, v) = −µp(t, x, v) + µ ∂t

T (v, v ′ )p(t, x, v ′ )dv ′ .

(4)

Here µ is the turning rate or turning frequency, and τ = 1/µ is the mean run time. The kernel T (v, v ′ ) describes ; the probability for the new velocity v given the previous velocity v ′ , hence T (v, v ′ )dv = 1. The one dimensional Goldstein-Kac model (1) occurs as a special case in one space dimension with v ∈ {±γ} and T (v, v ′ ) ≡ 1/2. 1.3 Models with Reaction Reaction diffusion equations are the standard models for spread in space and interaction of particles, e.g. density-dependent birth and death processes (e.g. Murray [Mur89]). If the diffusion process is replaced by a more detailed transport process, then reaction transport models result, pt + v · ∇p = −µp + µ T (v, v ′ )p(t, x, v ′ )dv ′ + F [p], (5) where the functional F [p] describes the growth dynamics (see [Had96]). Whereas in earlier papers on reaction transport equations (and on correlated random walk models with reactions) the nonlinearity has not been further specified, later, with respect to ecological modeling, a clear distinction between production and removal events has been made. If, for example, the newly produced individuals have a uniform distribution of velocities, then the nonlinearity has the form F [p] =

1 b(m0 )m0 − g(m0 )p, |V |

with non-negative rates b(m0 ), g(m0 ) whereby 0 m (t, x) = p(t, x, v) dv

(6)

V

is the total population density (we will use mj for moments later). This model preserves positivity. In the isotropic case, where the non-linearity depends only on m0 , F [p] = f (m0 ), positivity can no longer be guaranteed if f changes sign. The isotropic case has been studied in detail in [Hil98], where it has been shown that in certain situations all ω limit sets are steady states. For isotropic and non-isotropic models boundary value and spectral problems have been studied in [Had99b]. The travelling front problems in the case of one space dimension (which covers also the case of Cattaneo systems in any space dimension) have been studied in a sequence of papers, where [Had99a] gives the most detailed analysis. For the stability of hyperbolic fronts see also [GR00]. Schwetlick [Sch00, Sch02] has obtained rather general results on the minimal propagation speed in reaction transport equations (which depends on space dimension in a natural way) and on the existence of fronts.

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1.4 The Langevin or Kramers approach In the Langevin approach it is not the position of the particle but √ its velocity that is subject to Brownian motion, i.e., the equation dx = DdW (with the Wiener process W ) is replaced by the deterministic transport process √ dx = vdt and the Ornstein-Uhlenbeck process dv = −γvdt + ddW for the velocity, with γ > 0, whereby the term −γvdt pulls the velocity towards the origin. The density u(t, x, v) satisfies the Kramers equation ut + v · ∇x u − γ∇v (vu) = d∆v u.

(7)

This equation can be easily understood in terms of its three constituents ut + v · ∇x u (transport), ut = d∆v u (Brownian motion in velocity space) and ut = −γ∇v (vu) (drift in velocity space). In [HHL04] a reaction Kramers equation has been proposed ut + v · ∇x u − γ∇v (vu) = d∆v u + H(v)m(¯ u)¯ u − g(¯ u)u with u ¯(t, x) =

(8)

u(t, x, v)dv.

The function H(v) ; describes the velocity distribution of new particles, it satisfies H(v) ≥ 0, H(v)dv = 1. The equation (8) has been studied in great detail in [HHL04]. Moment approximations yield damped wave equations and diffusion equations with an effective diffusion rate D = d/γ 2 .

2 Models for Chemotaxis 2.1 The Classical Patlak-Keller-Segel (PKS) Model If the movement of a population or an individual is biased by a chemical signal then the response is termed chemotaxis (or more generally chemosensitive movement). Models for chemotaxis have been successfully applied to bacteria, slime molds, skin pigmentation patterns, leukocytes and many other examples. As mentioned in the introduction the first model for chemosensitive movement has been developed by Patlak [Pat53] and Keller and Segel [KS70]. Patlak’s model is based on a detailed random walk description [Pat53]. The Keller-Segel model in its general form consists of four coupled reaction diffusion equations. In most publications it has been reduced to two essential variables, the population density u(t, x) and the concentration of a chemical signal v(t, x). Then the Keller-Segel model reads ut = ∇(k1 (u, v)∇u − k2 (u, v)∇v) vt = kc ∆v − k3 (v)v + uf (v).

(9)

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This system has been studied on unbounded and on bounded domains with various boundary conditions (Dirichlet, Neumann, mixed). In his survey Horstmann [Hor03] gives an ample review of the many now available analytical results, in particular on blow-up in finite time. In the case of constant coefficient functions k1 , k2 , k3 , and f it is known that the qualitative behavior strongly depends on the space dimension. In 1-D the system has globally existing solutions. In the 2-D case there may be thresholds: solutions with large total initial mass blow up in finite time, and solutions with small initial mass exist globally in time. The blow-up solutions of the system (9) show the existence of a very strong instability and a large aggregational force. In certain situations, however, it is desirable to obtain stable aggregation patterns, which do not blow up in finite time. There are various mechanisms which prevent blow up. These can be classified as follows: 1. Saturation effects in k2 (u, v) occur very naturally if cell surface receptor kinetics is taken into account. Chemotaxis models with saturation effects have been studied analytically and have been used in many applications (Othmer and Stevens [OS97], Biler [Bil98], Rivero et al. [RTBL89], Ford et al. [FPQL91]). 2. A volume filling effect was introduced by Hillen and Painter [HP01, PH02]. Here it is assumed that particles have a finite volume and that cells cannot move into regions which are already filled by other cells. A simple version of the volume filling method leads to a term k2 (u, v) = χu(1 − u). It was shown analytically that this form of k2 leads to globally existing solutions in all space dimensions. 3. Quorum sensing occurs if the cells release an extra chemical which is repulsive to other cells [HP01]. The resulting equation has two competing drift terms, chemotactic attraction and quorum sensing repulsion. It is an open mathematical problem to find general conditions for blow-up or global existence. 4. Also a finite sampling radius leads to global existence, at least in 2-D, as was shown by Hillen et al. [HPS04]. Here it is assumed that individuals measure the chemical substance on a disc (modeling measurement near the cell surface). 2.2 Hyperbolic Models for Chemotaxis The general hyperbolic chemotaxis model in one space dimension is based on the Goldstein-Kac model (1): + + + − − u+ t + (γ(S, St , Sx )u )x = −µ (S, St , Sx )u + µ (S, St , Sx )u ,

− + + − − u− t − (γ(S, St , Sx )u )x = µ (S, St , Sx )u − µ (S, St , Sx )u ,

τ St = αSxx + f (S, u+ + u− ) , τ ≥ 0,

u± (0, .) = u± 0,

S(0, .) = S0 .

(10)

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Here the rates µ± are turning rates which depend on the signal strength S(t, x) and on the space and time-variations Sx , St . Note that in (1) µ is a stopping rate (not a turning rate) and each direction will be chosen with probability of 1/2. The speed γ might also depend on S, St , Sx , as supported by experiments of Soll and Wessels [SW98], or Fisher et al. [FMG89]. The function f (S, u+ + u− ) describes production and decay of the external signal. The system (10) requires additional assumptions to exclude backward diffusion (via the dependence on Sx ) and to ensure a well-defined Cauchy problem. A possible set of sufficient conditions is given in [HRL01] or in [HKS03]. Special cases of (10) were studied by Segel [Seg77], Rivero et al. [RTBL89], Greenberg and Alt [GA87], and Chen et al. [CFC99, CFC98]. In [HS01] local and global existence of solutions has been proved for the case of constant speed and for turning rates depending on S and Sx . In [HRL01] global existence in time has been shown for γ = γ(S), where the signal distribution was assumed to be in quasi-equilibrium (τ = 0). The results of [HS01] and [HRL01] have been extended recently in [HKS03] to include St and Sxx dependence in the turning rates and τ = 0 for the case studied in [HRL01]. Using “parabolic scaling” the hyperbolic model (10) can be related to the classical PKS-model (9). We reformulate (10) in terms of the total density u = u+ + u− and the difference q = u+ − u− : ut + (γq)x = 0, qt + (γu)x = −(µ+ − µ− )u + (µ+ + µ− )q, τ St = αSxx + f (S, u).

(11)

Similar to the analysis of the linear model (1), we can derive a telegraph equation (12) utt + hut − (γ(γu)x )x − (γ(µ+ − µ− )x + hx q = 0, where the function h is given by h = µ+ (S, St , Sx ) + µ− (S, St , Sx ) −

γt (S, St , Sx ) . γ(S, St , Sx )

(13)

To obtain the parabolic limit we assume that h does not depend explicitly on the space variable x. We give examples later. We introduce a small scaling parameter ε > 0 as γ=

γ0 , ε

and µ± =

µ± 0 , ε2

(14)

where γ0 and µ± 0 are of order 1 with respect to ε. We use the order one parameters to define − 2 γ0,t h0 = µ+ . 0 + µ0 − ε γ0 Then we find that

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h(t) = For ε → 0 we obtain

h0 (t) . ε2

(15)

γ2 γ2 = + 0 −. ε→0 h(t) µ0 + µ0

D = lim

(16)

If we scale the corresponding telegraph equation accordingly then we obtain for ε → 0 the parabolic limit equation ut = (Dux − uΦ)x ,

(17)

with a diffusion coefficient given by (16) and the chemotactic velocity Φ given by γ0 1 + − Φ=− + γ0,x + lim (µ0 − µ0 ) . (18) ε→0 ε µ0 + µ− 0 These relations were derived in special cases in Rivero et al. [RTBL89] and Segel [Seg77] and in general in [HS01]. If Φ has the form Φ(S, Sx ) = χ(S)Sx then (17) is the classical chemotaxis model with chemotactic sensitivity χ(S). The chemotactic velocity (18) consists of two terms which indicate that aggregation can be caused by two different mechanisms. Either the individuals slow down at high concentrations of S (i.e. γ ′ < 0), or individuals which move up a gradient turn less often than individuals moving down the gradient (i.e. µ+ Sx < µ− Sx ). These effects have been discussed in detail in [HS01]. Schnitzer [Sch93] found the same effects in a similar one-dimensional hyperbolic model for bacterial movement, where memory effects have been included. 2.3 Transport Models for Chemotaxis There is a rich literature on transport models applied to populations. We refer to the articles of Alt [Alt80, Alt81], Othmer, Dunbar and Alt [ODA88], Chen et al. [CFC99], Dickinson and Tranquillo [DT95], Dickinson [Dic00], and Hillen and Othmer [HO00, OH02]. In [HO00, OH02] a general theory has been developed to obtain the parabolic limit (diffusion limit) for a general transport equation which describes movement of populations. We present the basic result. We consider (4) on Ω = IRn . We assume that V ⊂ IRn is compact and symmetric such that v ∈ V implies −v ∈ V . Let K denote the cone of nonnegative functions in L2 (V ). We define the following operators on L2 (V ): T p(v) = T (v, v ′ )p(v ′ )dv ′ , T ∗ p(v) = T (v ′ , v)p(v ′ )dv ′ , V

V

L = −µ(I − T ),

where I denotes the identity. For the kernel T we assume:

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;;

(T1) T (v, v ′ ) ≥ 0, T (v, v ′ )dv = 1, and T 2 (v, v ′ )dv ′ dv < ∞. (T2) There exists some u0 ∈ K with u0 ≡ 0, some integer N and a constant ρ > 0 such that for all (v, v ′ ) ∈ V × V u0 (v) ≤ T N (v ′ , v) ≤ ρu0 (v), where the N -th iterate of T is T N (v, v ′ ) := . . . T (v, w1 )T (w1 , w2 ) · · · T (wN −1 , v ′ )dw1 . . . dwN −1 . (T3) T 1⊥ < 1, where 1⊥ denotes the orthogonal complement of the subspace 1 ⊂ L2 (V ) of functions constant in v. ; (T4) V T (v, v ′ )dv ′ = 1.

Under these assumptions the turning operator L has the following KreinRutman properties: Theorem 1. Assume (T1)-(T4). Then 1. 0 is a simple eigenvalue of L with eigenfunction φ(v) ≡ 1. 2. There exists an orthogonal decomposition L2 (V ) = 1 ⊕ 1⊥ and for all ψ ∈ 1⊥ holds ψLψdv ≤ −ν2 ψ2L2 (V ) , with ν2 ≡ µ(1 − T 1⊥ ). 3. Each eigenvalue λ = 0 satisfies −2µ < Re λ ≤ −ν2 < 0, and there is no other positive eigenfunction. 4. LL(L2 (V ),L2 (V )) ≤ 2µ. 5. L restricted to 1⊥ ⊂ L2 (V ) has a linear inverse F with norm F L(1⊥ ,1⊥ ) ≤

1 . ν2

For a proof and more details see [HO00]. Similar to the one-dimensional case we study the parabolic scaling. In the previous section the scaling was expressed through a scaling of γ and µ. Here we choose another, equivalent approach. For a small parameter ε > 0 we consider macroscopic time and space scales τ = ε2 t

and ξ = εx.

(19)

We rescale equation (4) ε2 pτ + εv · ∇ξ p = Lp and for k > 2 we consider an expansion of p:

(20)

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p(τ, ξ, v) =

k

pi (τ, ξ, v)εi + pk+1 (τ, ξ, v)εk+1 .

i=0

Comparing orders of ε we are led to the diffusion limit for the leading order approximation p0 (τ, ξ): p0,τ = ∇ · D∇p0 , p0 (ξ, 0) = p(ξ, v, 0)dv, (21) V 1 with diffusion tensor D≡− vFv T dv. (22) ω V The procedure is outlined in detail in [HO00]. It can be continued to higher orders in ε. In [HO00] it has been shown that the residuum of this approximation can be controlled: As a consequence the asymptotic behavior of solutions of (4) is described by the diffusion equation in (21). The proof of this result uses an induction argument. In particular property (T3) is important to show that the limiting equation in (21) is parabolic. As seen in the above diffusion limit (22), the general diffusion coefficient D is a 2-tensor. Under certain conditions, however, the diffusion tensor is isotropic, i.e., it is a scalar multiple of the identity. To formulate this condition for isotropy we compare three properties. (S1): There is an orthonormal basis {e1 , . . . , en } of IRn such that for each i = 1, . . . , n the coordinate mappings πi : V → IR, πi (v) = vi are eigenfunctions of L with common eigenvalue ν ∈ (−2µ, 0). (S2): There;is a constant γ ∈ (−1, 1) such that for each v ∈ V the expected velocity v¯(v) ≡ T (v, v ′ )v ′ dv ′ satisfies v¯(v) v

and

v¯(v) · v = γ. |¯ v (v)||v|

(S3): There is a constant d > 0 such that D = d In . Theorem 2. Let (T1)-(T4) hold and assume that V is symmetric with respect to SO(n). Then we have (S1)

⇐⇒

(S2)

=⇒

(S3)

whereby the constants ν, γ and d are related as follows. γ=

ν+µ , µ

d=−

KV KV = , ων ωµ(1 − γ)

with

KV I =

vv T dv.

V

If T also satisfies (T5): There is a matrix M such that v¯(v) = M v for all v ∈ V , then all three statements are equivalent. This Theorem is proved in [HO00].

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All statements (S1)-(S3) are true if T has the symmetric form of T (v, v ′ ) = t(|v − v ′ |) (see also Alt [Alt80]), hence diffusion is isotropic. Reference [HO00] gives an example for non-isotropic diffusion. In the case of chemosensitive movement in (4) the turning rate µ and the velocity distribution kernel T (v, v ′ ) depend on the signal distribution S(t, x), on its gradient ∇S(t, x), or on other properties of S (e.g. non-local dependence can be included). µ = µ(S, ∇S, . . . ),

T (v, v ′ ) = T (v, v ′ , S, ∇S, . . . )

(23)

In [OH02] we systematically study perturbations, which come form chemical cues, of the form ˆ = T0 (v, v ′ ) + εk T˜(v, v ′ , S), ˆ T (v, v ′ , S)

ˆ = µ0 + εl µ ˆ µ(v, S) ˜(v, v ′ , S),

for k = 0, 1 and l = 1, where Sˆ denotes dependence on the function S and not only on the local value S(t, x). Perturbations of higher order k, l ≥ 2 will not affect the parabolic limit equation. Perturbations of the turning rate µ0 of order one (l = 0) do not fit into the framework developed here. But that case can be handled in the theory of moment closure as illustrated in Section 4. There, it is shown that also order one perturbations in the turning rate lead to PKS-type models. In [OH02] a large number of examples is given. Some relate directly to specific applications (bacteria, amoebae), others show how the classical PKS model appears as a diffusion limit. Moreover an example of non-isotropic diffusion is given. The above choice of turning rates µ and velocity distributions T can be extended to include non-local gradients, which might be measured by amoeba along their surface. Let R > 0 denote an effective sampling radius, then we define a non-local gradient by ◦ n σ S(x + Rσ, t) dσ, (24) ∇ S(x, t) = ω0 R S n−1 For R → 0 this expression approximates the local gradient ∇S. For chemosen◦

sitive movement we treat the non-local gradient ∇ S in exactly the same way as ∇S. From the mathematical point of view the non-local gradient has better regularity properties. This alone is sufficient to show global existence of solutions to a PKS-model, which would show finite time blow-up if the local gradient is used [HPS04].

3 Coupled Dynamics and Quiescent States 3.1 Splitting the Dynamics If the evolution of populations is modelled by differential or partial differential equations it is often assumed that processes react simultaneously. For example

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in reaction diffusion equations the transport and the interaction processes run parallel. However, in many realistic situations relevant processes alternate, either periodically or with random transitions. Periodically alternating actions lead to non-autonomous problems ([HW90]) while random transitions lead to a new class of dynamical systems which we call coupled dynamical systems. We study the following types of models: Suppose v˙ = f (v) and w˙ = g(w) are two dynamics in IRn . These dynamics are diffusively coupled in the system v˙ = f (v) − γ2 v + γ1 w w˙ = g(w) − γ1 w + γ2 v where γ1 and γ2 are positive constants. If one of the kinetic terms is zero, say g = 0, then we call w the quiescent state and the resulting system reads: v˙ = f (v) − γ2 v + γ1 w w˙ = γ2 v − γ1 w.

(25)

In a particle interpretation the system (25) says that particles are governed by the dynamics f and move at a rate γ2 to a quiescent state from where they return at a rate γ1 . Examples include predators which rest in their den and go out for foraging from time to time, or cells which spend a significant amount of time in a quiescent state. Other examples are systems derived from the Fisher equation ut = D∆u+ f (u). In ([LS96], [HL02]) the dynamics has been split to vt = D∆v − γ2 v + γ1 w wt = f (w) − γ1 w + γ2 v

(26)

with f (w) = w(1 − w), where v is a moving state and w an interaction state. This system is essentially equivalent to a wave equation with nonlinear diffusion and with viscous damping (with τ = 1/(γ1 + γ2 ) and ρi = τ γi ) τ wtt + (1 − τ f ′ (w))wt − τ D∆wt

= ρ1 D∆w + ρ2 f (w) − τ D∆f (w).

(27)

In this equation again a “parabolic limit” τ → 0 can be taken which results in another rescaled Fisher equation. The wave operator in this problem, i.e., wtt −ρ1 D∆w +τ D∆f (w), shows a transition from hyperbolic to elliptic for large τ . Models of this form also play a role in the theory of infectious diseases, where the moving and sedentary states correspond to long range and short range infections [Had03]. In another extension of the Fisher equation the reaction-diffusion dynamics is coupled to a quiescent state vt = D∆v + f (v) − γ2 v + γ1 w wt = −γ1 w + γ2 v.

(28)

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This system leads also to a wave equation which is however hyperbolic for all choices of the parameters. For both systems (26) and (28) one can study the dynamics in a bounded domain (e.g. with zero Dirichlet conditions for the variable v) as well as the invasion or travelling front problem. Using recent results of Lewis, Li, and Weinberger [Wei75] one can exploit the cooperative structure of the system (in view of γ1 , γ2 > 0) and show, for concave f , that the spread rate (essentially the speed of travelling fronts) can be determined by a linear analysis at the leading edge of the front [HL02]. Also in lattice dynamics or in coupled oscillators one studies systems with diffusive coupling with an arbitrary number of constituents which have identical or very similar structure. 3.2 Quiescent States Following the particle interpretation, introducing a quiescent state as in (25) should have a similar effect as introducing a delay, i.e., the system (25) should behave as the equation u(t) ˙ = f (u(t − τ )) (29)

with τ > 0. Surprisingly enough, the effect is quite different. For the scalar delay equation, (29) with n = 1, with negative feedback (f (0) = 0, f ′ (0) = α < 0) sufficiently large delays (|α|τ > π/2) destabilize the stationary solution and typically lead to stable periodic oscillations. Quite on the contrary, introducing a quiescent state suppresses oscillations. Indeed, in (25), with n ≥ 1, assume f (¯ v ) = 0. Then (¯ v , w) ¯ with w ¯ = v ) and the corresponding γ2 v¯/γ1 is a stationary state. The eigenvalues µ of f ′ (¯ eigenvalues λ of the Jacobian at (¯ v , w) ¯ are connected by the equation λ2 + λ(γ1 +γ2 −µ)−µγ1 = 0. To each eigenvalue µ there are two eigenvalues λ1 , λ2 . We choose the convention ℜλ2 ≤ ℜλ1 . In [HH04] we explore the relationships between these eigenvalues in detail. The result is: Theorem 3. (a) Let µ = α ∈ IR. Then λ1 , λ2 are real. (a.i) If α < 0 then λ2 < α < λ1 < 0. (a.ii) If α = 0 then λ2 = −(γ1 + γ2 ) < 0 = λ1 . (a.iii) If α > 0 then λ2 < 0 < λ1 < α. (b) Let µ = α ± iβ, β > 0. Then ℜλ2 < 0. (b.i) If α ≤ 0 then ℜλ1 < 0. (b.ii)) If α > 0 then ℜλ1 < α. (b.iii) If α ≤ 0 and β 2 + (γ1 + γ2 + α)2 + 4αγ2 > 0 and β 2 (γ1 + α) + α(γ1 + γ2 + α)2 > 0, then ℜλ1 < α. (b.iv) If α > 0 and β 2 > 4αγ1 − (γ1 + γ2 − α)2 and β(γ2 − α) > α(γ1 + γ2 − α)2 , then ℜλ1 < 0.

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The result shows that the stability with respect to real eigenvalues is not affected while purely imaginary eigenvalues move into the left half plane of the complex plane and high frequency oscillations are damped. Hence in general it can be stated that introducing a quiescent state into a given dynamics tends to remove oscillations, a delay (together with negative feedback) tends to enhance oscillatory behavior. Further results on coupled dynamics and quiescent states are presented in [HH04]. We apply the concept of a quiescent state to transport equations in the following subsection. 3.3 Transport Equations with Resting Phases The theory on parabolic scaling of transport equations becomes very useful if we study transport equations with resting phases. Transport equations with resting phases are an adequate model for populations whose individuals switch between an active, moving phase, and a resting phase. Moreover we assume that reproduction can only occur during resting periods, which is certainly true for most mammals. Applications of resting-phase transport models occur for predator-prey systems (e.g. rabbits which are subject to predation when outside, but safe while inside a den), stabilization of oscillations in predator-prey systems (Neubert et al. [NKvdD02]), movement to better habitats, (Cosner and Lou [CL03]), the river-drift paradox (Pachepsky et al. [PLNL04]), territoriality and home ranges, invasion of microbes (Lewis and Hadeler [HL02]), and also movement of proteins in the cell nucleus, (Carrero et al. [CMC+ 03]). We split the population into a moving compartment p(t, x, v) and a resting compartment r(t, x), with a total population density of p(t, x, v)dv + r(t, x). N (t, x) = V

Then the resting phase transport model as studied in [Hil03b] reads: pt + v · ∇p = Lp − γ;2 (x)p + γω1 r − l(N )p rt = γ2 (x) V p(., ., v)dv − γ1 r + g(N )r − l(N )r,

(30)

where γ2 (x) denotes a spatially dependent stop rate, γ1 denotes a start rate, g(N ) is the gain rate, and l(N ) the loss rate, and L is the linear turning operator used earlier. For later use we introduce the kinetic function: f (u) := g(u)u − l(u)u. Again we study the parabolic scaling τ = ε2 t, ξ = εx, and we assume that the reproduction and death occur on the macroscopic time scale, i.e. f (u) → ε2 f (u) = ε2 (g(u)u − l(u)u). With this scaling the above system (30) becomes ε2 pτ + εv · ∇ξ p = Lp − γ;2 (x)p + γω1 r − ε2 l(N )p ε2 rτ = γ2 (x) V p(., ., v)dv − γ1 r + ε2 g(N )r − ε2 l(N )r,

(31)

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We study expansions in ε: p(τ, ξ, v) = p0 + εp1 + ε2 p2 + . . . , N (τ, ξ) = N0 + εN1 + ε2 N2 + . . . r(τ, ξ) = r0 + εr1 + ε2 r2 + . . . If we compare the leading order terms in ε we obtain 0 = Lp;0 − γ2 p0 + γω1 r0 ε0 : 0 = γ2 p0 dv − γ1 r0

The second equation is solved to give r0 =

γ2 γ1

p0 dv,

which we substitute into the first equation: γ2 0 = Lp0 − γ2 p0 + p0 dv =: Lγ2 p0 . ω The right hand side defines an effective turning operator: Lγ2 (ξ)[ψ(v)] = −(µ + γ2 (ξ))ψ(v) µ γ2 (ξ) ′ T (v, v ) + +(µ + γ2 (ξ)) ψ(v ′ ) dv ′ µ + γ2 (ξ) (µ + γ2 (ξ))ω V with effective turning kernel Tγ2 (ξ, v, v ′ ) :=

γ2 (ξ) µ T (v, v ′ ) + . µ + γ2 (ξ) (µ + γ2 (ξ))ω

In [Hil03b] it is shown that for each ξ ∈ IRn this effective kernel Tγ2 (ξ) satisfies the above conditions (T1)-(T4). Hence the spectral theorem (Theorem 1) applies and we can define a pseudo-inverse -−1 , . Fγ2 (ξ) := Lγ2 (ξ)|1⊥

If we also study order ε and ε2 terms, then we arrive at the following parabolic limit for the leading order approximation of the total population, N0 (see details in [Hil03b]): N0 ∇γ2 (ξ) N0,τ = ∇ Dγ2 ,γ1 (ξ)∇ N0 − Dγ2 ,γ1 (ξ) γ1 + γ2 (ξ) γ2 (ξ) g(N0 )N0 − l0 (N0 )N0 (32) + γ2 (ξ) + γ1 with diffusion tensor

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Dγ2 ,γ1 (ξ) := −

γ1 ω(γ2 (ξ) + γ1 )

vFγ2 (ξ)v dv.

(33)

Note that the spatially dependent stop-rate γ2 (ξ) gives rise to a taxis term towards better habitats (regions where the individuals stop more often). A special case of this model has been studied by Cosner and Lou [CL03] to understand if it is always a good strategy for a population to move in direction of a better habitat. In (32) the birth rate is reduced by the mean fraction at rest, while the diffusion coefficient (33) is scaled by the mean proportion moving. If only a small proportion of the population is in resting phase (i.e. γ2 → 0) then individuals do not stop and hence cannot reproduce. Then the limit equation reduces to N0,τ = ∇D∇N0 − l(N0 )N0 .

Example: To give a specific example we study V = sS n−1 and T = 1/ω. Then Tγ2 = 1/ω and the pseudo-inverse Fγ2 is a multiplication operator by −(µ + γ2 )−1 . Then the diffusion tensor is isotropic Dγ2 ,γ1 =

γ1 s2 I. (µ + γ2 )n γ2 + γ1 2

s For γ2 = 0 the diffusion constant reduces to d0 = µn . Hence increasing γ2 reduces the motility. The scaling analysis for the transport model with resting phase (30) can also be applied to the parabolic model with resting phases

ut = d∆u − γ2 (x)u + γ1 r − l(N )u

rt = γ2 (x)u + g(N )r − γ1 r − l(N )r.

Here we assume that the movement is described by a diffusion process. In this situation the total population is N (t, x) = u(t, x) + r(t, x), which to leading order satisfies the limit equation γ1 d γ1 d N0,τ = ∇ ∇N0 − ∇γ N 2 0 γ2 + γ1 (γ2 + γ1 )2 γ2 + g(N0 )N0 − l(N0 )N0 , γ2 + γ1 which is the isotropic version of (32).

4 Moment Closure The theory of Extended Thermodynamics [MR98] provides a systematic approach to moment closure for the Boltzmann equation based on entropy maximization. In the applications to biology considered here there is no functional

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that would naturally assume the role of an entropy while the classical physical entropy does not satisfy an H-Theorem. In [Hil04] the negative L2 (V )-norm is used as an entropy. It satisfies an H-Theorem (Theorem 4 below), and it can be used to derive the moment closure for transport equations in applications to biology. We close the moment system by maximizing the negative L2 (V )-norm under the constraint of fixed first n moments. This minimization approach flattens oscillations, high frequencies in space and time will be smoothed out and the global structure of the solution is emphasized. We illustrate this procedure here to close the system for the first two moments (total population density and population flux). The closed system is a Cattaneo system, which is well known in heat transport theory. In [Hil03a] we generalized this approach to close the moment system at any order. We summarize some results in subsection 4.2. We apply this method to the transport equation for chemosensitive movement in subsection 4.1. We demonstrate the moment closure method for the example of a velocity jump process with fixed speed, but variable direction (Pearson walk [Pea05]). In this case V = sS n−1 with s > 0 and ω = |V | = sn−1 ω0 , where ω0 = |S n−1 |. The turn angle distribution is assumed to be constant T (v, v ′ ) = ω1 . As presented in [Hil03a], the method developed here can be generalized to more general kernels T and more general velocity sets V . The initial value problem for the linear transport equation reads 0 m −p , (34) pt + v · ∇p = µ ω p(0, x, v) = ϕ0 (x, v). (35) In [Hil04] we proved the following H-Theorem: Theorem 4. d p(t, x, .)22 + ∂j dt

v j p(t, x, .)2 dv

V

≤ 0.

(36)

The velocity-moments of p are defined by mα , where m0 is defined by (6) and the higher moments of p are denoted by i v i p(t, x, v) dv, i = 1, . . . , n (37) m (t, x) = V v i v j p(t, x, v) dv, i, j = 1, . . . , n. (38) mij (t, x) = V

Note that m0 is scalar, (mi ) is a vector and (mij ) is a 2-tensor. We stress the usual summation convention on repeated indices. To derive the equations for the first two moments m0 and mi we integrate (34) over V to obtain the conservation law m0t + ∂j mj = 0.

(39)

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Multiplication of (34) with v i and integration gives mit + ∂j mij = −µmi .

(40)

To close this system of two moment equations (39) and (40) we replace mij (p). We derive a function umin (t, x, v) which minimizes the L2 (V ) norm u(t, x, .)22 under the constraint that umin has the same first moments m0 and mi as p has. Once we have such a function umin we replace mij (p) by mij (umin ). We introduce Lagrangian multipliers Λ0 ∈ IR and Λi ∈ IR for i = 1, . . . , n and minimize 1 2 i i 0 H(u) := u dv − Λ0 v udv − m . udv − m − Λi 2 V V V We obtain an explicit representation of the minimizer (see [Hil04] for details) umin (t, x, v) =

' 1& 0 n m (t, x) + 2 (vi mi (t, x)) . ω s

(41)

It turns out that umin is the projection of p onto the linear subspace 1, v 1 , . . . , v n ⊂ L2 (V ). If we minimize the functional 1 i i 0 2 Ha (u) := v udv − m udv − m − Λi (u − a) dv − Λ0 2 V V V for some arbitrary a ∈ IR with the same constraints as above we arrive at the same minimizer (41). For fixed a ∈ IR the norm u(t, x, .) − a2 is a measure of the oscillation around the level a. Hence, umin minimizes oscillations. The extremum umin is indeed a minimum, since the second variation of H is δ 2 H(u) = 1 > 0. To derive the moment closure we consider the second moment of the minimizer umin : s2 mij (umin ) = m0 I. (42) n Now we close the system of the first two moments (39), (40) by assuming that mij (u) ≈ mij (p). Then, replacing mij in (40) together with (39) gives a linear Cattaneo system Mt0 + ∂j M j = 0, (43) 2 Mti + sn ∂i M 0 = −µM i , with initial conditions M 0 (0, .) = m0 (0, .),

M i (0, .) = mi (0, .).

(44)

We introduce capital letters to distinguish between the moments (m0 , mi ) of p and the solutions (M 0 , M i ) of the Cattaneo system (43). Of course, if

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mij (u) = mij (p) then (M 0 , M i ) = (m0 , mi ). The error which occurs in this approximation can be controlled. For that purpose we define r := m0 − M 0 and an energy es (r, q) :=

1 2

and q i := mi − M i ,

r2 +

IRn

n i q qi dx. s2

(45)

In [Hil04] the following error estimate has been shown. Theorem 5. es (r(t, .), q(t, .)) ≤ nb2n

s2 ∇x m0 2L2 ([0,t]×IRn ) , 2µ

(46)

with an appropriate constant bn > 0. Cattaneo [Cat48] used systems of the form (43) to model heat transport with finite speed. Then M 0 is the temperature and (M 1 , . . . , M n ) the heat flux and the ratio d = s2 /µn is the effective diffusion constant (see Joseph, Preziosi [JP88] or Gurtin, Pipkin [GP68] for the physical interpretations and [Had96], or [Hil98] for a biological interpretation). The derivation of the Cattaneo model from a moment closure approach gives a new understanding of the role of the Cattaneo system in biological applications. The relevant parameters are related to the individual movement behavior of the underlying species. 4.1 A Chemotaxis Model with Density Control Hillen and Painter [HP01] have studied a diffusion based model for chemosensitive movement where at high population densities the chemotaxis is turned off and pure diffusion dominates. This model can be constructed (from a transport equation) via a Cattaneo approximation. Solutions exist globally and no blow-up occurs. As an example consider a turning rate of the form & ' n µ(S, δv S) := µ0 1 − 2 β(m0 )χ(S)δv S , s

where β(m0 ) is a density dependent sensitivity. The function β is assumed to have a zero at some m ¯ 0 > 0 and β(m) > 0 for 0 < m < m ¯ 0 . With turning ′ −1 kernel T (v, v ) := ω µ(S, δv′ S) the moment closure procedure leads to a Cattaneo model for chemosensitive movement with density control Mt0 + ∂j M j = 0 Mti +

s2 0 n ∂i M

, = −µ0 1 −

n 0 s2 β(M )χ(S)St

-

M i + β(M 0 )χ(S) M 0 ∂i S.

(47)

This model has been used in [DH03] to describe pattern formation in slime molds and in bacteria. Moreover, a numerical scheme has been developed to solve (47).

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4.2 Higher Order Moment Closure The higher order moment closure requires rigorous bookkeeping of all the relevant tensor indices which cannot be presented here. We refer to [Hil03a] for details. The H-Theorem (Theorem 4) of the previous section can be generalized to turning kernels T which satisfy the general assumptions (T1)-(T4) defined above. The higher order moment closure can be derived in the framework of Lagrangian multipliers. It turns out that the steady states of the two moment closure (Cattaneo system) and of the three moment closure are determined by an elliptic equation (i.e., steady states of a corresponding diffusion problem). We conjecture that this is the case for all higher order moment closures.

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Travelling Waves in Systems of Hyperbolic Balance Laws J¨ org H¨ arterich and Stefan Liebscher Free University Berlin, Institute of Mathematics I, Arnimallee 2-6, D - 14195 Berlin, Germany haerter|[email protected] Summary. We discuss several results on the existence of continuous travelling wave solutions in systems of conservation laws with nonlinear source terms. In the ﬁrst part we show how waves with oscillatory tails can emerge from the combination of a strictly hyperbolic system of conservation laws and a source term possessing a stable line of equilibria. Two-dimensional manifolds of equilibria can lead to Takens-Bogdanov bifurcations without parameters. In this case there exist several families of small heteroclinic waves connecting diﬀerent parts of the equilibrium manifold. The second part is concerned with large heteroclinic waves for which the wave speed is characteristic at some point of the proﬁle. This situation has been observed numerically for shock proﬁles in extended thermodynamics. We discuss the desingularization of the resulting quasilinear implicit diﬀerential-algebraic equations and possible bifurcations. The results are illustrated using the p-system with source and the 14-moment system of extended thermodynamics. Our viewpoint is from dynamical systems and bifurcation theory. Local normal forms at singularities are used and the dynamics is described with the help of blowup transformations and invariant manifolds.

1 Introduction The influence of source terms on the structure of solutions to hyperbolic conservation laws recently has attracted much attention. While the only travelling-wave solutions of hyperbolic conservation laws are single shock waves, systems of balance laws may possess a variety of different continuous and discontinuous travelling waves. In this paper we concentrate on continuous travelling waves and study two different types of interplay between the flux of the conservation law and the dynamics due to the source term. In both situations we encounter the presence of manifolds consisting of equilibria. Although this looks rather special and non-generic, it will turn out that manifolds of equilibria occur naturally in the travelling-wave problem of conservation laws with source terms. In section 2, we focus on a combination

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of conservation laws and balance laws which gives rise to subspaces of equilibria. Here the existence of a large number of small heteroclinic waves can be proved. Section 3 deals with the bifurcation of large heteroclinic waves. Here the manifold of equilibria is obtained from a rescaling of the travelling-wave system. Our viewpoint is from dynamical systems and bifurcation theory. Local normal forms at singularities are used and the dynamics is described with the help of blow-up transformations and invariant manifolds.

2 Oscillatory profiles of stiff balance laws This section is devoted to a phenomenon in hyperbolic balance laws, first described by Fiedler and Liebscher [FL00], which is similar in spirit to the Turing instability. The combination of two individually stabilizing effects can lead to quite rich dynamical behaviour, like instabilities, oscillations, or pattern formation. Our problem is composed of two ingredients. First, we have a strictly hyperbolic conservation law. The second part is a source term which, alone, would describe a simple, stable kinetic behaviour: all trajectories eventually converge monotonically to some equilibrium. The balance law, constructed of these two parts, however, can support profiles with oscillatory tails. They emerge from singularities in the associated travelling-wave system. 2.1 Travelling waves We are interested in profiles of balance laws of the form ut + f (u)x =

1 g(u), ε

with x ∈ R and u ∈ RN . Travelling-wave solutions x − st u(t, x) = u , lim u(ξ) = u± ∈ RN ξ→±∞ ε

(1)

(2)

are heteroclinic orbits of the dynamical system (Df (u) − s · id) u′ = g(u).

(3)

In particular, the asymptotic states have to be zeros of the source term: g(u± ) = 0.

(4)

Choose a fixed wave speed s. As long as the speed of the wave does not coincide with one of the characteristic speeds, i.e. s ∈ spec Df (u), the travelling-wave equation (3) yields a system of ordinary differential equations: −1

u′ = (Df (u) − s · id)

g(u).

(5)

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2.2 Manifold of equilibria Usually one expects the zeros of a generic function g to form a set of isolated points. However, typical systems of balance laws are combinations of pure conservation laws and balance laws. For example, conservation of mass and momentum are often assumed to hold strictly. Let us consider a system of K pure conservation laws and N − K balance laws. Then, generic source terms g give rise to K-dimensional zero-sets. Near points of maximal rank of Dg, the zeros of g and the equilibria of (3) form a K-dimensional manifold. In the following sections, we shall investigate the dynamics near such a manifold of equilibria and describe the resulting structure of travelling waves of the system of balance laws (1). The asymptotic behaviour of profiles u(ξ) of (3) for ξ → ±∞ depends on the linearization −1 (6) L = (Df (u) − s · id) Dg(u). of the vectorfield (5). Of particular interest are fixed points where the stability of L changes. At these points the linearization L has non-maximal rank less than N − K and normal hyperbolicity of the equilibrium manifold breaks down. Depending on the type of the singularity, a very rich set of local heteroclinic connections can emerge and leads to small-amplitude travelling waves of (1). We call this phenomenon “bifurcation without parameters”, because it does not depend on the variation of some additional parameter. 2.3 Hopf point Let us start with the case K = 1 of a one-dimensional curve of equilibria. Bifurcations without parameters along lines of equilibria have been studied in [FLA00, FL00, Lie00]. Typical singularities are of codimension one. They are characterized by a simple eigenvalue of (6) crossing zero (simple-zero point) or by a pair of conjugate complex eigenvalues crossing the imaginary axis (Hopf point). The more interesting Hopf point is described as follows. Theorem 1. [FLA00] Let F : RN → RN be a C 5 -vectorfield with a line of fixed points along the u1 -axis, F (u1 , 0, . . . , 0) ≡ 0. At u1 = 0, we assume the Jacobi matrix DF (u1 , 0, . . . , 0) to be hyperbolic, except for a trivial kernel vector along the u1 -axis and a complex conjugate pair of simple, purely imaginary, nonzero eigenvalues µ(u1 ), µ(u1 ) crossing the imaginary axis transversely as u1 increases through u1 = 0: µ(0) = iω(0), Re µ′ (0) = 0.

ω(0) > 0,

(7)

Let Z be the two-dimensional real eigenspace of F ′ (0) associated to ±iω(0). By ∆Z we denote the Laplacian with respect to variations of u in the eigenspace Z. Coordinates in Z are chosen as coefficients of the real and imaginary parts

284

J. H¨ arterich, S. Liebscher z1

z2

u1

b) Fig. 1. Dynamics near a Hopf point along a line of equilibria: a) hyperbolic, η = +1, b) elliptic, η = −1.

of the complex eigenvector associated to iω(0). Note that the linearization acts as a rotation with respect to these not necessarily orthogonal coordinates. Let P0 be the one-dimensional eigenprojection onto the trivial kernel along the u1 -axis. Our final nondegeneracy assumption then reads ∆Z P0 F (0) = 0.

(8)

Fixing orientation along the positive u0 -axis, we can consider ∆Z P0 F (0) as a real number. Depending on the sign η := sign (Re µ′ (0)) · sign (∆Z P0 F (0)),

(9)

we call the Hopf point u = 0 elliptic if η = −1 and hyperbolic for η = +1. Then the following holds true in a neighbourhood U of u = 0 within a three-dimensional centre manifold to u = 0. In the hyperbolic case, η = +1, all non-equilibrium trajectories leave the neighbourhood U in positive or negative time direction (possibly both). The stable and unstable sets of u = 0, respectively, form cones around the positive/negative u1 -axis, with asymptotically elliptic cross section near their tips at u = 0. These cones separate regions with different convergence behaviour. See Fig. 1(a). In the elliptic case all non-equilibrium trajectories starting in U are heteroclinic between equilibria u± = (u± 1 , 0, . . . , 0) on opposite sides of the Hopf point u = 0. If F (u) is real analytic near u = 0, then the two-dimensional strong stable and strong unstable manifolds of u± within the centre manifold intersect at an angle which possesses an exponentially small upper bound in terms of |u± |. See Fig. 1(b). Note that the heteroclinic connections which fill an entire neighbourhood in the centre manifold of an elliptic Hopf point then lead to travelling waves of

Travelling Waves in Systems of Hyperbolic Balance Laws

285

u

z1

u−

x u

+

u+

u− u z2

u1

u−

x u+ Heteroclinic orbit near the Hopf point.

Proﬁle for two values of ε in (1).

Fig. 2. Oscillatory travelling wave emerging from an elliptic Hopf point.

the balance law (1). In Fig. 2, such a wave is shown, and a generic projection of the n-dimensional space of u-values onto the real line was used. For stiff source terms, ε ց 0, the oscillations imposed by the purely imaginary eigenvalues now look like a Gibbs phenomenon. But here, they are an intrinsic property of the analytically derived solution. In [FL00, Lie00] simple examples of Hopf points in systems of viscous balance laws have been provided. The following result goes beyond these examples and emphasizes the possibility of Hopf points in systems with arbitrary flux functions when combined with a stabilizing source term. Theorem 2. Let f : R3 → R3 be a generic C 6 -vectorfield such that Df (u) has only real distinct eigenvalues λ1 (u) < λ2 (u) < λ3 (u) for all u in a neighbourhood of the origin u = 0. Then, for every value s ∈ {λ1 (0), λ2 (0), λ3 (0)} there exists a C 5 -vectorfield g : R3 → R2 × {0}

(10)

such that 1. the kinetic part g stabilizes the line of equilibria near the origin, i.e. the linearization Dg(0) has one (trivial) zero eigenvalue and two negative real eigenvalues, 2. the travelling-wave equation (5) admits a Hopf point in the sense of Theorem 1. Proof. Without loss of generality, we choose s = 0 and require the eigenvalues of Df (0) to be nonzero. We shall provide a particular source g with a straight line of equilibria.

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First, we construct a suitable linearization Dg(0) that creates the purely imaginary eigenvalues of Df (0)−1 Dg(0). Secondly, we continue this linearization along the line of equilibria such that the transversality (7) holds. Finally, we use genericity to satisfy the nondegeneracy condition (8). The main problem of the construction is the constraint (10) imposed by the structure of one conservation law and two balance laws. Let S be the transformation of Df (0) into diagonal form: Df (0) = S Λ S −1 ,

Λ = diag(λ1 , λ2 , λ3 ).

(11)

A Hopf point of system (5) at the origin requires the existence of a transformation T ∈ GL(3), such that ⎞ ⎛ 0 0 0 Dg(0) = S Λ S −1 T −1 ⎝ 0 0 1 ⎠ T. (12) 0 −1 0

Here we have normalized the imaginary part of the Hopf eigenvalue to one. On this linear level, the constraint (10) yields 0 = eT 3 Dg(0) which is equivalent to , (13) Λ S T e3 ⊥ S −1 T −1 {0} × R2 , where e3 = (0, 0, 1)T denotes the third standard unit vector. Aside from (13) we can define T arbitrarily in order to construct the two negative eigenvalues of Dg(0) defined by (12). This is done as follows. We start with two arbitrary, linearly independent vectors (a1 , a3 ), (a2 , a4 ) ∈ R2 . (The actual choice will be made later on.) As a first genericity condition of f we require eT k = 1, 2, 3. (14) 3 Sek = 0,

Then we can obtain a basis of R3 by: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ∗ ∗ 1 v1 = ⎝ 0 ⎠ , v2 = ⎝ a1 ⎠ , v3 = ⎝ a2 ⎠ , a4 a3 0

v2 , v3 ∈ (ΛS T e3 )⊥ .

(15)

We define T by the equation

(T S)−1 =

&

v1 v2 v 3

and insert it into (12) to obtain

'

⎛

⎞ 1 ∗ ∗ = ⎝ 0 a1 a2 ⎠ . 0 a3 a4

(16)

Travelling Waves in Systems of Hyperbolic Balance Laws

T Dg(0) T −1

287

⎛

⎞ 0 0 0 = T S Λ S −1 T −1 ⎝ 0 0 1 ⎠ 0 −1 0 ⎞ ⎞⎛ ⎞⎛ ⎛ 0 0 0 λ1 1 ∗ ∗ 1 ⎠ ⎝ 0 −a2 a1 ⎠ ⎝ 0 a4 −a2 ⎠ ⎝ λ2 = a1 a4 − a2 a3 0 −a4 a3 λ3 0 −a3 a1 ⎞ ⎛ 0 ∗ ∗ 1 ⎝0 (λ3 − λ2 )a2 a4 λ2 a1 a4 − λ3 a2 a3 ⎠ = a1 a4 − a2 a3 0 λ2 a2 a3 − λ3 a1 a4 (λ3 − λ2 )a1 a3

(17)

The lower right (2 × 2)-block has trace (λ3 − λ2 )(a1 a3 + a2 a4 )/(a1 a4 − a2 a3 ) and determinant λ2 λ3 . The trace can be made negative of arbitrary size regardless of λ2 , λ3 by choice of a1 , ..., a4 . We conclude: if λ2 λ3 > 0 then we can find parameters a1 , ..., a4 in (15) such that the resulting matrix Dg(0) has two negative real eigenvalues. In fact, there is an open region of admissible parameters. The requirement λ2 λ3 > 0 can be fulfilled without loss of generality by a permutation of λ1 , λ2 , λ3 , since at least two of them must have the same sign. From now on, let v1 , v2 , v3 , T be fixed according to the above considerations. Then we continue ⎧ ⎛ ⎞⎫ ⎧ ⎛ ⎞⎫ 1 ⎬ 1 ⎬ ⎨ ⎨ ker Dg(0) = span S ⎝ 0 ⎠ = span T −1 ⎝ 0 ⎠ (18) ⎩ ⎩ ⎭ ⎭ 0 0

to a straight line of equilibria by the definition ⎛ ⎞ ⎞⎛ ⎞ ⎛ w1 0 0 0 & ' w1 g ◦ T −1 ⎝ w2 ⎠ = S Λ S −1 T −1 ⎝ 0 cw1 1 ⎠ ⎝ w2 ⎠ w3 0 −1 cw1 w3

(19)

The transversality of the Hopf eigenvalue (7) can be achieved by an appropriate choice of the parameter c dependent on the higher order terms of f . Again, there is in fact an open region of admissible parameter values. The required nondegeneracy of the Hopf point (8) is the second nondegeneracy condition needed for f . That finishes the proof. Note that (10) yields # Df (0)−1 ∆Z g(u)#u=0 ∈ Z = span {Sv2 , Sv3 } (20)

and does not contribute to (8). Therefore, the inclusion of higher order terms in (19) would not enter the nondegeneracy condition. The value of P0 ∆Z Df (0)−1 g(0) is specified by first-order terms of g (that have been defined only using Df (0)) and second-order terms of f . Indeed, (8) is a genericity condition on f .

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Remark 3. The nondegeneracy condition (8) is equivalent to the requirement, that every flow-invariant foliation transverse to the line of equilibria breaks down at the Hopf point already to second order. In terms of our system of conservation laws and balance laws, it requires in particular that the flux couples the component with source terms back to the pure conservation law. Without such a coupling, the conservation law gives rise to a foliation, such that in each fibre only finitely many of the equilibria remain. That happens for instance in the systems of extended thermodynamics that are one of the motivating examples of the second part of this article, see section 3.5. 2.4 Takens-Bogdanov points Along two-dimensional surfaces of equilibria, we expect singularities of codimension two to occur. The possible cases are characterized by the critical eigenvalues of the linearization in directions transverse to the surface of equilibria: a geometrically simple and algebraically double eigenvalue zero (Takens-Bogdanov point), a pair of purely imaginary eigenvalues accompanied by a simple eigenvalue zero (Hopf-zero point), or two non-resonant pairs of purely imaginary eigenvalues (double Hopf point). Additionally, simplezero points and Hopf points with a degeneracy in the higher order terms are possible. Takens-Bogdanov points have been studied in [FL01]. Theorem 4. [FL01] Let F : R4 → R4 be a vectorfield with a plane of fixed points, F (0, 0, u3 , u4 ) ≡ 0. At u = 0, we assume the Jacobi matrix to be nilpotent. Then, for generic F , the vectorfield can be transformed to the normal form u˙ 1 u˙ 2 u˙ 3 u˙ 4

= au1 (−u3 + u4 ) − u2 u3 + abu22 , = u1 , = u2 + u1 (c1 u3 + c2 u4 ), = c3 u1 (c1 u3 + c2 u4 ),

(21)

written up to second order terms. Depending on the value of b three qualitatively different cases occur. The structure of the set of heteroclinic connections between different equilibria near the Takens-Bogdanov point is depicted in Fig. 3. In each case, the TakensBogdanov point is the intersection of the line {u3 = 0} of simple-zero points and the line {u3 = u4 > 0} of Hopf points of either hyperbolic or elliptic type. Similar to Theorem 2, Takens-Bogdanov points can occur in systems with at least two conservation laws and two balance laws. Note that travelling waves corresponding to heteroclinic orbits starting or ending near the Hopf line have oscillatory tails.

Travelling Waves in Systems of Hyperbolic Balance Laws u4

289

hyp. Hopf

(2) (1)

(0)

(1)

(0) (1)

u3

(0)

(A) u4 hyp. Hopf

(1)

(2) (0)

(1)

u3 (1)

(0) (B) u4 ell. Hopf

(1)

(2) (0)

(1)

u3 (1)

(0) (C)

Fig. 3. Three cases of Takens-Bogdanov bifurcations without parameters, see (21). (A) b < −17/12; (B) −17/12 < b < −1; (C) −1 < b. Unstable dimensions i of trivial equilibria (0, y) are denoted by (i). Arrows indicate heteroclinic connections between diﬀerent regions of the manifold of equilibria.

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2.5 Discussion In summary, Hopf point as well as Takens-Bogdanov points are possible in systems of stiff hyperbolic balance laws. For all generic strictly hyperbolic flux functions and a suitable number of pure conservation laws and balance laws there exist appropriate source terms such that these bifurcations occur in a structurally stable fashion. The bifurcations are generated by the interaction of flux and source. In particular, Hopf points can be constructed for generic fluxes and stabilizing sources. For Takens-Bogdanov points at least one example is given in [FL01]. This holds true under small perturbations of the system, for instance in numerical calculations. In particular, an additional viscous regularisation ut + f (u)x = g(u) + δuxx

(22)

still yields the bifurcation scenario for small positive δ. In [FL00, Lie00, FL01] viscous oscillatory profiles are constructed for specific examples of f, g. The treatment of the viscous terms is still applicable in the general case presented here. In particular, the proof of convective stability of the oscillatory profiles near an elliptic Hopf point in [Lie00] is applicable for systems given by Theorem 2 with additional viscosity. For numerical calculations on bounded intervals in co-moving coordinates this implies nonlinear stability of the corresponding oscillatory travelling waves. For hyperbolic conservation laws one usually expects viscous shock profiles to be monotone. In particular, in numerical simulations small oscillations near the shock layer are regarded as numerical artefacts due to grid phenomena or unstable numerical schemes. In many schemes “artificial viscosity” is used to automatically suppress such oscillations as “spurious”. Near elliptic Hopf points as well as near the elliptic Hopf line of Takens-Bogdanov points, in contrast, all heteroclinic orbits correspond to travelling waves with necessarily oscillatory tails. Numerical schemes should therefore resolve this “overshoot” rather than suppress it.

3 Bifurcation of heteroclinic waves In this section, we study heteroclinic travelling waves of (1) with ε = 1, i.e. x ∈ R, u ∈ RN .

ut + f (u)x = g(u),

(23)

Travelling-wave solutions u(t, x) = u (x − st) ,

lim u(ξ) = u± ∈ RN ,

ξ→±∞

are again orbits of the dynamical system

(24)

Travelling Waves in Systems of Hyperbolic Balance Laws

A(u, s)u′ := (Df (u) − s · id) u′ = g(u).

291

(25)

As before we concentrate on heteroclinic waves connecting two equilibria of the reaction dynamics. In contrast to section 2 we will now consider the wave speed s as a bifurcation parameter and study the bifurcation of heteroclinic waves. As long as s does not coincide with one of the characteristic speeds for all u on the heteroclinic orbit, (25) is equivalent to the explicit ordinary differential equation (5). However, in general there are also orbits containing points where s coincides with one of the characteristic speeds such that det(Df (u)−s·id) vanishes at some point on the heteroclinic profile. 3.1 Quasilinear implicit DAEs In general, the travelling-wave equation (25) is a differential-algebraic equation. In contrast to the common setting in differential-algebraic equations the rank of the matrix A(u, s) is non-maximal only on a codimension-one surface Σs := {u ∈ RN ; det A(u, s) = 0}

(26)

of the phase space. One might suspect that solutions can never cross this surface. Rabier and Rheinboldt [RR94] have studied solutions in a neighbourhood of Σs and shown that they typically reach Σs in finite (forward or backward) time and cannot be continued. For this reason Σs is often referred to as the impasse surface. However, there may exist parts of Σs where it is possible to cross from one side to the other. Heteroclinic solutions passing through Σs have been found in applications [MNP00, Wei95] and, as will be shown below, their behaviour differs from that of heteroclinic orbits in ordinary differential equations. To identify points on Σs where crossing is possible, one needs to desingularize the vector field near the hypersurface Σs . To this end one uses the adjugate matrix adj A(u, s) which is defined as the transpose of the matrix of cofactors of A(u, s) and which satisfies the identity (adj A(u, s))A(u, s) = A(u, s)(adj A(u, s)) = det A(u, s) · id.

(27)

Solution curves of (25) coincide outside Σs with trajectories of the desingularized system u′ = adj A(u, s)g(u) = adj (Df (u) − s · id) g(u).

(28)

This system is obtained by multiplying (25) from the left with adj A(u, s) and rescaling time by using det A(u, s) as an Euler multiplier. Also, the direction of the orbits is reversed in that part of the phase space where det A(u, s) < 0. In addition to the equilibria of (25), equation (28) may possess additional fixed points on Σs . Since they are not equilibria of the original system they

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are called pseudo-equilibria. As we will see, they play an important role in the bifurcations. The time rescaling is singular at the impasse surface Σs , so trajectories of (28) that need an infinite time to reach a pseudo-equilibrium correspond to solutions of the original system (25) which reach the pseudoequilibrium in finite time. A solution of (25) may therefore consist of a concatenation of several orbits of (28). The dynamics near the impasse surface is strongly affected by the interaction between “true” equilibria and pseudo-equilibria, when equilibria cross the impasse surface as s is varied. In the context of differential-algebraic equations, such a passage of a non-degenerate equilibrium U0 through the impasse surface was first studied by Venkatasubramanian et al. in [VSZ95]. Their Singularity-Induced Bifurcation Theorem states that under certain nondegeneracy conditions one eigenvalue of the linearization of (25) at U0 moves from the left complex half plane to the right complex half plane or vice versa by diverging through infinity, while all other eigenvalues remain bounded and stay away from the origin. 3.2 Scalar balance laws Let us very briefly consider the simplest situation of a scalar balance law to describe some of the features that show up in larger systems, too. Let f : R → R be a convex flux function with f ′ (0) = 0 and g : R → R be a nonlinear source term with three simple zeroes uℓ < um < ur and the sign condition g(u) · u < 0 outside [uℓ , ur ]. Looking for travelling waves of (25) with speed s then leads to the scalar equation (∂u f (u) − s)u′ = g(u).

(29)

It is easy to check that the “impasse surface” consists here of a single point us where ∂u f (us ) = s. No trajectory can pass through this impasse point except when us = um , i.e. s = ∂u f (um ). For this exceptional wave speed there is a heteroclinic orbit from uℓ to ur which consists of the concatenation of two heteroclinic orbits of the desingularized system u′ = g(u).

(30)

Note that the flow has to be reversed for u > um such that the two heteroclinic orbits of (30) from uℓ to um and from ur to um can indeed be combined to yield a single heteroclinic orbit of (29). 3.3 The p-system with source While in scalar balance laws heteroclinic waves crossing Σs occur only for isolated values of s, already in (2×2)-systems of balance laws such heteroclinic waves may occur for an open set of wave speeds.

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Instead of studying general (2 × 2)-systems with arbitrary source terms we are going to illustrate our results for this case using the well-known psystem. This does not change the results in an essential way, however, it has the advantage that the impasse surface Σs is a straight line u = const. Consider therefore the system ut + vx = g1 (u, v) vt + p(u)x = g2 (u, v).

(31)

We assume that p′ (u) > 0 such that the conservation-law part is strictly hyperbolic. Moreover we require that there exists a non-degenerate equilibrium, i.e. a point (u0 , v0 ) with g1 (u0 , v0 ) = g2 (u0 , v0 ) = 0 and det Dg(u0 , v0 ) = 0. The travelling-wave equation corresponding to this balance law is ′ g1 (u, v) u −s 1 (32) = v′ g2 (u, v) p′ (u) −s such that for fixed s the impasse set Σs is either empty or consists of the line Σs := {(u, v); p′ (u) = s2 }. While orbits which do not cross this line can be treated by standard methods, some care is needed for orbits which reach the line Σs . The Singularity-Induced Bifurcation Theorem tells that the stability type of the% equilibrium (u0 , v0 ) changes when it crosses the impasse surface at s = s0 = p′ (u0 ). To describe more precisely what happens at this bifurcation, we perform the desingularization via the adjugate matrix. This leads to the desingularized system ′ −sg1 (u, v) − g2 (u, v) u (33) = v′ −p′ (u)g1 (u, v) − sg2 (u, v) The implicit-function theorem can now be applied to this equation restricted to Σs to find for |s − s0 | small a branch of pseudo-equilibria (˜ u(s), v˜(s)) with u ˜(s0 ) = 0 and v˜(s0 ) = v0 if s0 ∂v g1 (u0 , v0 ) + ∂v g2 (u0 , v0 ) = 0.

(34)

Assuming that this condition holds, one can describe the dynamics close to (u0 , v0 ) for |s − s0 | sufficiently small by using classical bifurcation theory for system (33) and translating the results back to the original system (32). Lemma 5. Consider the p-system with a source term which possesses a nondegenerate equilibrium at (u0 , v0 ) for all wave speeds s. Then the desingularized travelling-wave system (33) undergoes a transcritical bifurcation at s = s0 . The trivial branch of equilibria crosses a branch (˜ u(s), v˜(s)) of equilibria which are pseudo-equilibria of system (32). For |s − s0 | sufficiently small the pseudoequilibrium (˜ u(s), v˜(s)) and the equilibrium (u0 , v0 ) are connected by a heteroclinic orbit.

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J. H¨ arterich, S. Liebscher Σs

Σs

Σs

(u0 , v0 )

s < s0

s = s0

s > s0

Fig. 4. A Singularity Induced Bifurcation occurs when a non-degenerate equilibrium crosses the impasse surface (dotted line). The pseudo-equilibrium involved in the transcritical bifurcation of the desingularized system is drawn in grey. For s = s0 there exist orbits which pass through Σs .

There are different cases depending on the eigenvalue structure at the equilibria. One of them is depicted in Fig. 4. Remark 6. Recall that system (33) and system (32) are related via a rescaling of time with the factor det A(u, v, s) which is singular at the impasse surface Σs . For this reason the trajectory of (32) corresponding to the heteroclinic u(s), v˜(s)) needs only a finite time to reach the orbit between (u0 , v0 ) and (˜ pseudo-equilibrium (˜ u(s), v˜(s)). 3.4 Heteroclinic waves in the p-system Since we are basically interested in heteroclinic travelling waves, we will now assume that there exists some heteroclinic orbit of (32) asymptotic to the equilibrium (u0 , v0 ) at s = s0 . We restrict our attention to heteroclinic orbits which connect some equilibrium (u− , v− ) to (u0 , v0 ) and which are structurally stable. There are three cases which may occur: • Case I: (u− , v− ) is of source type while (u+ , v+ ) is a saddle equilibrium of (25) • Case II: (u− , v− ) is a saddle while (u+ , v+ ) is a sink. • Case III: (u− , v− ) is of source type while (u+ , v+ ) is a sink. In the first two cases we may think of the heteroclinic orbit for instance as coming from a saddle-node bifurcation. As the parameter s is varied across s0 the stationary point (u0 , v0 ) moves through Σs . The following lemma tells what happens to the heteroclinic connection for s > s0 when the two equilibria lie on different sides of Σs .

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Theorem 7. Assume that (32) possesses two stationary points (u− , v− ) and (u0 , v0 ) which are on the same side of Σs for s < s0 . Assume furthermore that there is a heteroclinic connection from (u− , v− ) to (u0 , v0 ) at s = s0 and that the tangent vector to this heteroclinic orbit at (u0 , v0 ) is transverse to Σs0 . Then for s − s0 > 0 sufficiently small the following holds: (i) In case I the desingularized system (33) possesses a unique heteroclinic orbit from the pseudo-equilibrium (˜ u(s), v˜(s)) to the saddle (u− , v− ) and a unique heteroclinic orbit from (˜ u(s), v˜(s)) to the saddle (u0 , v0 ). The concatenation of these two orbits provides a heteroclinic orbit from (u− , v− ) to (u0 , v0 ) in the original system (25). See Fig. 5(a). (ii) In case II the pseudo-equilibrium (˜ u(s), v˜(s)) is of saddle-type and possesses unique heteroclinic connections to (u− , v− ) and (u0 , v0 ). A heteroclinic orbit of the original system (25) is obtained by piecing these two orbits together. See Fig. 5(b). (iii) Case III is similar to case I with the difference that there exist infinitely many heteroclinic orbits from the pseudo-equilibrium sink (˜ u(s), v˜(s)) to the source (u− , v− ). This in turn yields infinitely many heteroclinic orbits of (25) from (u− , v− ) to (u0 , v0 ). Remark 8. When the pseudo-equilibrium is of saddle type (case II), the heteroclinic orbit between the source and a sink is as smooth as the vector field. In contrast, if the pseudo-equilibrium is of source/sink-type and the heteroclinic wave connects two saddle equilibria, then the heteroclinic orbit has in general only a finite degree of smoothness which depends on the ratio of the eigenvalues at the pseudo-equilibrium. An analogous result for general (2 × 2)-systems can be obtained using Lyapunov-Schmidt reduction. Under a certain non-degeneracy condition the passage of a non-degenerate equilibrium through the impasse surface corresponds to a transcritical bifurcation of the desingularized system. Close to the bifurcation point there exist solutions which pass through the impasse surface and converge to the equilibrium. It turns out that the situation is similar for N -dimensional systems (28) associated with (N × N )-systems of hyperbolic balance laws. Here, the impasse surface Σs is of codimension one and generically within Σs there is a codimension one set of pseudo-equilibria. The linearization of (28) in such a pseudo-equilibrium possesses 0 as an eigenvalue of multiplicity at least N − 2 corresponding to the (N − 2)–dimensional set of pseudo-equilibria. 3.5 Shock profiles in extended thermodynamics Extended thermodynamics comprises a class of systems of hyperbolic balance laws which describe for instance the thermodynamics of rarefied gases under the physical assumption that the propagation speed of heat flux and shear

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s < s0

s < s0

(u− , v− ) (u− , v− )

s = s0

s = s0

s > s0

s > s0

a)

b)

Fig. 5. Continuation of heteroclinic orbits in the p-system: a) Case I, b) Case II The impasse surface is depicted as a dotted line the pseudo-equilibrium is shown in grey, the heteroclinic connection from (u− , v− ) to (u0 , v0 ) is the bold curve.

stress is finite. We concentrate on one specific model, the 14-moment system, as described in [Wei95], [MR98]. It is one in a hierarchy of models based on the kinetic theory of gases. In particular, they are used to get a better resolution of the internal structure of shock waves in rarefied gases by taking into account more moments. For brevity, we do not write down the full system consisting (in one space dimension) of three conservation laws for mass, momentum, and energy and of three balance laws. Since it is invariant under Galilean transformations it suffices to look for stationary solutions instead of travelling waves with arbitrary speed. Integrating the three conservation laws allows to eliminate three variables and

Travelling Waves in Systems of Hyperbolic Balance Laws

297

to replace them by integration constants. Moreover, by scaling the variables suitably, it is possible to reduce the system to a DAE in the three variables v, p and ∆ with a single real parameter α which can be related to the Mach number. In quasilinear implicit form the travelling-wave equation then reads ⎞ ⎛ ′⎞ ⎛ −(1 − v − p) v ⎠ . (35) −(4α + 2v(1 − 6p − 2v))/3 A(v, p, ∆, α) ⎝ p′ ⎠ = ⎝ ′ 3 2 2 ∆ −(4v + 4v + 36pv − 16αv − 2∆)/3

where A(v, p, ∆, α) is a polynomial matrix function. We omit here most of the (lengthy) calculations and formulas and concentrate on the geometric situation. A more detailed treatment will be performed elsewhere. The impasse surface Σα = {(v, p, ∆); det A(v, p, ∆, α) = 0} is a graph over the v-p-plane. For any α < 25/32, there are precisely two equilibria √ √ 5 ∓ 25 − 32α 3 ± 25 − 32α E1,2 = , ,0 8 8

which bifurcate at α = 25/32 in a subcritical saddle-node bifurcation. The main object of interest are continuous heteroclinic orbits from E2 to E1 alias shock profiles. It is clear that for α close to the bifurcation value there exists a unique heteroclinic connection between E2 and E1 . It has been observed numerically by Weiss [Wei95] that in the 14-moment system this heteroclinic orbit can be continued to values of α where the shock profile has to cross the impasse surface Σα because E1 and E2 lie on different sides of Σα . However, in this parameter regime, the dimension of the unstable manifold of E1 is one while the stable manifold of E2 is two-dimensional. Without some additional structure one cannot explain that a heteroclinic connection between these two saddle-type equilibria persists for a whole range of α. In the following we propose a scenario how a one-dimensional manifold E of pseudo-equilibria can be responsible for a structurally stable heteroclinic connection between E1 and E2 in a way similar to case I in the p-system with source. Let α1 be the parameter value where E1 lies in Σα . Proposition 9. For α < α1 the one-dimensional stable manifold of E1 connects to some pseudo-equilibrium Epseudo (α) on E. The two-dimensional unstable manifold of E2 connects to a whole interval of points on E containing Epseudo (α). The concatenation of the two heteroclinic orbits of the desingularized system involving Epseudo (α) yields a heteroclinic orbit from E2 to E1 in the original system (35). The scenario is in accordance with numerical calculations performed for the 14-moment system, although we do not have an analytic proof that the heteroclinic orbit created in the saddle-node bifurcation at α = 25/32 can be continued down to α = α1 without intersecting the impasse surface Σα .

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However, assuming the existence of such a heteroclinic profile at α = α1 proposition 9 is able to explain why the heteroclinic shock profile persists for α < α1 . Let us remark that the bifurcation is connected to a change of stability along the line E of pseudo-equilibria, similar to the situation considered in section 2. 3.6 Viscous profiles In many situations systems of balance laws include a small viscous term: ut + f (u)x = εuxx + g(u),

x ∈ R, u ∈ RN .

(36)

The travelling-wave equation now becomes a singularly perturbed equation of the form εu′′ = (Df (u) − s · id)u′ − g(u) (37) where the prime denotes differentiation with respect to the comoving coordinate ξ := x − st. Note that, unlike in viscous conservation laws, the viscosity ε is still present in the travelling-wave equation. For scalar balance laws, the travelling-wave equation is a planar system with one fast and one slow variable involving the small parameter ε and the wave speed s as an additional parameter. Returning to the setting of section 3.2 where the flux was convex and the source term had three simple zeroes uℓ < um < ur one might ask whether (36) possesses a travelling wave close to the monotone solution of (23) that connects uℓ to ur . However, it turns out that such a solution necessarily has to pass close to a non-hyperbolic point on the slow manifold such that standard techniques in geometric singular perturbation theory can give no answer. For this reason, recent blow-up techniques [KS01] have to be used to establish the following existence result: Theorem 10. [H¨ ar03] Consider a scalar viscous balance law (36) with a convex flux f : R → R and a source term g : R → R which possess three simple zeroes uℓ < um < ur . Let s0 = f ′ (um ) be the velocity of the heteroclinic wave that connects uℓ to ur for ε = 0. Then for ε > 0 sufficiently small there is a unique velocity s(ε) such that a unique monotone heteroclinic wave uε of (37) connects uℓ to ur . To first order the wave speed s(ε) depends linearly on the viscosity ε: ′ # g (u) ## 1 d ε + O(ε3/2 ). s(ε) = s0 − 2 du f ′′ (u) #u=um

Since the heteroclinic travelling wave uε follows both stable and unstable parts of the slow manifold, it is a so-called canard trajectory. For larger systems the viscous travelling-wave equation (37) can be written as a fast-slow-system with N slow and N fast variables:

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εu′ = w + f (u) − su w′ = −g(u)

The N -dimensional slow manifold {(u, w) ∈ R2N ; w + f (u) − su = 0} is a graph over the subspace {w = 0} spanned by the variables of the hyperbolic balance laws. A short calculation shows that points on the slow manifold where normal hyperbolicity fails correspond exactly to the impasse surface Σs . This implies that the problem of finding heteroclinic travelling waves of the viscous system which are close to travelling waves of the hyperbolic system intersecting Σs will necessarily lead to a rather difficult singularly perturbed problem involving Canard solutions. An interesting and completely open question is the stability of such viscous travelling waves. In particular, as we have seen in the p-system with source, “ordinary” heteroclinic waves can become rather singular when one of the asymptotic states crosses the impasse surface Σs as s is varied. In the viscous setting this would correspond to a transition from an “ordinary” heteroclinic orbit to a canard orbit. It is not clear whether this transition affects the stability of heteroclinic waves.

References [FL00]

B. Fiedler and S. Liebscher. Generic Hopf bifurcation from lines of equilibria without parameters: II. Systems of viscous hyperbolic balance laws. SIAM Journal on Mathematical Analysis, 31(6):1396–1404, 2000. [FL01] B. Fiedler and S. Liebscher. Takens-Bogdanov bifurcations without parameters, and oscillatory shock proﬁles. In H. Broer, B. Krauskopf, and G. Vegter, editors, Global Analysis of Dynamical Systems, Festschrift dedicated to Floris Takens for his 60th birthday, pages 211–259. IOP, Bristol, 2001. [FLA00] B. Fiedler, S. Liebscher, and J. C. Alexander. Generic Hopf bifurcation from lines of equilibria without parameters: I. Theory. Journal of Differential Equations, 167:16–35, 2000. [H¨ ar03] J. H¨ arterich. Viscous Proﬁles of Traveling Waves in Scalar Balance Laws: The Canard Case. Methods and Applications of Analysis, 10:97–118, 2003. [KS01] M. Krupa and P. Szmolyan. Extending geometric singular perturbation theory to nonhyperbolic points – fold and canard points in two dimensions. SIAM Journal on Mathematical Analysis, 33:266–314, 2001. [Lie00] S. Liebscher. Stable, Oscillatory Viscous Profiles of Weak, non-Lax Shocks in Systems of Stiff Balance Laws. Dissertation, Freie Universit¨ at Berlin, 2000. [MNP00] B. P. Marchant, J. Norbury, and A. J. Perumpanani. Traveling shock waves in a model of malignant invasion. SIAM Journal on Applied Mathematics, 60:463–476, 2000. [MR98] I. M¨ uller and T. Ruggeri. Rational Extended Thermodynamics, volume 37 of Tracts in Natural Philosophy. Springer, 1998.

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P. Rabier and W. C. Rheinboldt. On impasse points of quasi-linear diﬀerential-algebraic equations. Journal of Mathematical Analysis and Applications, 181:429–454, 1994. [VSZ95] V. Venkatasubramanian, H. Sch¨ attler, and J. Zaborsky. Local bifurcations and feasibility regions in diﬀerential-algebraic systems. IEEE Transactions on Automatic Control, 40:1992–2013, 1995. [Wei95] W. Weiss. Continuous shock structure in extended thermodynamics. Physics Review E, 52:R5760–R5763, 1995.

The Role of the Jacobian in the Adaptive Discontinuous Galerkin Method for the Compressible Euler Equations Ralf Hartmann Institute of Aerodynamics and Flow Technology, German Aerospace Center (DLR), Lilienthalplatz 7, 38108 Braunschweig, Germany [email protected] Summary. We provide a full description of the Jacobian to the discontinuous Galerkin discretization of the compressible Euler equations, one of the key ingredients of the adaptive discontinuous Galerkin methods recently developed in [7, 8]. We demonstrate the use of this Jacobian within an implicit solver for the approximation of the (primal) stationary ﬂow problems as well as in the adjoint (dual) problems that occur in the context of a posteriori error estimation and adaptive mesh reﬁnement. In particular, we show that the (stationary) compressible Euler equations can eﬃciently be solved by the Newton method. Full quadratic Newton convergence is achieved on higher order elements as well as on locally reﬁned meshes.

1 Introduction In this paper we consider the adaptive discontinuous Galerkin method for the numerical approximation of the compressible Euler equations recently developed in [7] and [8] (for an overview of the development of the discontinuous Galerkin method we refer the reader to [2]). The purpose of the current paper is to provide a detailed description of the Jacobian, which is the key ingredient of the Newton iteration for solving the compressible Euler equations but also for the construction of adjoint (dual) problems arising in the context of duality-based a posteriori error estimation and adaptivity. In [7] only parts of the Jacobian are described in detail. In particular, there is no description given of the boundary terms and their Jacobian. Furthermore, the Jacobians of the numerical fluxes are approximated by simply neglecting some of the more complicated terms. In the present paper we now describe and assemble all terms and we demonstrate that the additional terms are actually necessary for obtaining an optimal convergence of the Newton iteration. Furthermore, we show that convergence of the Newton iteration is optimal also for higher order elements and on locally refined grids.

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Finally, we demonstrate that the duality-based a posteriori error representation, see [7, 8], gives sharp error estimation of specific target functionals, when evaluated by solving the so-called adjoint (dual) problem numerically. Again, the key ingredient is the Jacobian that governs the adjoint problem. The outline of this paper is as follows. In the following two subsections we motivate the importance of the Jacobian in the two fields of application already mentioned, namely the Newton iteration and the duality-based a posteriori error estimation and adaptivity. Then, in Section 2, we introduce the compressible Euler equations and formulate their discontinuous Galerkin finite element approximation including the treatment of various different types of boundary conditions. In Section 3, we give the Jacobian of the scheme, including the Jacobian of two specific numerical fluxes, and the treatment of the boundary conditions. The performance of the resulting Newton iteration is then studied in Section 4.1 through a series of numerical experiments on an inviscid flow around the NACA0012 airfoil. In Section 4.2 we summarize some recent results of the a posteriori error estimation and goal-oriented grid refinement applied to the discontinuous Galerkin discretization of a wide range of hyperbolic problems. Then we demonstrate the accuracy of the a posteriori error estimation on a sequence of adaptively refined meshes on the example problem already considered in Section 4.1. Finally, in Section 5, we summarize the work presented in this paper and give an outlook. 1.1 The Newton iteration In this section we introduce the Newton iteration for solving the finite element discretization of the following nonlinear variational problem: find u in X such that N (u, v) = 0 ∀v ∈ Y, (1) where N (·, ·) denotes a semi-linear form (nonlinear in its first argument, but linear in its second), and X and Y denote two Hilbert spaces. In order to discretize (1) we consider finite–dimensional spaces Xh ⊂ X and Yh ⊂ Y that, for the purposes of this paper, may be thought of as finite element spaces consisting of piecewise polynomial functions on a partition of granularity h. The Galerkin approximation uh of u is then sought in Xh as the solution of the finite–dimensional problem N (uh , vh ) = 0 ∀vh ∈ Yh .

(2)

For simplicity of presentation, we assume that Xh and Yh are suitably chosen finite element spaces to ensure the existence of a unique solution uh to (2). The nonlinear equation (2) may be solved using the Newton method. This nonlinear iteration generates a sequence of iterates unh ∈ Xh as follows. Given an iterate unh , the update dnh of unh to get to the next iterate un+1 = unh + ω n dnh h

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303

is given by the following problem: find dnh ∈ Xh such that Nu′ [unh ](dnh , vh ) = R(unh , vh ) ≡ −N (unh , vh ) ∀vh ∈ Yh .

(3)

Here, 0 < wn ≤ 1 denotes a damping parameter that is wn = 1 for a full (undamped) Newton iteration. Nu′ [w](·, v) denotes the derivative of u → N (u, v), for v ∈ Y fixed, at some w in X, and represents the Jacobian of the scheme. 1.2 Duality-based a posteriori error estimation In order to highlight the importance of this Jacobian in the context of dualitybased a posteriori error estimation, in this section we shortly give the main results of its general theoretical framework. For a more detailed discussion see the series of articles [1, 3, 4] and the references cited therein. Let J(·) be a target or error functional of the solution; for example, J(·) may represent the mean flow across a line, a point value of the solution, or the drag and lift coefficients of a body immersed into an inviscid fluid. For simplicity of presentation, here we assume that J(·) is linear. By employing a duality argument, the following error representation formula may be deduced, see [7, 8] for example, J(u) − J(uh ) = −N (uh , z − zh ),

(4)

where zh ∈ Yh , and z is the solution to the following dual or adjoint problem: find z ∈ Y such that M(u, uh ; w, z) = J(w)

∀w ∈ X.

(5)

Here, M(u, uh ; ·, ·) denotes the mean–value linearization of the semi–linear form N (·, ·) given by M(u, uh ; u − uh , v) = N (u, v) − N (uh , v) 1 Nu′ [θu + (1 − θ)uh ](u − uh , v) dθ = 0

for all v in Y . In order to obtain a computable a posteriori estimate of the error J(u) − J(uh ) in (4) we need to replace the – in general – unknown solution z to the dual problem (5) by a numerical approximation z˜h ∈ Y˜h . To this end, the dual problem (5) is linearized : find z˜ ∈ Y such that M(uh , uh ; w, z˜) = Nu′ [uh ](w, z˜) = J(w)

∀w ∈ X,

(6)

and then discretized: i.e. find z˜h ∈ Y˜h such that ˜h, Nu′ [uh ](wh , z˜h ) = J(wh ) ∀wh ∈ X

(7)

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˜ h and Y˜h yielding the required numerical approximation z˜h ∈ Y˜h to z. Here, X denote spaces that are chosen to be richer than the spaces Xh and Yh , respectively, as for example realized by function spaces which include polynomials of higher degree than the spaces Xh and Yh . In common with each Newton iteration step (3), the linearized and discretized dual problem (7) includes the Jacobian of the scheme. In fact, problem (7) gives rise to a linear problem whose matrix is simply the transpose ˜ h and Y˜h which are of the matrix of a Newton iteration step, assembled on X potentially larger spaces than Xh and Yh ,.

2 Discontinuous Galerkin discretization of the compressible Euler equations We consider the two–dimensional steady state compressible Euler equations of gas dynamics. Writing ρ, (v1 , v2 ), p and E to denote the density, Cartesian velocity, pressure and total energy per unit mass, respectively, the equations of motion are given by ∇ · F(u) ≡

2 ∂ Fk (u) = 0 in Ω, ∂xk

(8)

k=1

where Ω is an open bounded domain in R2 , supplemented with appropriate boundary conditions on the boundary Γ of the domain. Here, the vector of conservative variables u and the fluxes Fk , k = 1, 2, are defined by u = (ρ, ρv1 , ρv2 , ρE), F1 = (ρv1 , ρv12 + p, ρv1 v2 , ρHv1 ) and F2 = (ρv2 , ρv1 v2 , ρv22 + p, ρHv2 ), respectively. Additionally, H is the total enthalpy defined by H = E + p/ρ. The equation of state of an ideal gas is given by p = (γ − 1)ρ(E − (u2 +v 2 )/2), where γ is the ratio of specific heats which, for dry air, is γ = 1.4. To formulate the discontinuous Galerkin finite element method (DGFEM, for short) for (8), we first introduce some notation. Let Th = {κ} be an admissible subdivision of Ω into open quadrilateral domains κ; here h is a piecewise constant mesh function with h(x) = diam(κ) when x is in element κ. For p ∈ N0 , we define the following finite element space 6 5 4 4 Sh,p = v ∈ [L2 (Ω)] : v|κ ◦ σκ ∈ [Qp (ˆ κ)] ∀κ ∈ Th , where σκ denotes a smooth bijective image of the reference element κ ˆ = (0, 1)2 to the element κ ⊂ Ω, and Qp (ˆ κ) is the set of tensor product polynomials of degree at most p in each coordinate direction over κ ˆ . The DGFEM for (8) is defined as follows: find uh ∈ Sh,p such that 0 < + − + N (uh , vh ) ≡ − F(uh ) · ∇vh dx + H(uh , uh , nκ ) vh ds κ ∂κ\Γ κ∈Th (9) , + - + + + H uh , uΓ (uh ), nκ vh ds = 0 ∀vh ∈ Sh,p . Γ

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Here, H(·, ·, ·) denotes a numerical flux function, which depends on both the inner– and outer–trace of uh on ∂κ, κ ∈ Th , and the unit outward normal vector nκ to ∂κ. H(·, ·, ·) is assumed to be Lipschitz continuous, consistent and conservative; for example, we give descriptions of the (local) Lax–Friedrichs flux and the Vijayasundaram flux in the following. The (local) Lax–Friedrichs flux HLF (·, ·, ·), is defined by − HLF (u+ h , uh , nκ )|∂κ =

-, − 1, + − , (10) Fk (u+ h ) · nκ + Fk (uh ) · nκ − α uh − uh 2

for κ in Th , where α = α(u) denotes the largest eigenvalue (in absolute value) of the Jacobi matrix A(w, n) = F ′ (w) · n in the neighbourhood of ∂κ, i.e. α(u) =

max

w=u+ ,u−

α ˜ (w) ≡

where vn (u) = v · n and c(u) =

=

max {|vn (w)| + c(w)},

w=u+ ,u− γp ρ

(11)

denote the normal velocity and the

speed of sound, respectively. The Vijayasundaram flux HV (·, ·, ·), is defined by − + − HV (u+ uh , nκ )u+ uh , nκ )u− h , uh , nκ )|∂κ = A (ˆ h + A (ˆ h

for κ ∈ Th ,

(12)

where A+ (ˆ uh , nκ ) and A− (ˆ uh , nκ ) denote the positive and negative parts of the Jacobi matrix A(ˆ uh , nκ ), respectively, i.e. A± = PΛ± P −1 ,

Λ± = diag{λ± i , i = 1, . . . , 4},

(13)

with λ+ = max{λ, 0} and λ− = min{λ, 0}. Here, λi , i = 1, . . . , 4, and the columns of P are the eigenvalues and eigenvectors A(ˆ uh , nκ ), , of the−matrix ˆ h = 21 u+ . respectively, evaluated at the mean value u + u h h Finally, the boundary function uΓ (u) is given according to the type of boundary condition applied. We set uΓ (u) = uD on the Dirichlet parts of the boundary with a prescribed boundary function uD , uΓ (u) = u on superpout + 21 ̺v 2 ) sonic outflow parts of the boundary, and uΓ (u) = (̺, ̺v1 , ̺v2 , γ−1 on subsonic outflow parts of the boundary with prescribed pressure pout . Furthermore, we set uΓ (u) = urefl (u) on reflective (slip-wall) boundaries, where urefl (u) originates from u by simply inverting the sign of the normal velocity component of u, i.e. v = (v1 , v2 ) is replaced by v− = (v − 2(v · n)n) leading to ⎛ ⎞ 1 0 0 0 ⎜0 1 − 2n21 −2n1 n2 0⎟ ⎟ urefl (u) = urefl ′ (u)u = ⎜ (14) ⎝0 −2n1 n2 1 − 2n22 0⎠ u, 0 0 0 1 where ni , i = 1, 2, are the components of the unit outward normal vector n = (n1 , n2 ) to the boundary Γ .

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3 The Jacobian of the numerical scheme In this section we give a detailed description of the Jacobian Nu′ [w](·, v) of the semi-linear form N (u, v) defined in (9). In particular, Nu′ [w](φ, v) = − (F ′ (w)φ) · ∇v dx κ

κ∈Th

+ +

Γ

∂κ\Γ

,

,

< Hu′ + (w+ , w− , nκ )φ+ + Hu′ − (w+ , w− , nκ )φ− v+ ds

Hu′ + (w+ , wΓ (w+ ), nκ ) + Hu′ − (w+ , wΓ (w+ ), nκ )wΓ′ (w+ ) φ+ v+ ds,

where w → Hu′ + (w+ , w− , nκ ) and w → Hu′ − (w+ , w− , nκ ) denote the derivatives of the flux function H(·, ·, ·) with respect to its first and second arguments, respectively. Furthermore, u′Γ (u) denotes the derivative of the boundary function uΓ (u) with respect to the conservative variables in (i.e. the components of) u. We have u′Γ (u) = 0 on the Dirichlet parts of the boundary, u′Γ (u) = id ∈ R4,4 on supersonic outflow parts of the boundary, and ⎞ ⎛ 1 0 0 0 ⎜ 0 1 0 0⎟ ⎟ u′Γ (u) = ⎜ (15) ⎝ 0 0 1 0⎠ − 12 v 2 v1 v2 0

on subsonic outflow parts of the boundary. Finally, we have u′Γ (u) = u′refl (u), see (14), on reflective (slip-wall) boundary parts. It remains to give an expression for the derivatives Hu′ + and Hu′ − of the numerical flux function H. Clearly, they depend on the specific choice of the flux function and do – strictly speaking – not exist for many flux functions, as they typically include some non-differentiable terms such as ‘min’ and ‘max’ or absolute value functions, for example. Nevertheless, they can be approximated in practice, and, provided the approximation is sufficiently good, the resulting Jacobian is still capable of delivering an optimal convergence of the Newton iteration as demonstrated in Section 4. First, we consider the local Lax-Friedrichs flux. According to (10) its ith component is defined by 1, − Fki (u+ )nk + Fki (u− )nk + α(u)(u+ Hi (u+ , u− , n) = i − ui ) . 2 Its derivative with respect to its first argument is then given by , ′ Hu+ (u+, u−, n) ij = ∂u+ Hi (u+, u−, n) j , -' (16) 1& − ∂uj Fki (u+ )nk +α(u)δij +(αu′ + (u))j u+ , = −u i i 2

The Role of the Jacobian in the Adaptive DGM for the Compr. Euler Eqs.

where αu′ + (u) might be approximated by 0 sign (vn (u+ )) vn′ (u+ ) + c′ (u+ ) ′ αu+ (u) = 0

for α(u+) ˜ ≥ α(u−), ˜ else,

307

(17)

with a ˜(u) = |vn (u)| + c(u), see also (11). Similarly, we compute the derivative Hu′ − (u+ , u− , n) with respect to the second argument,. Finally, we consider the Vijayasundaram flux. According to (12) the ith component of this flux is given by − u, n)u− u, n)u+ Hi (u+ , u− , n) = A+ j , j + Aij (ˆ ij (ˆ

where the derivative with respect to the first argument is given by , ′ − Hu+ (u+ , u− , n) ij = A+ u, n) + ∂u+ A+ u, n)u+ u, n)u− ij (ˆ ik (ˆ k + ∂u+ Aik (ˆ k. j

j

Due to the involved dependence of A± (ˆ u, n) on u+ and u− , see definition in ± (13), its derivative ∂u+ Aik may be very complicated if computed explicitly. j Instead, we approximate it by difference quotients, i.e. ,1 , + - − ,n u, n) = ∂u+ A± ∂u+ A± ij (¯ ij 2 u + u k k , ± 1, + − 1 (18) = 2ε Aij ( 2 u + u + εek , n) , + , 2± 1 − −Aij ( 2 u + u − εek , n) + O ε , where ek , i = 1, 2, are the unit vectors and 0 < ǫ ≪ 1. Again, Hu′ − (u+ , u− , n) is computed accordingly.

4 Numerical results As described in the last section the Jacobian of the discontinuous Galerkin discretization (9) can be assembled approximately only. In the following two subsections we now present some numerical examples to show that in spite of this approximation the Jacobian is still accurate enough for its two applications mentioned in Sections 1.1 and 1.2 of the introduction. In particular, we will demonstrate the performance of the Newton iteration based on the Jacobian described in Section 4.1. Finally we show in Section 4.2 that the duality-based a posteriori error representation gives sharp error estimations when evaluated by solving the adjoint (dual) problem numerically. 4.1 Performance of the Newton iteration In order to demonstrate the performance of the Newton iteration we first consider a subsonic flow around a NACA0012 airfoil. The length of the profile is

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Fig. 1. M = 0.5, α = 0 ﬂow around the NACA0012 airfoil: Mach number isolines i M = 100 , i = 1, 2, . . .

rescaled to one. The outer boundary of the computational domain Ω consists of a circle of radius 10 units. There, we prescribe a Mach 0.5 flow at zero angle of attack, with farfield density ρ = 1 and pressure p = 1. The solution to this problem consists of a strictly subsonic flow, symmetric about the x–axis, cf. Figure 1. The first computation uses piecewise bilinear ansatz and test functions uh , vh ∈ Sh,1 , i.e. DG(1), and the Vijayasundaram flux, see (12). The computation is performed on a sequence of 5 globally refined quadrilateral meshes, starting on a coarse mesh with 39 cells and 624 degrees of freedom up to the finest mesh with 9984 cells and 159744 degrees of freedom. Starting on the coarsest mesh with freeflow conditions the nonlinear problem is solved by the Newton iteration as described in Section 1.1. When the nonlinear residual gets below 10−10 , the mesh is globally refined once, the discrete solution is interpolated onto the new mesh and is taken as start solution for the Newton iteration on the new mesh. For each Newton step a linear problem must be solved. Employing the GMRES solver with Block-Gauss-Seidel preconditioning the residual of each linear problem is reduced by a factor of 10−8 . Table 1 shows the history of the nonlinear solution process. The Newton iteration proceeds with full Newton steps (damping parameter wn = 1) leading to a very fast convergence of the Newton iteration. On each mesh the rate of convergence increases significantly clearly indicating a superlinear convergence. In fact, on several single Newton steps the residual is reduced by a factor of more than 103 . On all meshes the nonlinear residual is reduced below the tolerance in at most five Newton steps. Additionally, this history is displayed in Figure 2 which shows a plot of the nonlinear residuals over the total number of Newton steps. In this plot the increase in the rate of convergence is clearly seen in the increase of the slope

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Table 1. Nonlinear residual and convergence rates for DG(1). mesh 1 res rate

mesh 2 res. rate

mesh 3 res. rate

mesh 4 res. rate

mesh 5 res. rate

1.8-01 1.3-02 14 5.5-04 23 9.1-07 610 7.6-10 1192 4.2-13 1820

5.8-02 1.8-03 32 1.6-05 110 1.0-08 1599 4.6-12 2218

2.1-02 1.2-03 19 5.7-05 20 7.5-08 760 5.5-11 1361

8.5-03 6.8-04 12 6.2-05 11 1.1-07 540 1.6-10 695 1.5-13 1060

3.5-03 2.4-04 15 1.6-05 15 3.9-08 423 5.4-11 718

of the convergence on each mesh. Each time the residual crosses the 10−10 tolerance line the solution is interpolated onto the next mesh and the residual jumps back up as it is then measured on the new mesh. Table 2 collects the number of steps the Newton iteration requires for reaching the given tolerance on meshes 1-4 for DG(p), 1 ≤ p ≤ 4. We see that only about 4 to 5 steps are required for linear elements but similarly few steps also for higher order elements. Naturally, the number of iterations on the coarsest mesh is higher than on the following meshes as the Newton iteration first needs to get into the range of fast (quadratic) convergence. In order to accelerate this process, a DG(1) pre-iterated solution was taken as start solution for DG(4). Furthermore, we see that the number of Newton steps required stays virtually constant while the mesh is successively refined. A variance of at most one step as for example on the fourth mesh in DG(1) occurs by chance as sometimes the fourth iteration step happens to be just below or above the tolerance, see also Figure 2. In the latter case an additional step is needed for converging below the tolerance. Table 2. Number of Newton steps on each global reﬁnement level for DG(p), 1 ≤ p ≤ 4 (⋆ pre-iteration on DG(1)). mesh DG(1) DG(2) DG(3) DG(4) 1 2 3 4

5 4 4 5

5 4 4 4

6 5 5 5

6⋆ 5 5 6

Several numerical tests have been performed for decreasing the computing time of the Newton iteration. In the following we shortly summarize a few of the experiences made. 1. Simplifying the Jacobian of the numerical fluxes by e.g. neglecting the u, n) terms in (18), as done in [7, 8], αu′ + (u) term in (16) or the ∂uk A± ij (¯

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R. Hartmann 1

DG(1), global refinement 1e-10

0.01

Residual

0.0001

1e-06

1e-08

1e-10

1e-12

1e-14 0

5

10 15 number of Newton steps

1

20

25

DG(1), global refinement 1e-10

0.01

Residual

0.0001

1e-06

1e-08

1e-10

1e-12

1e-14 0

0.1

0.2

0.3

0.4 0.5 0.6 fraction of total time

0.7

0.8

0.9

1

Fig. 2. Convergence of the residual under global reﬁnement plotted over (a) the number of Newton steps (b) the fraction of the total time.

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leads to small time savings in the assembly of the Jacobian but also to a significant increase of the numbers of Newton steps required. As the overall time is dominated by the time of the linear solver rather than by the assembly time of the Jacobian, this leads to an significant increase of computing time; numerical tests showed an increase of more than 50%. Consequently, the Jacobian should be approximated as accurate as possible. 2. In the numerical tests performed above the linear systems are solved by reducing the linear residual by a factor of 10−8 . Numerical tests show that reducing the linear residual on each step by a factor of 10−3 or 10−4 only, two or three additional Newton steps are required, but the overall computing time is decreased significantly. In numerical tests time savings of up to 65% have been encountered. 3. When the solution on the finest mesh is of interest only, the solutions on the coarser meshes do not need to be iterated below the given tolerance. In fact, performing only one Newton step on all but the finest mesh in the numerical example above gives a start solution for the Newton iteration on the finest mesh which is already in the range of fast convergence, see Figure 3. But, the gain of solving the solution on coarser meshes incorrectly by performing one Newton step only, is bounded by the time of the (exact) solution process on the coarser meshes. In the numerical test above, see Figure 2 (b), this is less than 15% of the total computing time only. Finally, we demonstrate the performance of the Newton iteration solving the DG(1) discretization on a sequence of locally refined meshes. Figure 4 (a) shows the convergence of the residual plotted over the number of Newton steps. Again, as in the case of global refinement, see Figure 2, the residual converges below the tolerance in about 4 steps. On the coarsest mesh it takes an additional step for reaching the range of fast convergence, on the finer meshes only 3 steps are required. 4.2 Accuracy of the a posteriori error estimation In this section we demonstrate the accuracy of the a posteriori error estimation with respect to arbitrary target functionals J(·). Starting from the error representation formula (4) the unknown solution z of the dual problem (5) is replaced by an approximate solution z˜h of the linearized and discretized dual problem (7) which results in following approximate error representation ηκ =: η, (19) J(u) − J(uh ) ≈ −N (uh , z˜h − zh ) = κ∈Th

which can be decomposed as a summation of local error indicators ηκ , κ ∈ Th , over the elements κ in the computational mesh Th . Employing these so-called weighted indicators |ηκ |, κ ∈ Th , for driving an adaptive mesh refinement algorithm, very economical meshes can be produced, that are specifically tailored

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DG(1), global refinement 1e-10

0.1 0.01 0.001

Residual

0.0001 1e-05 1e-06 1e-07 1e-08 1e-09 1e-10 1e-11 0

2

4

6 8 number of Newton steps

10

12

14

Fig. 3. Convergence of the residual plotted over the number of Newton steps on a sequence of globally reﬁned meshes. On all meshes but the ﬁnest only one Newton step is performed.

to the efficient computation of the quantity J(u) of interest. First results of the a posteriori error estimation and goal-oriented adaptivity approach described above applied to simple hyperbolic problems have been published in [5] which includes test cases for the linear advection equation and 1D inviscid Burgers equations. In a sequence of publications, see [7, 8, 10, 11], these results have been extended to the 2D Euler equations, on a variety of problems including the Ringleb flow problem, supersonic flow past a wedge, flows through a nozzle, and sub-, trans- and supersonic flows around different airfoil geometries; for an overview of typical test cases also including the Buckley-Leverett equation and the 1D Euler equations, see the thesis [6]. Furthermore, this approach has been generalized to the case of multiple target functionals, see [9]. In these publications it has been demonstrated that the meshes produced using the weighted indicators are much more efficient in computing accurate values of the target quantities than meshes that simply rely on ad hoc indicators that do not require the solution of a dual problem. Here, efficiency is measured in terms of number of cells but also in terms of total computing time that includes the time for solving the additional (but linear) dual problem. Furthermore, it has been shown that the approximate error representation yields a good approximation to the true error measured in terms of the target functional.

The Role of the Jacobian in the Adaptive DGM for the Compr. Euler Eqs. 1

313

DG(1), local refinement 1e-10

0.01

Residual

0.0001

1e-06

1e-08

1e-10

1e-12

1e-14 0

5

10

15

20 25 30 35 number of Newton steps

1

40

45

50

55

0.9

1

DG(1), local refinement 1e-10

0.01

Residual

0.0001

1e-06

1e-08

1e-10

1e-12

1e-14 0

0.1

0.2

0.3

0.4 0.5 0.6 fraction of total time

0.7

0.8

Fig. 4. Convergence of the residual under local reﬁnement plotted over (a) the number of Newton steps (b) the fraction of the total time.

314

R. Hartmann

In the following, we give an examples of the accuracy of the a posteriori error estimation. To this end, we consider the inviscid Mach 0.5 flow at a zero angle of attack around the NACA0012 airfoil, i.e. the test case already considered in the last subsection. Here, we consider the case of estimating the drag on the surface of the airfoil. As for this subsonic flow the exact value of the drag is known to be zero, the true value of the target quantity is J(u) = 0. The primal (nonlinear) problem is approximated with discontinuous piecewise bilinear functions, i.e. uh ∈ Sh,1 , and the dual (linear) problem is approximated with discontinuous piecewise biquadratic functions, i.e. z˜h ∈ Y˜h ≡ Sh,2 . Starting on a coarse mesh the mesh is successively refined using the weighted indicators, where 20% and 10% of the cells are flagged for refinement and coarsening, respectively; for a more detailed discussion of the adaptive refinement algorithm, see [7] or [6]. In Table 3, we demonstrate the performance of the adaptive algorithm. Here, we show the number of elements, the true errorin the target functional, J(u)−J(uh ), the computed error representation η = κ η˜κ , and the effectivity index θ = η/(J(u) − J(uh )). We see that initially on the coarsest meshes the quality of the computed error representation formula η, see (19), is poor, in the sense that θ is not close to one; however, as the mesh is refined the effectivity index θ is very close to unity indicating that the computed error representation gives a very close approximation to the true error. Table 3. Subsonic ﬂow around a NACA0012 airfoil. Adaptive algorithm for the evaluation of the drag coeﬃcient. Linear residuals of the dual problems are reduced by factor of 10−8 . # el. # DoFs J(u) − J(uh ) η = κ ηκ θ 39 624 66 1056 120 1920 204 3264 348 5568 570 9120 927 14832 1557 24912 2538 40608 4152 66432 6747 107952

-3.902e-02 -2.005e-02 -1.359e-02 -9.147e-03 -4.244e-03 -1.583e-03 -5.760e-04 -2.287e-04 -9.725e-05 -4.850e-05 -2.359e-05

-2.727e-02 -5.485e-03 -1.794e-03 -4.600e-03 -3.321e-03 -1.441e-03 -5.531e-04 -2.256e-04 -9.592e-05 -4.674e-05 -2.138e-05

0.70 0.27 0.13 0.50 0.78 0.91 0.96 0.99 0.99 0.96 0.91

By employing a block-Gauss-Seidel preconditioned GMRES solver, see [6], the linear residual of each dual problem has been reduced by a factor of 10−8 . Numerical tests have shown that it is not necessary to solve the dual problem up to this accuracy. Reducing the linear residual by a factor of 10−3 only, mainly retains the accuracy of the error estimation and the efficiency of

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315

the locally refined meshes, see Table 4. In fact, the values of the computed error representation are changed slightly only; also the sequence of adaptively refined meshes is almost the same. In this test case, time savings of up to 90% have been encountered for computing the dual solution when its linear residual is reduced by a factor of 10−3 instead of reducing it by a factor of 10−8 . Table 4. Subsonic ﬂow around a NACA0012 airfoil. Adaptive algorithm for the evaluation of the drag coeﬃcient. Linear residuals of the dual problems are reduced by factor of 10−3 . # el. # DoFs J(u) − J(uh ) η = κ ηκ θ 39 624 66 1056 120 1920 204 3264 348 5568 570 9120 930 14880 1566 25056 2565 41040 4203 67248 6816 109056

-3.902e-02 -2.005e-02 -1.359e-02 -9.147e-03 -4.244e-03 -1.583e-03 -5.747e-04 -2.277e-04 -9.523e-05 -4.767e-05 -2.331e-05

-2.727e-02 -5.481e-03 -1.796e-03 -4.589e-03 -3.322e-03 -1.440e-03 -5.547e-04 -2.244e-04 -9.403e-05 -4.456e-05 -2.055e-05

0.70 0.27 0.13 0.50 0.78 0.91 0.97 0.99 0.99 0.93 0.88

5 Concluding Remarks In this paper we have provided a full description of the Jacobian to the discontinuous Galerkin discretization of the compressible Euler equations which is key ingredient of the Newton iteration for solving the compressible Euler equations and for the construction of adjoint (dual) problems arising in the context of duality-based a posteriori error estimation and adaptivity. In various numerical tests we demonstrated the performance of the Newton iteration. We encountered full Newton convergence on globally as well as on locally refined grids. Similar results were obtained for higher order elements. Furthermore, we demonstrated the accuracy of the a posteriori error estimation for the error in the drag of an airfoil immersed in an inviscid fluid. Numerical tests have shown that solving the linear systems of the Newton iteration steps and the dual problems with lower accuracy, only, may lead to a significant speed up in the solution process without reducing the accuracy of the numerical solution and the a posteriori error estimation.

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Acknowledgement. The author acknowledges the ﬁnancial support of the DFG Priority Research Program “Analysis and Numerics of Conservation Laws”, the SFB 359 “Reactive Flows, Diﬀusion and Transport” at the IWR, University of Heidelberg, and the German Aerospace Center (DLR), Braunschweig.

References 1. R. Becker and R. Rannacher. An optimal control approach to error estimation and mesh adaptation in ﬁnite element methods. Acta Numerica, 10:1–102, 2001. 2. B. Cockburn, G. Karniadakis, and C.-W. Shu. The development of discontinuous Galerkin methods. In B. Cockburn, G. Karniadakis, and C.-W. Shu, editors, Discontinuous Galerkin Methods, volume 11, pages 3–50. Springer, 1999. 3. K. Eriksson, D. Estep, P. Hansbo, and C. Johnson. Introduction to adaptive methods for diﬀerential equations. Acta Numerica, pages 105–158, 1995. 4. M. Giles and E. S¨ uli. Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numerica, 2002. 5. R. Hartmann. Adaptive FE Methods for Conservation Equations. In H. Freist¨ uhler and G. Warnecke, editors, Hyperbolic Problems: theory, numerics, applications: eighth international conference in Magdeburg, February, March 2000, volume 2 of International series of numerical mathematics; Vol. 141, pages 495–503. Birkh¨ auser, Basel, 2001. 6. R. Hartmann. Adaptive Finite Element Methods for the Compressible Euler Equations. PhD thesis, University of Heidelberg, 2002. 7. R. Hartmann and P. Houston. Adaptive discontinuous Galerkin ﬁnite element methods for nonlinear hyperbolic conservation laws. SIAM J. Sci. Comp., 24:979–1004, 2002. 8. R. Hartmann and P. Houston. Adaptive discontinuous Galerkin ﬁnite element methods for the compressible Euler equations. J. Comp. Phys., 183:508–532, 2002. 9. R. Hartmann and P. Houston. Goal-oriented a posteriori error estimation for multiple target functionals. In T. Y. Hou and E. Tadmor, editors, Hyperbolic problems: theory, numerics, applications, pages 579–588, Springer, 2003. 10. P. Houston and R. Hartmann. Goal–oriented a posteriori error estimation for compressible ﬂuid ﬂows. In F. Brezzi, A. Buﬀa, S. Corsaro, and A. Murli, editors, Num. Mathematics and Advanced Applications, pages 775–784. Springer, 2003. 11. P. Houston, R. Hartmann, and A. S¨ uli. Adaptive discontinuous Galerkin ﬁnite element methods for compressible ﬂuid ﬂows. In M. Baines, editor, Numerical methods for Fluid Dynamics VII, ICFD, pages 347–353, 2001.

The Multi-Scale Dust Formation in Substellar Atmospheres Christiane Helling1,2 , Rupert Klein2,3,4 , and Erwin Sedlmayr1 1

2

3

4

Zentrum f¨ ur Astronomie und Astrophysik, TU Berlin, Hardenbergstraße 36, D-10623 Berlin chris|[email protected] Konrad-Zuse-Zentrum f¨ ur Informationstechnik Berlin, Takustraße 7, D-14195 Berlin [email protected] Fachbereich Mathematik und Informatik, Freie Universit¨ at Berlin, Takustraße 7, D-14195 Berlin Potsdam Institut f¨ ur Klimafolgenforschung, Telegrafenberg A31, D-14473 Potsdam

Summary. Substellar atmospheres are observed to be irregularly variable for which the formation of dust clouds is the most promising candidate explanation. The atmospheric gas is convectively unstable and, last but not least, colliding convective cells are seen as cause for a turbulent ﬂuid ﬁeld. Since dust formation depends on the local properties of the ﬂuid, turbulence inﬂuences the dust formation process and may even allow the dust formation in an initially dust-hostile gas. A regime-wise investigation of dust forming substellar atmospheric situations reveals that the largest scales are determined by the interplay between gravitational settling and convective replenishment which results in a dust-stratiﬁed atmosphere. The regime of small scales is determined by the interaction of turbulent ﬂuctuations. Resulting lane-like and curled dust distributions combine to larger and larger structures. We compile necessary criteria for a subgrid model in the frame of large scale simulations as result of our study on small scale turbulence in dust forming gases.

1 Introduction The astrophysical field of substellar objects – of which brown dwarfs shall be considered as an example – has gained considerable attention during the last few years since more and more brown dwarfs and extrasolar planets could be detected. The need for appropriate models has increased with increasing observational resolution power which, however, can only provide information about the largest scale structures. The major challenge has been the observational evidence for the presence of dust (i.e. small solid or fluid particles) in substellar atmospheres where energy is mainly transported by convection.

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These gas parcels collide after their ascend through the atmosphere during which they adiabatically expand. Such adiabatic collisions cause turbulence, and the turbulent kinetic energy is transfered cascade-like from large to small scales. Due to its chemical nature, the dust formation process itself (like combustion) depends on local quantities like temperature, density and chemical composition of the gas, which can vary on much smaller spatial scale lengths than those accessible by observations. Photon interactions, molecular collisions and friction influence the dust formation process and proceed on scales comparable to or even less than one mean free path of the fluid, which belongs to the smallest relevant scale regime. The dust formation is hence initiated and controlled by small-scale processes, but produces consequences on the largest, observable scales. The immediate couplings between chemistry, hydrodynamics and thermodynamics due to the presence of dust can cause an amplification of initially small perturbations resulting in the formation of large-scale dust clouds. The formation of such clouds is one of the major candidates to explain the observed non-periodic variability [BJM99, BJM01b, BJM01b, BJM01a, MZL01, GM00, Cla03] which is - from the physical point of view - very alike to what we know from weather-like variation in the Earth atmosphere. However, substellar object like brown dwarfs and known extraterrestrial planets show more extreme physical conditions (warmer and hotter or cooler and thinner) and the direct transfer of Earth’s knowledge has to be considered with care. It was therefore the aim of this work to investigate the multi-scale problem of astrophysical dust formation in brown dwarf stars which requires an appropriate description of the chemical and turbulent processes and their interactions. The challenge in modeling turbulence in reactive gas flows lies in an adequate description of all relevant scale regimes. This work follows the scale hierarchy from the smallest to the largest scales. At first, turbulent dust formation will be studied on small scales (Sect. 3) which can be computationally resolved. These investigations intend to provide the basis for a sub-grid closure model for the next (larger) scale regime where it can be applied and so on. This procedure requires a detailed, scale-dependent view on the problem [HLSK01], or in other words, needs to adopt different windows of perception [Sed02].

2 The model of a dust forming gas flow Redefining the variables of the system of model equations like α → α/αref transforms the equations in their scaled analogues wherein all quantities are dimensionless and can be compared by number. The non-dimensional characteristic numbers describe the importance of the source terms and general features can already be derived by investigating the dimensionless numbers for physical meaningful reference values.

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Complex A: The specification of the source terms in the equations of momentum and energy conservation results in the following dimensionless system of equations which describe the fluid field in a substellar atmosphere which is influenced by the stellar radiation field [HOL01]: (ρ)t + ∇ · (ρv) = 0

(1)

Sr (ρv)t + ∇ · (ρv ◦ v) = −

1 1 ∇P − ρg γM 2 F r2

(2)

4 (ρe)t + ∇ · (v[ρe + P ]) = Rd1 κ(TRE − T 4 ),

(3)

with the caloric equation of state 2 ρv 1 P 2 ρ e = γM + ρgy + 2 F r2 γ−1

(4)

with g = {0, g, 0}. Radiative heating/cooling is treated by an relaxation ansatz (r.h.s. (3)). For the characteristic numbers M , F r, Sr, and Rd1 , which relate certain physical processes, see Table 2. Complex B: The dust formation process is considered as a two step process. At first, seed particles form out of the gas phase (nucleation) which provide the first surfaces. Subsequent growth by surface reactions results in the formation of (chemically) macroscopic µm-sized particles. Nucleation, growth, evaporation, drift and element depletion/enrichment are physical and chemical processes which occur simultaneously in an atmospheric gas flow and may be strongly coupled. Following the classical work of [GS88], partial differential equations which describe the evolution of the dust component by means of moments of its size distribution function, f (V ), were derived in [WH03a] explicitly allowing for v gas = v dust . The conservation form of the dust model equations allows for a fast numerical solution in the frame of extensive model simulations: , ∂, ρLj +∇ v gas ρLj = ∂t

∞

Vℓ

◦

Rk V

k

>

?@ Aj

j/3

∞ ◦ dV − ∇ f (V ) V j/3 v dr (V ) dV , (5) A

>

Vℓ

?@ Bj

A

with v dr (V ) the grain size dependent equilibrium drift velocity (for more details please consult [WH03a]). The j-th moment of the dust size distribution j function Lj [cm /g] is defined by ρLj (x, t) =

∞

Vℓ

f (V, x, t) V j/3 dV .

(6)

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The source term Aj expresses the effects of nucleation and surface chemical reactions on the dust moments. Compared to the classical moment equations [GS88], Bj is an additional, advective term in the new dust moment equations which comprises the effects caused by a size-dependent drift motion of the grains. The meaning of the dust variables is summarizes in Table 1. Table 1. Deﬁnitions, meanings, and units of the dust and chemical quantities (CE = chemical equilibrium). Quantity v gas v dust f (V ) V Rk Lj ρL 0 = nd = 3

√ 3

3 L1 4π L0 L2 36π L 0 L3 = L0

nH

Unit [cm s−1 ] [cm s−1 ] [cm−6 ] [cm3 ] [s−1 ] [cmj g−1 ] [cm−3 ]

= a [cm]

2

= A [cm ] V [cm3 ] ǫx [–] [cm−3 ] nr = 1.427ρamu [cm−3 ] [cm s−1 ] vrel,x

νi,0 νi,r

Meaning Variables hydrodynamic gas velocity hydrodynamic dust velocity grain size distribution function volume of the dust particle surface chemical reaction rates dust moments (j ∈ N) number of dust particles mean radius of dust particles mean dust surface mean dust volume element abundance relative to hydrogen number density of gas species r total hydrogen density thermal relative velocity of the gas species x Constants stoichiometric ratios of homogeneous nucleation stoichiometric ratios of surface reaction r

The motion of a dust grain with a velocity v dust is determined by an equilibrium between the force of gravity, F grav , and the frictional force, F fric , ◦ (equilibrium drift ⇒ v dr (V ); see discussion in [WH03a]). Depending on the particle size and the density of the surrounding fluid, the character of the hydrodynamic situation changes which also influences the growth process of the particles. The dimensionless dust moment equations for nucleation, growth, evaporation, and equilibrium drift write, e. g. for the case of a subsonic free molecular flow, are ! !, Sr ρLj t + ∇(vgas ρLj ) = Sr · Danuc (7) d · Sej J(Vℓ ) 3j 2 χnet ρLj−1 + Sr · Dagr d,lKn 3 lKn & ' πγ 1/2 Sr · M · Dr Lj+1 ξlKn ∇ + er , 32 cT KnHD · Fr

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with j = 0, 1, 2 . . . the order of the dust moments. The first source term on the r.h.s. describes the seed formation (nucleation), the second the growth process, and the third describes the transport of already existing dust particles by drift. In those cases, where dust and gas are positionally coupled and drift effects are negligible, (7) simplifies to ! !, gr j net χ ρLj−1 . (8) Sr ρLj t +∇(vgas ρLj ) = Sr·Danuc d · Sej J(Vℓ )+[Sr· Dad ] 3

For the definition of the characteristic numbers in square brackets see Table 2. The element depletion is taken into account by evaluating the consumption of each involved element i with relative abundance to hydrogen ǫi by nucleation and growth R ! ! nuc Sr (ρǫx )t + ∇ · (v ρǫx ) = − Sr· El (νx,r Danuc Nl J∗ d r=1

+

gr νx,r

Dagr d

(9)

αr nx,r vrel,x ρL2 ).

ǫx is the element abundance of the chemical element x (e.g. Ti, Si, O) in mass fractions, which is consumed by nucleation (first term, r.h.s.) and growth (second term, r.h.s.) with the stoichiometric factor νx,r due to the reaction r. One equation needs to be solved for each element involved in the dust formation process. Characteristic behavior Adopting typical values for a brown dwarf atmosphere the following general characteristics of a dust forming fluid can be drawn (for definitions see Table 2): – High Reynolds numbers, Re ≈ 107 . . . 109 indicate that the viscosity in the brown dwarf atmosphere is too small to damp hydrodynamical perturbations and a turbulent hydrodynamic field can be expected. – Assuming a typical turbulence velocity of the order of one tenth of the velocity of sound leads to a Mach number of M ≈ O(0.1) for the whole fluid. Fluctuations on smaller scales can have M ≈ O(10−2 ). – The Froude number F r = O(10−3 . . . 10−1 ) shows that the gravity only gains considerable influence on the hydrodynamics for scale regimes lref Hρ . Drift terms are important in the macroscopic regime and may be neglected in the small scale regimes. An analysis of the combined characteristic drift number shows that the drift term in the dust moment equations is mainly influenced by the gravity and the bulk density of the grains. – The characteristic number for the radiative heating / cooling, Rd1 = 4 4κref σTref · Ptref gives the ratio of the radiative and the thermal energy ref content of a fluid. The scaling of the system influences Rd1 by the reference time tref which increases with increasing spatial scales.

Sedlma¨ yr number (j ∈ N0 )

Kn

=

Kn = Dr = ρref 4 Rd1 = 4κref σTref · tref 'j Pref & Sej = aaℓref Danuc = d

Damk¨ ohler no. of growth (lKn) Dagr d,lKn = Element consumption number

El =

ρref

[g/cm3 ]

temperature velocity

Tref vref

[K] [cm/s]

length

lref

[cm] [cm/s2 ] [s−1 ] [cm]

gravitational acceleration gref nucleation rate Jref /n,ref mean particle radius a ref element abundance

ǫref

tref Jref ρref L0,ref tref χref,lKn

)1/3 ( 43π a3 ref ρref L0,ref Nl n,ref ǫref

Combined characteristic number & πγ '1/2 Sr · M · Dr combined drift number (lKn) 32 KnHD· Fr

Dependent reference values thermal pressure total hydrogen density hydrodynamic time

Pref =

ρref kTref µ ¯

n,ref = ρref / tref = vlref ref

0th dust moment (= nd /ρ) L0,ref = χref lKn

[dyn/cm2 ] ǫ i mi

∆VSiO nref,SiO ρref 4π a3 3 ref

(∗ )

growth velocity ℓ¯ref mean free path abs,gas opacity κref = κref (σref , L3,ref ) molecular number density nref = nref,SiO ≈ ǫi n,ref (∗∗ )

[cm−3 ] [s] [1/g] [cm/s] [cm] [cm−1 ] [cm−3 ]

(∗ ) – approximation (compare (5)) (∗∗ ) – to be determined from chemical equilibrium calculations 3 2 erg 3 Parameters: σ - Stefan-Boltzmann constant 2 4 , al - hypothetical monomer radius [cm], ∆VSiO - monomer volume of SiO2 [cm ], s cm K −3 abs 2 ρd,ref - bulk density [g cm ], mi - element mass [g], σref - gas absorption cross section [cm ]

lKn = large Knudsen number (Kn ≫ 1)

Ch. Helling, R. Klein, E. Sedlmayr

Fr = Sr =

hydrodyn. Knudsen number

Damk¨ ohler no. of nucleation

2 vref 1 lref gref lref tref vref HD lref 2aref ℓ¯ref 2aref ρd,ref

(fundamental dimensions)

density

322

Froude number Strouhal number Knudsen number Drift number 1. Radiation number

Independent reference values vref cs

M=

Table 2. Characteristic numbers and reference values of the scaled model equations in Sect. 2.

Characteristic numbers Mach number

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– An analysis of the characteristic numbers in front of the source terms of the dust moment equations reveals a clear hierarchy of the processes of dust formation: nucleation → growth → drift. The governing equations of the model problem are those of an inviscid, compressible fluid which are coupled to stiff dust moment equations and an almost singular radiative energy relaxation if dust is present. The coupling between the processes becomes stronger with increasing time scales. If additionally v gas → 0, i. e. the dust-forming system reaches the static case, and t → ∞, the source terms in (7) must balance each other. In the S > 1 case (S - supersaturation ratio), this means that the gain of dust by nucleation and growth must be balanced by the loss of dust by rain-out. In the S < 1 case, just the opposite is true, i. e. the loss by evaporation must be balanced by the gain of dust particles raining in from above. Both control mechanisms (in the static limit) result in an efficient transport of condensible elements from the cool upper layers into the warm inner layers, which cannot last forever. If the brown dwarf ’s atmosphere is truly static for a long time, there is no other than the trivial solution for (7) where the gas is saturated (S ≡ 1) and dust-free (Lj ≡ 0).

3 Dust formation on small scales 3.1 Numerical approach The fully time-dependent solution of the model equations (1) – (3), (8), (9) has been obtained in the small scale regimes by applying a multi-dimensional hydro code [SMK97] which has been extended in order to treat the complex of dust formation and elemental conservation. Initial conditions: The initial conditions have been chosen as homogeneous, static, adiabatic, and dust free, i. e. ρ0 = 1, p0 = 1, u0 = 0, L0 = 0 (⇒ Lj = 0) in order to represent a (semi-)static, dust-hostile part of the substellar atmosphere. This allows us to study the influence of our variable boundaries on the evolution of the dust complex without a possible intersection with the initial conditions. Boundary conditions and turbulence driving: The Cartesian grid is divided in the cells of the test volume (inside) and the ghost cells which surround the test volume (outside). The state of each ghost cell is prescribed by our adiabatic model of driven turbulence for each time. The hydro code solves the model equation in each cell (test volume + ghost

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cells) and the prescribed fluctuations in the ghost cells are transported into the test volume by the nature of the HD equations. The numerical boundary occurs between the ghost cells and the initially homogeneous test volume and are determined by the solution of the Riemann problem. Material can flow into the test volume and can leave the test volume. The solution of the model problem is considered inside the test volume. Stiff coupling of dust and radiative heating / cooling Dust formation occurs on much shorter time scales than the hydrodynamic processes. Approaching regimes of larger and larger scales makes this problem more and more crucial. Therefore, the dust moment and element conservation equations (Complex B) are solved applying an ODE solver in the framework of the operator splitting method assuming T, ρ =const during ODE solution. In [HOL01] we have used the CVODE solver (Cohen & Hindmarsh 2000; LLNL) which turned out to be insufficient for the mesoscopic scale regime. CVODE failed to solve our model equations after the dust had reached its steady state. The LIMEX solver: The solution of the equilibrium situation of the dust complex is essential for our investigation since it describes the stationary case of Complex B when no further dust formation takes place. The reason may be that all available gaseous material has been consumed and the supersaturation rate S = 1 or the thermodynamic conditions do not allow the formation of dust. The first case involves an asymptotic approach of the gaseous number density of S = 1 which often is difficult to be solved by an ODE solver due to the choice of too large time steps. However, the asymptotic behavior is influenced by the temperature evolution of the gas/dust mixture which in our model is influenced by radiative heating/cooling. Since the radiative heating/cooling 4 − T 4 ) heating/cooling rate) depends on the absorption (Qrad = Rd κ(TRE coefficient κ of the gas/dust mixture which strongly changes if dust forms. Consequently, the radiative heating/cooling rate is strongly coupled to the dust complex which in turn depends sensitively on the local temperature which is influenced by the radiative heating/cooling. It was therefore necessary to include also the radiative heating/cooling source term in the separate ODE treatment for which we adopted the LIMEX DAE solver. LIMEX [DN87] is a solver for linearly implicit systems of differential algebraic equations. It is an extrapolation method based on a linearly implicit Euler discretization and is equipped with a sophisticated order and stepsize control [Deu83]. In contrast to the widely used multistep methods, e. g. OVODE, only linear systems of equations and no non-linear systems have to be solved internally. Various methods for linear system solution are incorporated, e. g. full and band mode, general sparse direct mode and iterative solution with preconditioning. The method has shown to be very efficient and robust in

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several fields of challenging applications in numerical [NEO98, ENOD99] and astrophysical science [Str02]. 3.2 Microscopic regime The smallest scale involved into the structure formation of a substellar atmosphere is the monomer volume, ∆V , by which a chemical reaction r increases the volume of a grain. For radiative transfer and drift effects, the grain size becomes important. On small hydrodynamic scales, drift effects are negliggibaly small. However, the aim is to investigate the role of turbulence in the formation of dust clouds and the possibly related variability of brown dwarfs. Therefore, the interesting hydrodynamic phenomena in the microscopic scale regime are acoustic waves. Carrying out numerical simulations of the whole time-dependent system of model equations (1) – (3), (8), (9), turbulence is seen as acoustic waves being by-products of colliding convective cells [HOL01] which undergo an adiabatic increase of size during their upward travel through the substellar atmosphere. A feedback loop establishes in a dust-forming system in which interacting expansion waves play a key role (Fig. 1): The interaction of small-scale disturbances of the fluid field can cause a local and temporarily limited temperature decrease low enough to initiate dust nucleation. These seed particles grow until they reach a size where the dust opacity is large enough to accelerate radiative cooling which causes the temperature to decrease again below a nucleation threshold (Ts ). Dust nucleation is henceforth re-initiated which

Fig. 1. Unstable feedback loop in dust forming systems: Dust, once formed, increases the total opacity which intensiﬁes radiative cooling. The temperature decreases and enters the dust formation window. Therefore, more dust is formed which increases the opacity further. An instability establishes where the dust improves the conditions for its own formation [Hel03].

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results in a further intensified radiative cooling. The nucleation rate and consequently also the amount of dust particles increase further. This run-away process is stopped if either the radiative equilibrium temperature of the gas is reached or all condensible material is consumed. Meanwhile, the seed particles have grown to macroscopic sizes. The result is a highly variable dust distribution in space and time on the microscopic scales investigated (e. g. ∆l ≈ 102 cm, ∆t ≈ 0.5 s) due to the occurrence of singular nucleation events. It is immediately clear, that such scales are not easily resolvable by observations, but these are the scales on which structure formation is likely to be initiated. However, such small scale pattern might move and successively initiate dust formation at various sites in the atmosphere. A larger and larger cloud structure may thereby form which is, hence, determined by a non-local hydrodynamic coupling whereas at each site the above outlined local feedback loop will act (compare Fig. 3). 3.3 Mesoscopic regime Following Kolmogoroff’s idea, the mesoscopic scales shall be considered as those where energy is only transfered through the turbulence cascade, but is not injected or dissipated. Convective elements are generated as long as the Schwarzschild criteria is fulfilled in the substellar atmosphere. The inertia of mass will drive the convective elements beyond Schwarzschild’s boundary, a phenomenon usually named as overshooting, which extends this zone further. The mass elements interact, collide, and transfer part of their energy into small scale structures, thereby producing a whole spectrum of them being viewed as waves or – turbulence elements. These waves propagate and enter even atmospheric regions which are not influenced by convective motions any more. The question arises how the dust formation process takes place in such a stochastically fluctuating thermo- and hydrodynamic environment, and therefore which influence such small scale disturbances may have on the whole atmospheric structure and the formation of large scale pattern possibly responsible for observed variabilities. A model for driven turbulence was proposed [HKWS03, HKWN04] which simulates a constantly occurring, small-scale energy input assumed to originate from a convectively active zone. Following Reynolds scale separation ansatz a background field α0 (x, t) is disturbed by a fluctuation δα(x, t) of some variable α(x, t) such that α(x, t) = α0 (x, t) + δα(x, t).

(10)

α(x, t) ǫ {u(x, t), P (x, t), S(x, t)} (u(x, t) - velocity, P - pressure, S - entropy). The velocity fluctuation δu(x, t) follows the Kolmogoroff spectrum in k-space in which a whole range of wavenumbers is excited (kmin = 2π/lmax , kmax = 2π/(3∆x), lmax = 5 × 104 cm – maximum scale considered, ∆x = 102 cm – spatial grid resolution). The model for driven turbulence relies on a superposition ansatz of different wave modes and therefore allows again to carry out

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direct simulation of the time and space evolution of the dust complex, now in a stochastically excited medium typical for the elswise dust-hostile part of a brown dwarf atmosphere. Nucleation events and nucleation fronts An initially hydrodynamically homogeneous and dust-free, 500m-sized gas parcel has been considered in a deep layer of a brown dwarf atmosphere initially too hot for nucleation. The gas is disturbed by superimposed waves entering through its boundaries. If these disturbances carry already a temperature below the nucleation threshold (Ts - nucleation threshold temperature), a nucleation front will develop: The nucleation peak moves inwards together with the wave and initiates the feedback loop already known from the microscopic regime by leaving behind first dust seeds. If the entering temperature disturbances is not enough to cross Ts , interaction with some expansion wave coming from another direction at some time and some site will take place thereby causing singular nucleation events to occur. A nucleation front tends to homogeneously fill the gas parcel with dust in 1D situations while nucleation events cause a more heterogeneous dust distribution due to their short life time. In more than 1D, also nucleation fronts cause a heterogeneous dust distribution because their ’parent’ waves will interact and influence the conditions for dust formation. Large variations in the dust quantities occur when the formation process starts to influence the hydro- and thermodynamics of the gas parcel. If all available material has been consumed, the mean dust quantities reach an almost constant values (Fig. 2) at a certain place in the atmosphere which is characterized by Tref . During this active time (∆t ≈ 3 s), the variation in the number of dust particles is O(105 cm−3 ) and in the mean particle size O(102 µm). The variation of the density is considerable (see standard deviations in Fig. 2) during the time of temperature decrease which assures the establishment of the pressure equilibrium. A convectively ascending, initially dust free gas element can be excited to form dust by waves running through it. A cloud can therefore be fully condensed at much higher temperatures than classically expected, i. e. in an undisturbed case.

Formation of large scale structures in 2D 2D simulations with the smallest eddy size being λ2D min = 5 m, the largest are of = 500 m, reveal a very intermittent distribution of the size of the gas box λ2D max the dust in space during the time of efficient dust formation (Fig. 3). The very inhomogeneous appearance of the dust complex is a result of nucleation fronts and nucleation events comparable to the 1D results. The nucleation is now

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Fig. 2. Fluctuating dust component in substellar atmospheres. The space-means α (t) (dotted/dashed) as α(t) x (solid) with the apparent standard deviations σN x −1 −4 −3 function of time for Tref = 2100K, ρref = 3.16 10 g cm , vref = cS /10. (α(t) x + α α (t) - dotted; α(t) x − σN (t) - dashed). σN x −1 x −1

triggered by the interaction of eddies coming from different directions. Large amounts of dust are formed and appear to be present in lane-like structures (large log nd ; dark/red areas in Fig. 3). The lanes are shaped by the constantly inward traveling waves. Our simulations show that some of the small scale structure merge thereby supporting the formation of lanes and later on even larger structures. Dust is also present in curled structure which indicates the formation of vortices due to the 2D waves driving turbulence. As the time proceeds in the 2D simulation, vortices develop orthogonal to the velocity field which show a higher vorticity (∇ × u(x, t)) than the majority of the background fluid field. Comparing the distribution of dust particles (false color background in Fig. 3) with the vorticity (contour lines in Fig. 3), shows that the vortices with high

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Fig. 3. Spatial appearance of the number of dust particles (log nd [cm−3 ]; false color background) and the vorticity (∇ × v; black and grey contour lines) of the 2D velocity ﬁeld for t = 0.8s of a simulation with Tref = 2100 K, ρref = 3.16 10−4 g cm−3 , vref = cS . (See also color ﬁgure, Plate 22.)

vorticity preferentially occur in dust free regions or regions with only little amounts of dust present. These vortices can efficiently drag the dust into regions with still material available for further condensation shaping thereby larger and larger structures.

4 Dust formation on large scales The atmospheric extension (≈ 100 km – several pressure scale heights Hp ) proposes a natural upper limit for the macroscopic scale regime. On this scale, the interplay between convective overshooting and gravitational settling (drift) is a major mechanism which influence the dust structure of the whole atmosphere. Since the dust is strongly coupled to the hydro- and thermodynamics of the atmosphere, gravitational settling and convective overshooting indirectly alter the atmospheric density and temperature structure which are important for the spectral appearance of the objects. The large scale structures are those which are usually accessible by observations. Therefore, the

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understanding and modeling of processes directly connected with macroscopic scale motions are of particular interest. 4.1 The model of a quasi-static dust layer A theoretical consistent description of dust formation, gravitational settling, and element depletion ((7) and (9)) was needed in order to understand the formation and the structure of dust cloud layers which is urgently needed for explaining the spectral appearance of Brown Dwarfs [WH03a]. A first application to the quasi-static case of a brown dwarf atmosphere calculation was carried out by providing a model of a quasi-static cloud layer, i.e. v gas = 0 and ∂Lj ∂t = 0 in (1) – (3), (7), (9) [WH03b]. However, if v gas = 0 Eqs. (7) have the trivial solution Lj ≡ 0 (see Sect. 2). The physical interpretation of this solution is that dust grains have once formed in the sufficiently cool layers, have consumed all available condensible elements up to the saturation level, and have finally left the model volume by gravitational settling. Consequently, a truly static atmosphere must be dust-free which – in this generality – contradicts the observations. This picture changes, however, if we take into account a mixing of the atmosphere caused by the convection. This mixing leads to an ongoing replenishment of the gas with fresh, uncondensed matter from the deep interior of the brown dwarf, which can counterbalance the loss of condensible elements via the formation and gravitational settling of dust grains. Thus, a convective mixing can maintain a stationary situation: Seed particles nucleate in metalrich, i. e. supersaturated upwinds. Dust particles grow on top of these nuclei by the accretion of molecules and rain out as soon as they have reached a certain size. The sinking grains will finally reach deeper atmospheric layers which are hot enough to cause their thermal evaporation, which completes the life cycle of a dust grain in a brown dwarf atmosphere. We therefore enlarge the static case of our model by a simple description of the convective mixing such that the stationary dust moment equations write j 1 ρLj d Lj+1 + Vℓ j/3 J⋆ + χnet ρL (11) = − − j−1 dz cT ξlKn τmix 3 lKn with algebraic auxiliary conditions R √ nH (ǫ0i − ǫi ) 1 3 rel = νi,0 Nℓ J⋆ + 36π ρL2 νi,r nr vr αr 1 − . τmix Sr r=1

(12)

According to this approach, the gas/dust mixture in the atmosphere is continuously replaced by dust-free gas of solar abundances on a depth-dependent mixing time-scale τmix (z), which can be adapted to a convection model (see discussion in [WH03b, WH03c]). ǫi is the actual abundance of element i in the gas phase and ǫ0i its solar value [AG89]. By fitting the mass exchange frequency of 3D dynamic simulations of surface convection by [LAH02] with an exponential we can apply a rough description of τmix (z).

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4.2 Numerical approach Equation (11) is a system of ordinary differential equations of first order which can be solved by standard numerical methods. The differential equations are integrated inward by means of the variable transformation z ′ = zmax − z using the Radau 5 – solver for stiff ordinary differential equations [HW91]). For test purposes, the equation of hydrostatic equilibrium dPgas /dz = −ρg is solved in additionto the three moment equations (11), where the actual gas pressure P = i ni kT results from the chemical equilibrium calculations. The integration is stopped as soon as one of the dust moments becomes negative, indicating that the dust has completely evaporated. 4.3 Structure of a quasi-static cloud layer The vertical structure of the cloud layer results from a competition between the four relevant physical processes: mixing, nucleation, growth/evaporation and drift. Following the cloud structures inward (from the left to the right in Fig. 4) roughly five different regions can be distinguished, marked by the Roman digits, which are characterized by different leading processes concerning the dust component. 0. Dust-poor depleted gas: High above the convection zone, the mixing time-scale is large and the elemental replenishment of the gas is too slow to allow for considerable amounts of dust to be present in the atmosphere. The few particles forming here are very small a < 10−2.5 µm and have drift velocities vdr ≈ 1 mm/s to 1 cm/s which causes these atmospheric layers to become dust-poor. The gas phase is strongly depleted in condensible elements. The Ti abundance in the gas phase ǫTi is reduced by a factor between 105 and 107 from its solar value. However, phase equilibrium (S = 1) is not achieved because even the very small disturbance of the atmosphere by mixing is sufficient to produce a solution which differs significantly from the trivial solution in the truly static case. I. Region of efficient nucleation: Nucleation takes place mainly in the upper parts of the cloud layer, where the temperatures are sufficiently low and the elemental replenishment by mixing is sufficiently effective. Although the gas is strongly depleted in heavy elements in these layers, it is nevertheless highly supersaturated (S > 1000) such that homogeneous nucleation can take place efficiently. Since very many seed particles are produced in this way, the dust grains remain small a < 0.01 µm and have small mean drift velocities vdr ≈ 0.1 mm/s ... 1 mm/s, which are even smaller than in region 0 because of the higher gas densities. II. Dust growth region: With the inward increasing temperature, the supersaturation ratio S decreases exponentially which leads to a drastic decrease of the nucleation rate J⋆ . Consequently, nucleation becomes unimportant at some point, i. e. the in-situ formation of dust particles becomes inefficient in

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Fig. 4. Calculated TiO2 cloud structures for models with log g = 5 and Teff = 1800 K. Note the scaling of the partly logarithmic and partly linear p-axis. 1st panel: prescribed gas temperature T (solid), according to Tsuji (2002), and mixing timescale τmix (dashed). 2nd panel: nucleation rate J⋆ (solid) and supersaturation ratio (solid) and in STiO2 (dashed). 3rd panel: Ti abundance in the dust phase ǫdust Ti the gas phase ǫTi (dashed-dotted). The solar value ǫ0Ti is additionally indicated by th panel: a thin straight line. The dashed line shows the growth velocity χnet lKn . 4 % 3 mean particle size a = 3/(4π) L1 /L0 (solid) and mean drift velocity vdr = √ πgρd a / (2ρcT ) (dashed, see Eq. (66) in Paper II). 5th panel: molecular particle densities of TiO (solid) and TiO2 (dashed). For comparison, the hypothetical total titanium nuclei density for solar abundances nH ǫ0Ti is depicted by a thin solid line.

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region II. Here, the condensible elements mixed up by convective overshoot are mainly consumed by the growth of already existing particles, which have formed in region I and have drifted into region II. The gas is still strongly supersaturated S ≫ 1, indicating that the growth process remains incomplete, i. e. the condensible elements provided by the mixing are not exhaustively consumed by growth. The dust component in region II is characterized by an almost constant degree of condensation (∝ ǫdust Ti ∝ L3 ≈ const), while the mean particle size a and the mean drift velocity vdr increase inward. Consequently, the total number of dust particles per mass (L0 ) and their total surface per mass (∝ L2 ) decrease. III. Drift dominated region: With the decreasing total surface of the dust particles (∝ L2 ), the consumption of condensible elements from the gas phase via dust growth becomes less effective. At the same time, due to the increase of the mean particle size a, the drift velocities increase. When the dust particles have reached a certain critical size, acr ≈ 15 µm to 50 µm, the drift becomes more important than the growth, and the qualitative behavior of the dust component changes. This happens at the borderline between region II and III, which we denote by rain edge. Although the gas is still highly supersaturated S ≈ 1 ... 10, the in-situ formation of dust grains is ineffective as in region II. The depletion of the gas phase vanishes in region III, i. e. the gas abundances approach close-to-solar values. The grains reach their maximum radii at the lower boundary of region III (the cloud base): a ≈ 30 µm to 400 µm at maximum drift velocities. IV. Evaporating grains: The gravitationally settling dust particles finally cross the cloud base and sink into the undersaturated gas situated below, where S < 1. Here, the dust grains raining in from above evaporate thermally. The evaporation of these dust particles, however, does not take place instantaneously, but produces a spatially extended evaporation region IV with a thickness of about 1 km. With decreasing altitude, the particles get smaller da/dz < 0 and slower dvdr /dz < 0. Consequently, their residence times increase, and a run-away process sets in which finally produces a very steep decrease of the degree of condensation, terminated by the point where even the biggest particles have evaporated completely. We note that region IV, in particular, cannot be understood by stability arguments, but requires a kinetic treatment of the dust complex.

5 Conclusions The dust formation in brown dwarfs atmospheres is a multi-scale problem where different physical mechanisms characterize the different scale regimes: a) In the microscopic scale regime, single acoustic waves interact, thereby initiate dust formation.

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b) The superposition of a whole spectrum of turbulence elements determine the mesoscopic scale regime resulting in a highly fluctuating fluid field. c) The interplay of gravitational settling of dust grains and the convective up-mixing of uncondensed material governs the macroscopic scale regime. The main scenario envisioned is a convectively ascending fluid element in a brown dwarf atmosphere, which is excited by turbulent motions and just reaches sufficiently low temperatures for condensation. The dust formation in a turbulent gas is found to be strongly influenced by the existence of a nucleation threshold temperature TS . The local temperature T must at least temporarily decrease below this threshold in order to provide the necessary supersaturation for nucleation, e.g. by eddy interaction. Depending on the relation between the local mean temperature T and TS , three different regimes can be distinguished (see l.h.s. of Fig. 5): (i) the deterministic regime (T < TS ) where dust forms anyway, (ii) the stochastic regime (T > TS ) where T < TS can only be achieved locally and temporarily by turbulent temperature fluctuations, and (iii) a regime where dust formation is impossible. The size of the stochastic regime depends on the available turbulent energy. This picture of turbulent dust formation is quite different from the usually applied thermodynamical picture (r.h.s. in Fig. 5) where dust is simply assumed to be present whenever T < Tsub , where Tsub is the sublimation temperature of a considered dust material. After initiation, the dust condensation process is completed by a phase of active particle growth until the condensible elements are consumed, thereby preserving the dust particle number density for long times. However, radiative cooling (as follow-up effect) is found to have an important influence on the

Fig. 5. Regimes of turbulent dust formation. TS : nucleation threshold temperature (supersaturation S ≪ 1 required), Tsub : sublimation temperature.

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subsequent dust formation, if the dust opacity reaches a certain critical value. This cooling leads to a decrease of T (t) which may re-initiate the nucleation. This results in a runaway process (unstable feedback loop) until radiative and phase equilibrium is achieved. Depending on the difference between the initial mean temperature T (t = 0) and the radiative equilibrium temperature TRE , a considerable local temperature decrease and density increase occurs. Since the turbulent initiation of the dust formation process is time-dependent and spatially inhomogeneous, considerable spatial variations of all physical quantities (hydro-, thermodynamics, dust) occur during the short time interval of active dust formation (typically a few seconds after initiation), which actually creates new turbulence. Thus, small turbulent perturbations have large effects in dust forming systems. Our 2D simulations show that the dust appears in lane-like and curled structures. Small scale dust structures merge and form larger structures. Vortices appear to be present preferentially in regions without or with only little dust. Non of these structures would occur without turbulent excitation. From our work on small-scale turbulence simulations of dust-hostile regions in substellar atmospheres, we compile necessary criteria for a subgrid model of a dust forming, turbulent system. Criteria on a subgrid model of a turbulent, dust-forming system: a) The subgrid model should describe the transition stochastic → deterministic regime in dust forming turbulent fluids. b) The dust formation process (nucleation + growth) is restricted to a short time interval since the dust formation time scales are much smaller than the large-scale hydrodynamic time scales. This involves that: • The nucleation does occur only locally and event-like in very narrow time slots. • The growth process continues as long as condensible material is available. • The condensation process freezes in and the inhomogeneous dust properties are preserved. • Almost constant characteristic dust properties result in the mean-long term behavior. c) The feedback loop with its fast radiative cooling should govern the transition from an almost adiabatic to an isothermal behavior of the dust/gas mixture. The largest, observable scale of a brown dwarf atmosphere has been investigated by attacking the drift problem. A consistent theoretical description was derived for dust formation and destruction, gravitational settling, and element depletion including the effect of convective overshoot. It was therewith

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possible for the first time to overcome the widely used concept of ad hoc dust presents in substellar atmospheres. Test calculations have shown that the dust will appear stratified in the atmosphere causing a corresponding depletion of the gas phase. Acknowledgement. This work has been supported by the DFG (grants SE 420/191,2; Kl 611/7-1; Kl 611/9-1) as part of the DFG-Schwerpunkt Analysis und Numerik von Erhaltungsgleichungen.

References [AG89]

E. Anders and N. Grevesse. Abundances of the elements: Meteoric and solar. Geochimica et Cosmochimica Acta, 53:197–214, 1989. [BJM99] C. A. L. Bailer-Jones and R. Mundt. A search for variability in brown dwarfs and L dwarfs. A&A, 348:800–804, 1999. [BJM01a] C. A. L. Bailer-Jones and R. Mundt. Eratum: Variability in ultra cool dwarfs: Evidence for the evolution of surface features. A&A, 374:1071, 2001. [BJM01b] C. A. L. Bailer-Jones and R. Mundt. Variability in ultra cool dwarfs: Evidence for the evolution of surface features. A&A, 367:218–235, 2001. [Cla03] F. Clarke. Variability in Ultra Cool Dwarfs. PhD thesis, Darwin College, Carmbridge, GB, 2003. [Deu83] P. Deuﬂard. Order and Stepsize Control in Extrapolation Methods. Numer. Math., 41:399–422, 1983. [DN87] P. Deuﬂhard and U. Nowak. Extrapolation Integrators for Quasilinear Implicit ODE’s. In P. Deuﬂhard and B. Engquist, editors, Large Scale Scientific Computing. Progress in Scientific Computing 7, pages 37–50. Birkh¨ auser, 1987. [ENOD99] E. Ehrig, U. Nowak, L. Oeverdieck, and P. Deuﬂhard. Advanced Extrapolation Methods for Large Scale Diﬀerential Algebraic Problems. In H.-J. Bungartz, F. Durst, and C. Zenger, editors, High Performance Scientific and Engineering Computing. Lecture Notes in Computational Science and Engineering, pages 233–244. Springer, 1999. [GM00] C. R. Gelino and M. S. Marley. Variability in an unresolved Jupiter. In C. A. Griﬃth and M. S. Marley, editors, From Giant Planets to Cool Stars, pages 322–330, 2000. [GS88] H.-P. Gail and E. Sedlmayr. Dust formation in stellar winds. IV. Heteromolecular carbon grain formation and growth. A&A, 206:153–168, 1988. [Hel03] Ch. Helling. Circuit of dust in substellar atmospheres. Reviews in Modern Astronomy, 19:114–131, 2003. [HKWN04] Ch. Helling, R. Klein, P. Woitke, and E. Nowak, U. Sedlmayr. Dust in Brown Dwarfs IV Dust formation and driven turbulence on mesoscopic scales. A&A, 423, 657-675, 2004. [HKWS03] Ch. Helling, R. Klein, P. Woitke, and E. Sedlmayr. Dust formation in brown dwarf atmosphers under conditions of driven turbulence. In N. E. Piskunov, W. W. Weiss, and D. F. Gray, editors, Modelling of Stellar Atmospheres, page in press. IAU Symp. 210, 2003.

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[HLSK01] Ch. Helling, M. L¨ uttke, E. Sedlmayr, and R. Klein. Dust formation in turbulent media. In Warnecke Freist¨ uhler, editor, Hyperbolic Problems: Theory - Numeric - Applications, pages 515–524. Birkh¨ auser, 2001. [HOL01] Ch. Helling, M. Oevermann, M.J.H. L¨ uttke, R. Klein, and E. Sedlmayr. Dust in Brown Dwarfs. I. Dust formation under turbulent conditions on microscopic scales . A&A, 376:194–212, 2001. [HW91] E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Springer, Berlin, 1991. [LAH02] H.-G. Ludwig, F. Allard, and P. H. Hauschildt. Numerical simulations of surface convection in a late M–dwarf. A&A, 395:99–115, 2002. [MZL01] E.L. Mart´ın, M.R. Zapatero Osorio, and H.J. Lehto. Photometric Variability in the Ultracool Dwarf BRI 0021–0214: Possible Evidence for Dust Clouds. ApJ, 557:822–830, 2001. [NEO98] U. Nowak, E. Ehrig, and L. Oeverdieck. Parallel Extrapolation Methods and Their Application in Chemical Engineering. In M. Sloot, P. amd Bubak and B. Hertzberger, editors, High-Performance Computing and Networking. Lecture Notes in Computer Science Vol. 1401, pages 419–428. Springer, 1998. [Sed02] E. Sedlmayr. Dynamical Stellar Dust Shells - diﬀerent perspectives. In C. Aerts, T. Bedding, and C. Christensen-Dalsgaard, editors, IAU Colloquium 185: Radial and Nonradial Pulsations as Probes of Stellar Physics, pages 524–533, Michigan, 2002. Astronomical Socity of the Paciﬁc. [SMK97] V. Smiljanovski, V. Moser, and R. Klein. A capturing/tracking hybrid scheme for deﬂagration discontinuities. Combustion Theory & Modelling, 1:183–215, 1997. [Str02] C.W. Straka. Thermonukleares Brennen und Mischen mit einer zeitanh¨ angigen Konvektionstheorie in masserecihen Population-IIISternen. PhD thesis, Ruprecht-Karls-Universit¨at Heidelberg, 2002. [WH03a] P. Woitke and Ch. Helling. Dust in Brown Dwarfs. II. The coupled problem of dust formation and sedimensation. A&A, 399:297–313, 2003. [WH03b] P. Woitke and Ch. Helling. Dust in Brown Dwarfs. III. Formation and structure of quasi-static cloud layers. A&A, accepted, 2003. [WH03c] P. Woitke and Ch. Helling. Formation and structure of quasi-static cloud layers in brown dwarf atmospheres. ZIB-Report, 3-11, 2003.

Meshless Methods for Conservation Laws D. Hietel1 , M. Junk2 , J. Kuhnert1 , and S. Tiwari1 1

2

Fraunhofer-Institut f¨ ur Techno-und Wirtschaftsmathematik, Gottlieb-Daimler-Straße, Geb. 49, 67663 Kaiserslautern, Germany Fachbereich Mathematik und Statistik, Universit¨ at Konstanz, Postfach D 194, 78457 Konstanz, Germany

Summary. In this article, two meshfree methods for the numerical solution of conservation laws are considered. The Finite Volume Particle Method (FVPM) generalizes the Finite Volume approach and the Finite Pointset Method (FPM) is a Finite Diﬀerence scheme which can work on unstructured and moving point clouds. Details of the derivation and numerical examples are presented for the case of incompressible, viscous, two-phase ﬂow. In the case of FVPM, our main focus lies on the derivation of stability estimates.

1 Introduction Meshfree techniques play an increasing role as solution methods for conservation laws. Practically, all meshfree methods are based on clouds of points, where each point carries the relevant information for the problem to be solved. One of the biggest advantages of these methods is that no mesh has to be established. Generating a grid can be very costly, sometimes it is even the dominating part in the problem. Compared to that, it is relatively simple to establish a point cloud, even within very complex geometries. Moreover, the cloud is easy to maintain or to adapt locally. Due to the free movement of the points, an optimal adaptivity of the cloud is provided towards changes in the geometry or towards movement of free surfaces as well as phase boundaries. Among the pioneering meshfree methods, Smoothed Particle Hydrodynamics (SPH) is certainly the most famous [10]. SPH is a Lagrangian idea, which is based on the movement of finite mass points. However, for a long time, SPH was suffering from several problems, among them stability and consistency. Facing these problems, the development of meshfree methods went into various directions, starting in the early nineties. On one hand, people tried to improve SPH. One idea was to avoid inconsistency problems of SPH by reproducing kernel methods [30], another idea was to improve the approximation properties of SPH by the introduction of so called Moving Least Squares (MLS) ideas [5, 15]. On the other hand, many new types of meshfree

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methods were developed. Widely used methods are the Element Free Galerkin (EFG) idea [23] or the Partition of Unity Method (PUM) [8]. As the EFG and PUM ideas provide the possibility to carry out Finite Element computations on gridfree structures, the present paper introduces meshfree Finite Volume (FV) and Finite Difference (FD) concepts. In section 2, we present the Finite Volume Particle Method (FVPM), which incorporates FV ideas into a meshfree framework [7]. In particular, the approach uses the concept of numerical flux functions and guarantees conservation on a discrete level. As we show in section 2, the similarity to classical Finite Volume schemes allows us to derive stability results using standard arguments. For numerical results obtained with FVPM, we refer to [7, 12, 24, 31]. In section 3, we concentrate on the Finite Pointset Method (FPM) which is a general Finite Difference Method for conservation laws on a meshfree basis [15, 16]. The latest FPM development which covers the discretization of the incompressible Navier-Stokes equations with multiple phases is presented.

2 The Finite Volume Particle Method 2.1 Derivation The Finite Volume Particle Method (FVPM) has been developed in an attempt to combine features of SPH (Smoothed Particle Hydrodynamics) with Finite Volume Methods (FVM) [7]. To explain the idea, we consider the problem to find a function u : [0, T ] × Rd → R which satisfies ∂u + divx F (u) = 0, ∂t

u(0, x) = u0 (x).

(1)

In the classical Finite Volume Method, conservation laws like (1) are discretized by introducing a Finite Volume mesh on Rd and integrating (1) over each volume element (see, for example, [6, 14]). To reformulate this procedure on a more abstract level, we note that a mesh gives rise to a particular partition of unity which is generated by the indicator functions of the elements. d More specifically, if Rd is the disjoint union of mesh cells Ci ⊂ R , then the functions χi = 1Ci have the property χi ≥ 0, and i χi = 1 where the sum is locally finite because at every x ∈ Rd , at most one function χi is nonzero (the one for which x ∈ Ci ). The Finite Volume scheme is then obtained by taking χi as test functions in a weak formulation of (1). In this context, the basic idea in FVPM is to choose a smooth partition of unity instead of a mesh-based partition. Let us therefore assume that a smooth, locally finite partition of unity {ψi }i∈I is given on Rd , i.e.smooth and compactly supported functions ψi : Rd → R with ψi ≥ 0 and i ψi = 1 where ψi (x) = 0 for only finitely many indices at every x ∈ Rd (the functions ψi are called particles). Such a partition can be constructed using a shape function W which is smooth, compactly supported around the origin, and strictly positive on its support,

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for example, a radially symmetric cubic spline, or the d-fold tensor product of a one-dimensional compactly supported function (in the first case, supp W is a d-dimensional ball, and in the second case, the support is an axis parallel cube). Then given a suitable set of points xi ∈ Rd , i ∈ I and some h > 0, we define scaled and shifted versions of W whose supports cover Rd 8 x − xi Wi (x) = W supp Wi = Rd . (2) , h i Finally, using Shephard’s method [22], the partition of unity is built ψi (x) =

Wi (x) , σ(x)

σ(x) =

Wk (x),

k

x ∈ Rd

(3)

and the partition functions ψi are used as test functions for equation (1). Multiplying (1) with ψi and integrating over Rd , we obtain after integration by parts d ψi u dx − F (u) · ∇ψi dx = 0. (4) dt Rd Rd

In order to split the flux integral into pairwise flux contributions between ψ particle i and its neighboring particles j, we use the fact that j j = 1 and ∇( j ψj ) = 0 which leads to d ψi u dx − F (u) · (ψj ∇ψi − ψi ∇ψj ) dx = 0. (5) dt Rd Rd j The first integral is obviously related to a local average of u 1 Vi = ψi u dx|t=tn , uni = ψi dx. Vi Rd Rd

(6)

To approximate the second integral in (5), we note that, if u varies only slightly around u ¯ on the intersection of the supports of ψi and ψj , we have F (u) · (ψj ∇ψi − ψi ∇ψj ) dx ≈ F (¯ u) · β ij , − j

Rd

where β ij =

Rd

ψi ∇ψj − ψj ∇ψi dx.

(7)

Following the usual procedure in the Finite Volume discretization we approximate the flux u) · nij , F (¯ u) · β ij = |β ij |F (¯

nij =

β ij if |β ij | = 0 |β ij |

in terms of the discrete values with the help of a numerical flux function

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F (¯ u) · nij ≈ g(ui , uj , nij ). Finally, using an explicit Euler discretization of the time derivative, we obtain a fully discrete approximation of (5) 1 n+1 n n n 0 ui Vi = ui Vi − ∆t |β ij |g(ui , uj , nij ), ui = ψi u0 dx. (8) V d i R j Observe that (8) has exactly the structure of the classical Finite Volume Method [6, 14]. The only difference appears in the definition of the geometric parameters Vi and β ij . While they are integral expressions involving the partition of unity functions ψi in the case of FVPM, they are mesh related quantities in the classical Finite Volume case: Vi is the volume of cell Ci and β ij = |Γij |nij is the product of interface area |Γij | and interface normal vector nij (see figure 1). In other words, FVPM is very similar to FVM – only grid

Ci Γij nij Cj

Fig. 1. Control volume Ci with interfaces Γij and outer normals nij .

generation is replaced by integration. From the update rule in (8) it is clear that the value ui at particle i is only influenced by those values uj for which β ij = 0. In view of (7), this can only happen if the particles i and j overlap. Hence, it is natural to call ψi and ψj (interacting) neighbors if β ij = 0. In particular, the sum in (8) only involves the particles from the neighbor list Ni = {j : β ij = 0}. Note also that particle i does not interact with itself since β ii = 0 according to (7). Before investigating the scheme (8) more closely, let us remark that the derivation works similarly in the case when the partition of unity is time dependent (moving particles). For simplicity, we assume here that the time dependence is such that ψi satisfy the advection equation ∂ψi + a · ∇ψi = 0 ∂t

(9)

where a is a smooth and bounded velocity field. Note that {ψi (t, ·)}i∈I is automatically a partition of unity if this is true initially. However, the parameters

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Vi and β ij now depend on time. If we use these functions ψi as test functions, the time derivative gives rise to an additional term ∂u d + divF (u) dx = ψi ψi u dx − (F (u) − ua) · ∇ψi dx ∂t dt Rd Rd Rd which is of the same form as (4) if F (u) is replaced by the Lagrangian flux G(t, x, u) = F (u)−ua(t, x). Another possibility to generate a time dependent partition of unity is to move the points xi according to the vector field a and to continuously apply Shephard’s method to Wi (x) = W ((x − xi )/h). Also in this case, the modifications can be incorporated into the flux function by going over from F to G (for details, we refer to [11]). In the following, we study mathematical properties of the Finite Volume Particle Method. For numerical results, we refer to [7, 12, 24, 31]. 2.2 Geometric parameters Inspecting the convergence proof for classical Finite Volume schemes (e.g. [1, 3, 14, 29]), one observes that there are only a few requirements on the geometric parameters β ij , Vi : the number of cell faces should be bounded, the cell surface area j |β ij | should be of the order hd−1 , and the cell volumes Vi should be bounded from below by αhd . Moreover, there are two important algebraic requirements. The first one reflects the fact that, if two cells Ci , Cj share a common interface, then the surface area |β ij | = |β ji | is equal but the orientation β ij /|β ij | = −β ji /|β ji | is opposite. The second requirement is related to the divergence theorem. If β ij is the product of interface area and interface normal, then the sum j β ij is nothing but the integral of the normal vector over the surface of cell Ci . According to the divergence theorem, this surface integral can be written as a volume integral of the divergence of constant fields so that j β ij = 0. In the following, we show that the same conditions are satisfied if the geometric parameters β ij , Vi are not based on a mesh but are obtained by integration from a partition of unity. As a consequence, many aspects of convergence proofs for classical Finite Volume schemes can be taken over without modification (examples are given in the following section). We start by a precise statement of our assumptions concerning the partition of unity. (P1) The initial partition of unity is constructed according to (2), (3), based on a locally finite point distribution {xi }. We assume that the maximal number of overlapping particles is bounded, i.e. max #{j : supp Wi ∩ supp Wj = ∅} = K < ∞. i∈I

Moreover, there should be a minimal overlap such that Wi (x) ≥ µ > 0 ∀x ∈ Rd . i∈I

(10)

(11)

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If we consider a sequence of partitions based on Wih (x) = W ((x−xhi )/h) with h → 0, we assume that K and µ are h-independent. (P2) The functions ψi (t, ·), t > 0 are obtained by solving (9) with initial values given by the functions of the initial partition. The field a is assumed to be bounded and continuously differentiable with bounded derivatives. Using standard results from the theory of ordinary differential equations, ˙ we conclude that the initial value problem x(t) = a(t, x(t)), x(τ ) = ξ admits a unique global solution which we denote by t → X(t, ξ, τ ). The function X is smooth and invertible with respect to the ξ-variable, where X(t, X(τ, y, t), τ ) = y

∀y ∈ Rd .

The Jacobian determinant of X(t, ξ, τ ) is given by t J(t, ξ, τ ) = exp (diva)(s, X(s, ξ, τ )) ds . τ

According to the method of characteristics, the solution of (9) satisfies ψi (t, x) = ψ(0, X(0, x, t))

(12)

which immediately implies that {ψi (t, ·)}i∈I is a partition of unity. We want to stress again that the restriction to the construction (9) is not compulsory. It only helps us to avoid additional technical arguments and assumptions which are necessary, for example, if we use moving points xi and construct the partition from the functions Wi which have moving supports of fixed shape. Also in that more technical situation, the following result can be shown. Proposition 1. Assume (P1) and (P2) and let Vi (t), β ij (t) be defined by (6) and (7) based on the partition functions {ψi (t, ·)}i∈I . Then there exist constants α, K, C > 0 such that for t ∈ [0, T ] the number of neighbors is bounded #Ni (t) ≤ K and Vi (t) ≥ αhd , |β ij | ≤ Chd−1 . (13) j

Moreover, the relations β ij = −β ji ,

i, j ∈ I

β ij = 0,

j

(14)

are satisfied. Proof. The algebraic conditions (14) follow directly from the skew-symmetric definition of β ij .;The vanishing sum over β ij is a consequence of the fact that ψ = 1, and ∇ψi dx = 0 j j ∇ψi dx = 0. ψj dx − ψi ∇ β ij = j

Rd

j

Rd

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While these arguments do not depend on the structure of the partition, the estimates (13) require more details. We begin with the investigation of the volume. ψi (t, x) dx = ψi (t, X(0, x, t)) dx = ψi (t, y)J(t, y, 0) dy. Vi (t) = Rd

Rd

Rd

If m is a lower bound for diva, then J(t, y, 0) ≥ exp(tm) and it suffices to estimate ψi (0, y) = ψi0 (y) from below. In view of (10) and (12), the maximal number of overlapping particles is K (which also proves the estimate of #Ni ) and in connection with (11), we conclude Wk (y) ≤ KW ∞ , ∀y ∈ Rd . (15) µ ≤ σ(y) = k

Continuing the estimate of Vi , we have Wi (y) exp(T m) d dy ≥ h W (z) dz = αhd Vi (t) ≥ exp(tm) KW ∞ Rd σ(y) Rd Next, we turn to the estimate ; of the sum over |β ij |. Allying integration by parts to (7), we have β ij = 2 ψj ∇ψi dx, so that |∇ψi | dx. |∇ψi | dx ≤ 2 |β ij | ≤ 2 Rd

Rd

j

Using chain rule and the same change of coordinates as above, we obtain |β ij | ≤ 2 |(∇X)T | |∇ψi0 |J dy. j

Rd

Gronwall’s lemma implies that the derivatives of X can be bounded in terms of the derivatives of a. If |∇a| ≤ M then |(∇X)T | ≤ exp(tM ) and J ≤ exp(tM d). Hence, it suffices to estimate the integral over |∇ψi0 |. Definition (3) implies ∇Wk ∇Wi 0 − ψi k , ∇ψi = σ σ so that with (15) 2 d−1 2 0 |∇W (z)| dz |∇W |(y/h) dy = h |∇ψi | dy ≤ hµ Rd µ Rd Rd and hence

j

|β ij | ≤ exp((d + 1)T M )

This completes the proof.

2(K + 1) d−1 h . µ ⊓ ⊔

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2.3 Stability The aim of this section is to demonstrate that local averages uni generated with FVPM can be estimated in the same way as in the mesh based Finite Volume approach. Our main assumption is that the flux function g is monotone. Note, however, that we allow for moving particles so that g should be an approximation of the Lagrangian flux F (u) − ua(t, x). Consequently, the flux between two particles will also depend on their location and time. Since the dependence on location can be introduced in several ways (see below), we do not specify it at this stage. More precisely, we incorporate the possible movement of the particles into (8) by assuming the form Vin+1 = uni Vin − ∆t Gnij (uni , unj ) (16) un+1 i j

with initial values

1 = 0 u0 (x)ψ(0, x) dx. Vi Rd where the flux function Gnij should satisfy the following requirements. (Z) Local interaction is ensured by u0i

Gnij (u, v) = 0 (S)

for j ∈ Nin .

The flux function should be antisymmetric in the following sense Gnij (u, v) = −Gnji (v, u).

(C) The relation to the Lagrangian flux is ensured by Gnij (u, u) = F (u) · β nij − uAnij · β nij

where Anij approximates a on the support of ψi |Anij − a(tn , xni )| ≤ ch,

j ∈ Nin

(17)

where xni is the barycenter of particle i 1 n xi = n xψ(tn , x) dx. Vi Rd

(L) The flux function should be locally Lipschitz continuous, i.e. for a ≤ u, v, w ≤ b we assume |Gnij (u, w) − Gnij (v, w)| ≤ |β ij |L(a, b)|u − v|.

(M) Finally, we assume monotonicity in the arguments of Gnij . For u = v Gnij (u, w) − Gnij (v, w) ≥ 0, u−v

Gnij (w, u) − Gnij (w, v) ≤ 0. u−v

In order to prove a conservation property of (16) we assume that the flux F (u) in (1) is defined for u = 0 (which can always be achieved by a simple transformation).

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Proposition 2. Assume (Z), (S), (C). If the initial value u0 of (1) has a compact support then uni = 0 for only finitely many indices i and Vin+1 = un+1 uni Vin . i i

i

Proof. According to (P1), the point distribution {xi } is locally finite which implies that the number of particles ψi (0, ·) whose support intersect with the particular, only finitely many average values u0i are one of u0 is finite. In non-zero and the sum i u0i Vi0 is well defined. Considering a particle i with vanishing average u0i = 0 and also u0j = 0 for all neighbors j ∈ Ni0 , we have u1i Vi1 = −∆t

j

G0ij (0, 0) = −∆t

, j

F (0) · β nij − 0 · Anij · β nij

-

where (C) has been used. Since the sum over β ij vanishes according to (14), we see that u1i = 0. This situation occurs for all but finitely many particles and hence u1i = 0 only finitely often so that i u1i Vi1 is well defined. Summing over (16), we get i

u1i Vi1 −

i

u0i Vi0 = −∆t

G0ij (u0i , u0j )

i,j

⎛

⎞ 1 1 = −∆t ⎝ G0 (u0 , u0 ) + G0 (u0 , u0 )⎠ . 2 i,j ij i j 2 j,i ji j i

Using (S), we conclude that the right hand side vanishes which yields conservation. Using an induction argument, the general statement follows. ⊓ ⊔ The second important property is monotonicity of the scheme. Proposition 3. Assume (L), (M) and let a ≤ uni , win ≤ b. If for some ξ ∈ (0, 1) the CFL condition ∆t ≤ min i∈I

is satisfied then un+1 ≤ win+1 . i

ξVin L(a, b) j |β ij |

Proof. Assuming first that uni = win , we obtain from (16) , Gnij (uni , wjn ) − Gnij (uni , unj ) )Vin+1 = −∆t (win+1 − un+1 i ⎛

+ ⎝Vin − ∆t

j

Gnij (win , wjn ) − Gnij (uni , wjn )

Using the Lipschitz property

j

win − uni

⎞

⎠ (win − uni )

(18)

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Vin − ∆t

Gnij (win , wjn ) − Gnij (uni , wjn ) j

win − uni

⎛

⎞ ∆t ≥ Vin ⎝1 − n |β ij |L(a, b)⎠ Vi j

so that the second term on the right of (18) is positive if the CFL condition is satisfied. The positivity of the first term follows immediately from (M). In the case uni = win , the second term is not present and non-negativity follows from the first term, again with the help of (M). ⊓ ⊔ An immediate consequence of monotonicity is L∞ -stability. Due to the possible movement of the particles, however, the maximum may increase in time. Proposition 4. There exist h-independent constants γ1 , γ2 such that under the assumptions of Proposition 2 together with (C) and the additional restriction ∆t ≤ α/(2cC), where α, C are from (13) and c from (17), we have ≤ exp(γ1 ∆t) max uni max un+1 i

if max uni ≥ 0,

≤ exp(γ1 ∆t) min uni min un+1 i i∈I i∈I ≤ exp(γ ∆t) max uni max un+1 2 i i∈I i∈I ≤ exp(γ2 ∆t) min uni min un+1 i i∈I i∈I

if min uni ≤ 0,

i∈I

i∈I

i

i

if max uni ≤ 0, i

if min uni ≥ 0. i

≤ Proof. Using Proposition 2 with win = maxj unj = M ≥ 0, we obtain un+1 i win+1 which yields the desired estimate if win+1 ≤ exp(γ1 ∆t)M . Since win = M , we obtain with (C) , F (M ) · β nij − M Anij · β nij win+1 Vin+1 = M Vin − ∆t ⎛

j

⎞ ∆t = Vin ⎝1 − n Anij · β nij ⎠ M. Vi j

The extra information about Anij in (C) implies # # # # # # ∆t n ∆t ## n ∆t ## n# n# n |β ij | Aij · β ij # ≤ n # a(tn , xi ) · β ij # + n ch n # Vi j Vi j Vi j

where the first term on the right vanishes because of (14). We conclude with (13) # # # cC ∆t ## n n# ∆t. · β A ij # ≤ ij n # Vi j α

Due to the movement of the particles, ψi (tn+1 , x) = ψi (tn , X(tn , x, tn+1 ) and since the Jacobian determinant follows the estimate J(tn+1 , bf y, tn ) ≥ exp(δ∆t) with δ being the minimum of diva, we have

Meshless Methods for Conservation Laws

Vin+1 =

Rd

349

ψi (tn , y)J(tn+1 , y, tn ) dy ≥ exp(δ∆t)Vin .

Combining these results and setting γ1 = cC/α − δ, we finally get win+1 ≤ exp(γ1 ∆t)M . Similarly, we can estimate the behavior of the minimum m = ≥ mini uni ≥ 0. Setting win = m, we conclude from uni ≥ win that also un+1 i win+1 , and as above, cC ∆t exp(δ∆t)m. win+1 ≥ 1 − α Setting D = α/(2cC), we obtain 1 − ∆t/(2D) ≥ exp(− ln(2)∆t/D) for 0 ≤ ∆t ≤ D so that the result follows with γ2 = δ − ln(2)/D. The remaining cases where maximum or minimum are negative can be shown in the same way. ⊓ ⊔. We remark that no exponential factors appear in the estimates for maximum and minimum if the volumes are approximately calculated from the formula ⎞ ⎛ ∆t Anij · β nij ⎠ . Vin+1 = V1n ⎝1 + n Vi j

This is an approximation to the true volume evolution because the sum over Anij · β nij can be regarded as approximation of diva if Anij is an evaluation of a at a suitable point xnij (see [12] for details). Apart from L∞ -estimates, the monotonicity can also be used to derive a discrete Krushkov-entropy estimate. Moreover, a weak BV-estimate can be shown along the lines of the proof in [1]. We will not go into further details here but conclude with some comments on the construction of the flux function Gnij . A particular example is based on the Lax-Friedrichs flux function Gnij (u, v) =

v F (u) + F (v) n u − v u ·β ij + − a(tn , xni )·β nij − a(tn , xnj )·β nij (19) 2 2λ 2 2

where xni , xnj are, for example, the barycenters of particles i and j at time tn . If F is locally Lipschitz continuous and if the parameter λ is chosen such that λ|F ′ (u) − a(t, x)| ≤ 1 for a ≤ u ≤ b, t ∈ [0, T ], and x ∈ Rd , then definition (19) satisfies the requirements (Z), (S), (C), (L), (M). More generally, if g0 (u, v, n) is any monotone and locally Lipschitz continuous numerical flux function which is consistent to F , then M + a(tn , xnj ) M − a(tn , xni ) −v Gnij (u, v) = |β nij |g0 (u, v, nnij ) + u · β nij 2 2 satisfies all the requirements on the numerical flux function if M ≥ |a(t, x)| for any t ∈ [0, T ], and x ∈ Rd . Instead of using two separate points, one can also base the construction on a common point xnij for each pair of particles such that xnij = xnji which replaces xni and xnj in the construction above.

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We close with the remark that a full convergence proof of the scheme (16) does not automatically follow from the estimates of the local averages uni . This is due to the fact that the reconstruction of an approximate solution uh from the discrete values is not so straightforward as in the classical Finite Volume case, where one sets (20) uni 1Ci (x)1[tn ,tn+1 ) (t). uh (t, x) = i∈I

The intuitive choice uh (t, x) =

uni ψi (x)1[tn ,tn+1 ) (t)

(21)

i∈I

seems to be promising because uh is bounded in L∞ by the maximum of |uni | and the reconstruction is conservative uh (tn , x) dx = uni Vin = u0i Vi0 = u0 (x) dx Rd

i

i

Rd

; (although locally uh (tn , x)ψi (tn , x) dx = uni Vin , in general). The main difficulty with the reconstruction (21) in the convergence proof is that it does not commute with nonlinear functions. While in the FV approach (20) the flux of the reconstruction is equal to the reconstruction of the fluxes, F (uni )1Ci (x)1[tn ,tn+1 ) (t), F (uh (t, x)) = i∈I

a similar relation is not available for the reconstruction (21). Therefore, the estimates in terms of uni cannot immediately be used for the reconstruction and further analysis is required for a complete convergence proof.

3 The Finite Pointset Method As we have seen in section 2.1, the Finite Volume Particle Method can be interpreted as a discretization of the weak form of conservation laws. The only difference to the classical Finite Volume Method is that the test functions are not constructed from mesh cells but are chosen as smooth partition of unity functions. The flux-divergence divF is approximated with the help of geometric parameters which depend on the relative location of the particles (similar to the classical Finite Volume Method where divF is replaced by an approximate surface integral over the flux – reflecting the weak formulation). While the weak approximation has the advantage of ensuring discrete conservation and stability, it also requires a large numerical effort for the evaluation of the geometric parameters and is, in its basic form, of low approximation order.

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In the Finite Pointset Method (FPM) these advantages and disadvantages are essentially reversed. The method is based on the strong form of the equation and therefore has the basic flavor of a Finite Difference scheme. Derivatives are approximated using a least squares approach which requires only little information about the relative location of the particles and easily allows high order approximations. However, as a consequence of the higher geometrical flexibility, discrete conservation cannot be proved (there is no volume associated to the particles and therefore integral values are not defined as naturally as in the Finite Volume approach), and also stability estimates are not available through standard techniques, even though the method has been successfully applied to a wide variety of problems. In the following, we describe the adaption of the method to the case of incompressible, viscous, multiphase flows. 3.1 Mathematical model and numerical scheme We consider two immiscible fluids, for example, liquid and gas in a situation where the motion can be described by the incompressible Navier-Stokes equations. They are, in Lagrangian form, 1 1 Dv = − ∇ p + ∇ · (2µD) + g, Dt ρ ρ ∇·v = 0

(22) (23)

where v is the fluid velocity vector, ρ is the fluid density, µ is the fluid viscosity, D is the viscous stress tensor D = 12 (∇v + ∇T v) and g is the body force acceleration vector. For a numerical simulation of the process, we assume that the flow domain is filled by a set of discretization points (point cloud). Each point (also referred to as particle) locally represents a lump of fluid. The points carry all necessary pieces of information (state variables) in order to completely describe the local state of flow, such as density, velocity, pressure, etc. The particles move exactly with fluid velocity, and, of course, the task of the numerical scheme is to evolve the state variables at the particle locations in an accurate way. The point cloud itself may be locally adaptive, i.e. the mean distance between points may change locally due to certain criteria. That also means that, during computation, new particles have to be created in sparse regions and removed in dense areas. In this paper, we initially provide each particle a flag representing the fluid phase it belongs to. These flags do not change throughout the computation. Moreover, the density and viscosity are constant on each particle path, so we have ∂ρ + v · ∇ρ = 0 ∂t ∂µ + v · ∇µ = 0. ∂t

(24) (25)

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Therefore, each fluid particle has constant ρ and µ. Since ρ and µ are discontinuous across the interface, the numerical scheme might have instabilities in that particular region. Therefore, we consider a smooth density and viscosity in the vicinity of an interface. The interface region can be detected by checking the flags of particles in the neighborhood. If there are flags of only one type in the neighbor list of some particle, then it is considered to be far from the interface region. Near the interface, particles will find both types of flags in the neighbor list. We modify the density and viscosity in each time step at each particle position x near the interface by using the Shepard interpolation n wi ρi (26) ρ˜(x) = i=1 n i=1 wi n wi µi µ ˜(x) = i=1 , (27) n i=1 wi where n is the total number of neighbor particles xi at x inside the compact support of the weight function wi which is defined by (38). Occasionally, the smoothing procedure of density and viscosity has to be repeated several times in order to gain stability. Equations (22–23) are solved together with some initial and boundary conditions. Numerical scheme Since the viscosity is smoothed near the interface, we can rewrite the momentum equations whose spatial components are given by du 1 µ ˜ 1 ∂µ ˜ ∂u ∂ µ ˜ ∂v ∂µ ˜ ∂w 1 ∂p = gx − + ∇˜ µ · ∇u + ∆u + ( + + ) dt ρ˜ ∂x ρ˜ ρ˜ ρ˜ ∂x ∂x ∂y ∂x ∂z ∂x dv 1 ∂p 1 µ ˜ 1 ∂µ ˜ ∂u ∂ µ ˜ ∂v ∂µ ˜ ∂w = gy − + ∇˜ µ · ∇v + ∆v + ( + + ) dt ρ˜ ∂y ρ˜ ρ˜ ρ˜ ∂x ∂y ∂y ∂y ∂z ∂y dw µ ˜ 1 ∂µ ˜ ∂u ∂ µ ˜ ∂v ∂ µ ˜ ∂w 1 ∂p 1 = gx − + ∇˜ µ · ∇w + ∆w + ( + + ). dt ρ˜ ∂z ρ˜ ρ˜ ρ˜ ∂x ∂z ∂y ∂z ∂z ∂z To ensure incompressibility, we use Chorin’s projection method [2]. Since we consider the fully Lagrangian method, we first move particles with their old velocities. The new positions are given by xn+1 = xn + ∆t v n . At each new particle position we modify the density and viscosity according to (26) and (27), respectively and then compute the intermediate velocities u∗ , v ∗ and w∗ implicitly by

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353

u∗ −

∂v n ∆t ∆t (∇˜ µ · ∇u∗ − µ ˜∆u∗ ) = un + ∆t gx + ∇˜ µ· ρ˜ ρ˜ ∂x

(28)

v∗ −

∂v n ∆t ∆t (∇˜ µ · ∇v ∗ − µ ∇˜ µ· ˜∆v ∗ ) = v n + ∆t gy + ρ˜ ρ˜ ∂y

(29)

∂v n ∆t ∆t (∇˜ µ · ∇w∗ − µ ∇˜ µ· . ˜∆w∗ ) = wn + ∆t gz + ρ˜ ρ˜ ∂z

(30)

w∗ −

Then, at the second step, we correct v ∗ = (u∗ , v ∗ , w∗ ) by solving the equation ∇pn+1 (31) v n+1 = v ∗ − ∆t ρ˜ for pn+1 , with the incompressibility constraint ∇ · v n+1 = 0.

(32)

By taking the divergence of equation (31) and by making use of (32), we obtain the Poisson equation for the pressure n+1 ∇p ∇ · v∗ ∇· . (33) = ρ˜ ∆t The boundary condition for p is obtained by projecting the equation (31) on the outward unit normal vector n to the boundary Γ . Thus, we obtain the Neumann boundary condition n+1 ∂p ρ˜ − v ∗Γ ) · n, (34) = − (v n+1 ∂n ∆t Γ where v Γ is the value of v on Γ . Assuming v · n = 0 on Γ , we obtain n+1 ∂p =0 ∂n

(35)

on Γ . We note that particle positions change only in the first step. The intermediate velocity v ∗ is obtained on the new particle positions. The pressure Poisson equation and the divergence free velocity vector are also computed on the new particle positions. We solve the above implicit equations (28–30) for the velocity vector v ∗ and pressure (33) by the constraint weighted least squares method which is described in the following section. 3.2 FPM for solving general elliptic partial differential equations To generalize the pressure Poisson problem, we consider the following linear partial differential equation of second order

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Aψ + B · ∇ψ + C∆ψ = f,

(36)

where A, B, C and f are given. Note that for the pressure Poisson equation (33), we have A = 0. The equation is solved with Dirichlet or Neumann boundary conditions ∂ψ = φ. (37) ψ=g or ∂n We use the method proposed in [26] to solve the elliptic equation in a meshfree framework. According to [9], it is more stable than the approach presented in [17] and it easily handles Neumann boundary conditions. Consider the computational domain Ω ∈ Rd , d ∈ {1, 2, 3}. Distribute N particles xj ∈ Ω, j = 1, · · · , N , which are the discretization points and might be irregular. Let x be an arbitrary particle in Ω, and we determine its neighboring cloud of points. We introduce the weight function w = w(xi − x, h) with small compact support of radius h. The weight function can be arbitrary but in our computation, we consider a Gaussian weight function of the following form 0 2 exp(−α xih−x ), if xih−x ≤ 1 2 (38) w(xi − x; h) = 0, else, where α is a positive constant. Usually we choose α = 6.25 but in case of Shepard interpolation α = 2. The size of h defines a set of neighboring particles around x. Let P (x, h) = {xi : i = 1, 2, . . . , m} be the set of m neighboring points of x in a ball of radius h. A basic idea in FPM is to construct approximate derivatives from point values using a least squares approach [15, 16, 25, 27, 28]. This leads to some obvious restrictions on the number and the location of points in P (x, h). Since we consider general second order equations, all derivatives up to order two (and the function value itself) should be constructable so that, in 3D, a minimal requirement is that P (x) contains at least ten points. To explain the least squares approach, consider the Taylor expansions of ψ(xi ) around x ψ(xi ) =

|j|≤2

∂ψ |j| 1 (xi − x)j1 (yi − y)j2 (zi − z)j3 + ei , ∂xj1 ∂y j2 ∂z j3 j!

(39)

for i = 1, . . . , m, where ei is the truncation error (we expand to second order for simplicity – higher order expansions are, of course, possible). Denote the coefficients a0 = ψ(x), a5 =

∂2ψ , ∂x∂y

a1 =

∂ψ , ∂x

a2 =

∂ψ , ∂y

a3 =

∂ψ , ∂z

a4 =

∂2ψ , ∂x2

a6 =

∂2ψ , ∂x∂z

a7 =

∂2ψ , ∂y 2

a8 =

∂2ψ , ∂y∂z

a9 =

∂2ψ . ∂z 2

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355

To the m equations (39) we add the equations (36) and (37) which are reexpressed as Aa0 + B1 a1 + B2 a2 + B3 a3 + C(a4 + a7 + a9 ) = f nx a1 + ny a2 + nz a3 = φ, where nx , ny , nz are the spatial components of the unit normal vector n on the boundary Γ . Note that, for the Dirichlet boundary condition, we have only m + 1 equations, where we directly prescribe the boundary conditions on the boundary particles. Now, we have to solve m + 2 equations. For m + 2 > 10 this system is over-determined with respect to the unknowns ai and can be written in matrix form as e = M a − b, where ⎛

1 h1,1 ⎜ .. .. ⎜. . ⎜ M = ⎜ 1 h1,m ⎜ ⎝ A B1 0 nx

⎞ h2,1 h3,1 12 h21,1 h1,1 h2,1 h1,1 h3,1 12 h22,1 h2,1 h3,1 12 h23,1 .. .. .. .. .. .. .. .. ⎟ . . . . . . . . ⎟ ⎟ 1 2 1 2 1 2 , h2,m h3,m 2 h1,m h1,m h2,m h1,m h3,m 2 h2,m h2,m h3,m 2 h3,m ⎟ ⎟ ⎠ B2 B3 C 0 0 C 0 C ny nz 0 0 0 0 0 0 T

T

T

wi e2i ,

(40)

a = (a0 , a1 , . . . , a9 ) , b = (ψ1 , . . . , ψm , f, g) , e = (e1 , . . . , em , em+1 , em+2 ) and h1,i = xi − x, h2,i = yi − y, h3,i = zi − z. The unknowns a are computed by minimizing a weighted error over the neighboring points. Thus, we have to minimize the following quadratic form J=

m+2 i=1

' & ∂ψ − φ and the corwhere em+1 = (Aψ + B · ∇ψ + C∆ψ − f ), em+2 = ∂n responding weights wm+1 = wm+2 = 1. The above equation (40) can be re-expressed in the form J = (M a − b)T W (M a − b) with the diagonal matrix W = diag (w1 , w2 , . . . , wm , 1, 1) . The minimization of J with respect to a formally yields ( if M T W M is nonsingular) (41) a = (M T W M )−1 (M T W )b.

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In (41) the vector (M T W )b is explicitly given by m m (M T W )b = wi h1,i ψi + B1 f + nx φ, wi ψi + Af, i=1

i=1

m

wi h2,i ψi + B2 f + ny φ,

i=1 m

1 2

m

wi h3,i ψi + B3 f + nz φ,

i=1

wi h21,i ψi + Cf,

m

wi h1,i h2,i ψi ,

wi h1,i h3,i ψi ,

i=1

i=1

i=1

m

m m m 1 1 wi h2,i h3,i ψi , wi h22,i ψi + Cf, wi h23,i ψi + Cf 2 i=1 2 i=1 i=1

T

.

Thus we obtain from equation (41) that m m ψ = Q11 wi ψi + Af + Q12 wi h1,i ψi + B1 f + nx φ + Q13 +

i=1 m

wi h2,i ψi + B2 f + ny φ i=1 m 1 2 Q15 wi h1,i ψi + Cf 2 i=1 m

+ Q17 + Q19

wi h1,i h3,i ψi

i=1 m

wi h2,i h3,i ψi

i=1

+

+ Q14 + Q16 + +

i=1 m

wi h3,i ψi + B3 f + nz φ

i=1 m

wi h1,i h2,i ψi i=1 m 1 2 wi h2,i ψi + Cf Q18 2 i=1 m 1 2 wi h3,i ψi + Cf , Q1,10 2 i=1

where Q11 , Q12 , . . . , Q1,10 is the first row of the matrix (M T W M )−1 . Rearranging the terms, we have ψ−

m i=1

wi

Q11 + Q12 h1,i + Q13 h2,i + Q14 h3,i + Q15

h21,i 2

h23,i h22,i + Q19 h2,i h3,i + Q1,10 +Q16 h1,i h2,i + Q17 h1,i h3,i + Q18 2 2

ψi

= (Q11 A + Q12 B1 + Q13 B2 + Q14 B3 + Q15 C + Q18 C + Q1,10 C) f + (Q12 nx + Q13 ny + Q14 nz ) φ. Hence, if x is one of the N particles, say xj and xji its neighbors of number m(j), where xj is distinct from xji , then we have the following sparse system of equations for the unknowns ψj , j = 1, . . . , N

Meshless Methods for Conservation Laws m(j)

ψj −

i=1

wji

Q11 + Q12 h1,ji + Q13 h2,ji + Q14 h3,ji + Q15

+Q16 h1,ji h2,ji + Q17 h1,ji h3,ji

357

h21,ji 2

h23,i h22,ji + Q19 h2,ji h3,ji + Q1,10 + Q18 2 2

ψji

= (Q11 A + Q12 B1 + Q13 B2 + Q14 B3 + Q15 C + Q18 C + Q1,10 C) f + (Q12 nx + Q13 ny + Q14 nz ) φ. and in matrix form LΨ = R.

(42)

We have solved the above sparse system (42) using iterative methods like Gauss-Seidel and SOR. Since the values of the velocities and the pressure from time step n can be taken as initial values in the iterative methods for the time step n + 1, only very few iteration steps are required if the variation in time is not too large. 3.3 Numerical Tests In the following we consider three examples in the two dimensional case. The test cases are given in dimensionless form but can be interpreted in SI-units. Rayleigh-Taylor instability In order to test our numerical scheme we first consider the Rayleigh-Taylor instability computed by the meshfree method SPH [4]. The authors have compared the SPH results with those from the VOF method. In this test case the heavy fluid lies above the light fluid. The computational domain is the rectangle [0, 1] × [0, 2]. The densities of two fluids are 1.8 and 1. The dynamical viscosity of both fluids is µ = 0.4286 × 10−3 . The gravity acts downwards with √ g = 1. The Reynolds number is Re = ρU L/µ = 420 with U = gL, L = 1. Initially, we have considered 5147 particles with smoothing length h = 0.6. The initial interface is given by 1 − 0.15 sin(2πx). The heavy particles (stars) lie on and above this interface and the rest are light ones (dots). No-slip boundary conditions are applied at the solid walls. The time evolutions of the simulation are plotted in Fig. 2. Qualitatively, the results seem to be better than those obtained by the SPH method in [4]. Breaking dam problem This is a classical and simple test case to validate numerical schemes for the simulation of free surface flows. In [19] experimental data are given and several authors have reported their numerical results [13, 18, 26, 27].

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2

2

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2

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Fig. 2. Rayleigh-Taylor instability from left to right at time t = 0, 3, 4, 5.

The computational domain is a rectangle with the size of [0, 0.6] × [0, 0.3]. Consider a rectangular column of water with a width of a = 0.1 and a height of 0.2. The rest of the domain is filled with air. In Fig. 3 the star particles represent the water and the dot particles represent the air. No-slip boundary conditions are applied at all boundaries. Initially, 5624 particles are distributed with smoothing length h = 0.015. The gravity with g = 9.81 acts downwards. The density of air is 1 and the viscosity is 1.81 × 10.−5 . For water the density is 1000 and the viscosity is 1.005 × 10−3 . In Fig. 3 we have plotted the simulation results at different times. Since the density ratio of the fluids is very high, we have to smooth the density and viscosity three times near the interface in order to avoid numerical instabilities. If the density ratio is 100:1, it is enough to smooth the density at the interface only once. For the viscosity, also three smoothing steps are used, but the number of smoothing steps is less significant in this case. In Fig. 4 the positions of the leading fluid front versus time as well as the heights of the water column versus time are compared with experimental results provided by [19]. The front position is computed as the maximum distance among the water particle from the origin. Similarly, the height is obtained from the maximum height of the water particles. The data are plotted before the particles hit the right wall. The numerical computations are performed with different values of h. The results are plotted for h = 0.04, 0.02, 0.01, 0.005 which correspond to the number of particles 400, 3117, 12593 and 49262 respectively. These figures show a good agreement between the numerical and experimental results for small values of h. The CPU time in a PC Pentium 4, 2.66GHz for different values of h are presented in Table 1. In all cases the time step ∆t = 0.002 has chosen and the program was terminated at time t = 0.4.

Meshless Methods for Conservation Laws 0.3

0.3

0.25

0.25

0.2

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0.15

0.15

0.1

0.1

0.05

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0

0

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Fig. 3. star: water, dot: air particles at time t = 0, 0.12, 0.24, 0.72. 8

1

Experiment h=0.04 h=0.02 h=0.01 h=0.005

7

Experiment h=0.04 h=0.02 h=0.01 h=0.005

0.9

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% Fig. 4. Left: dimensionless front position x/a versus dimensionless time % t 2g/a, right: dimensionless column height x/(2a) versus dimensionless time t g/a. Table 1. CPU time for the breaking dam problem. h 0.04 0.02 0.01 0.005

No of Particles 800 3117 12593 49262

CPU time 0 Min 31 Sec 2 Min 10 Sec 9 Min 48 Sec 46 Min 14 Sec

Droplet splash Finally, we consider a water droplet falling through air onto a water surface. This problem was originally proposed in [20] and later reconsidered in [21] to

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test numerical schemes for variable density flows. The computational domain and the resolution is the same as in the test case of the Rayleigh-Taylor instability. The particles inside the circle of radius 0.2 with center (0.5, 1.6) and below the line y = 1 are considered as water particles and the rest air particles. The other data are chosen as in the case of the breaking dam problem. These results are consistent with the results shown in [20, 21]. 2

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Fig. 5. Falling water droplet through air onto water surface from left to right and top to bottom at times t = 0.2, 0.32, 0.48, 0.64, 0.88, 1.0.

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Acknowledgement. This work has been carried out in the project Particle Methods for Conservation Systems NE 269/11-3 which is part of the DFG – Priority Research Program Analysis and Numerics for Conservation Laws.

References 1. C. Chainais-Hillairet. Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate. M 2 AN , 33:129–156, 1999. 2. A. Chorin. Numerical solution of the Navier-Stokes equations. J. Math. Comput., 22:745–762, 1968. 3. B. Cockburn, F. Coquel, and P. LeFloch. An error estimate for ﬁnite volume methods for multidimensional conservation laws. Math. Comput., 63:77–103, 1994. 4. S. J. Cummins and M. Rudmann. An SPH projection method. J. Comput. Phys., 152:284–607, 1999. 5. G. A. Dilts. Moving least squared particle hydrodynamics – i. consistency and stability. Int. J. Numer. Mech. Eng., 44:1115–1155, 1999. 6. E. Godlewski and P.-A. Raviart. Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer, 1996. 7. D. Hietel, K. Steiner, and J. Struckmeier. A ﬁnite-volume particle method for compressible ﬂows. Math. Models Methods Appl. Sci., 10:1363–1382, 2000. 8. J.M. Melenk I. Babushka, I. The partition of unity method. International Journal of Numerical Methods in Engineering, 40:727–758, 1997. 9. O. Iliev and Tiwari S. A generalized (meshfree) ﬁnite diﬀerence discretization for elliptic interface problems. In I. Dimov, I. Lirkov, S. Margenov, and Z. Zlatev, editors, Numerical Methods and Applications, Lecture notes in Computer Sciences, pages 488–497. Springer, 2002. 10. Monaghan J. J. Smoothed particle hydrodynamics 1990. Annu. Rev. Astron. Astrop, 30:543–574, 1992. 11. M. Junk and J. Struckmeier. Consistency analysis for mesh-free methods for conservation laws. Mitt. Ges. Angew. Math. Mech., 24:99–126, 2001. 12. R. Keck. The finite volume particle method. PhD thesis, Universit¨ at Kaiserslautern, 2003. 13. F. J. Kelecy and Pletcher R. H. The development of free surface capturing approach for multidimensional free surface ﬂows in closed containers. J. Comput. Phys., 138:939, 1997. 14. D. Kr¨ oner. Numerical Schemes for Conservation Laws. Wiley Teubner, 1997. 15. J. Kuhnert. General smoothed particle hydrodynamics. PhD thesis, Universit¨ at Kaiserslautern, 1999. 16. J. Kuhnert. An upwind ﬁnite pointset method for compressible Euler and Navier-Stokes equations. In M. Griebel and M. A. Schweitzer, editors, Meshfree methods for Partial Differential Equations, volume 26 of Lecture Notes in Computational Science and Engineering. Springer, 2002. 17. T. Liszka and J. Orkisz. The ﬁnite diﬀerence method on arbitrary irregular grid and its application in applied mechanics. Computers & Structures, 11:83–95, 1980.

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Simulations of Turbulent Thermonuclear Burning in Type Ia Supernovae W. Hillebrandt1 , M. Reinecke1 , W. Schmidt1 , F.K. R¨ opke1 , C. Travaglio1 , 2 and J.C. Niemeyer 1

2

Max-Planck-Institut f¨ ur Astrophysik, Garching, Germany [email protected] Institut f. Theor. Physik und Astrophysik, Univ. W¨ urzburg, Germany [email protected]

Summary. Type Ia supernovae, i.e. stellar explosions which do not have hydrogen in their spectra, but intermediate-mass elements such as silicon, calcium, cobalt, and iron, have recently received considerable attention because it appears that they can be used as ”standard candles” to measure cosmic distances out to billions of light years away from us. Observations of type Ia supernovae seem to indicate that we are living in a universe that started to accelerate its expansion when it was about half its present age. These conclusions rest primarily on phenomenological models which, however, lack proper theoretical understanding, mainly because the explosion process, initiated by thermonuclear fusion of carbon and oxygen into heavier elements, is diﬃcult to simulate even on supercomputers. Here, we investigate a new way of modeling turbulent thermonuclear deﬂagration fronts in white dwarfs undergoing a type Ia supernova explosion. Our approach is based on a level set method which treats the front as a mathematical discontinuity and allows for full coupling between the front geometry and the ﬂow ﬁeld. New results of the method applied to the problem of type Ia supernovae are obtained. It is shown that in 2-D with high spatial resolution and a physically motivated subgrid scale model for the nuclear ﬂames numerically “converged” results can be obtained, but for most initial conditions the stars do not explode. In contrast, simulations in 3-D do give the desired explosions and many of their properties, such as the explosion energies, lightcurves and nucleosynthesis products, are in very good agreement with observed type Ia supernovae.

1 Introduction Numerical simulations of any kind of turbulent combustion have always been a challenge, and thermonuclear supernova explosions are no exception to that rule. This is mainly because of the large range of length scales involved. In type Ia supernovae (SNe Ia), in particular, the length scales of relevant physical processes range from 10−3 cm for the Kolmogorov-scale to several 107 cm

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for typical convective motions. In the currently favored scenario the explosion starts as a deflagration near the center of the star. Rayleigh-Taylor unstable blobs of hot burnt material are thought to rise and to lead to shear-induced turbulence at their interface with the unburnt gas. This turbulence increases the effective surface area of the flamelets and, thereby, the rate of fuel consumption; the hope is that finally a fast deflagration might result, in agreement with phenomenological models of type Ia explosions. Despite considerable progress in the field of modeling turbulent combustion for astrophysical flows the correct numerical representation of the thermonuclear deflagration front has always been a weakness of the simulations. Methods used until recently were based on the reactive-diffusive flame model, which artificially stretches the burning region over several grid zones to ensure an isotropic flame propagation speed. However, the soft transition from fuel to ashes stabilizes the front against hydrodynamical instabilities on small length scales, which in turn results in an underestimation of the flame surface area and – consequently – of the total energy generation rate. Moreover, because nuclear fusion rates depend on temperature nearly exponentially, one cannot use the zone-averaged values of the temperature obtained this way to calculate the reaction kinetics. The front tracking method used in this project cures most of these weaknesses. It is based on the so-called level set technique which was originally introduced by [OS88]. They used the zero level set of a n-dimensional scalar function to represent (n − 1)-dimensional front geometries. Equations for the time evolution of such a level set which is passively advected by a flow field are given in [SSO94]. The method has been extended to allow the tracking of fronts propagating normal to themselves, e.g. deflagrations and detonations [SMK97]. In contrast to the artificial broadening of the flame in the reaction-diffusion-approach, this algorithm is able to treat the front as an exact hydrodynamical discontinuity.

2 The level set method The central aspect of our front tracking method is the association of the front geometry (a time-dependent set of points Γ ) with an isoline of a so-called level set function G: Γ := {r | G(r) = 0} . (1) Since G is not completely determined by this equation, we can additionally postulate that G be negative in the unburnt and positive in the burnt regions, and that G be a “smooth” function, which is convenient from a numerical point of view. This smoothness can be achieved, for example, by the additional constraint that |∇G| = 1 in the whole computational domain, with the exception of possible extrema and kinks of G. The ensemble of these conditions produces a G which is a signed distance function, i.e. the absolute value

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of G at any point equals the minimal front distance. The normal vector to the front is defined to point towards the unburnt material. The task is now to find an equation for the temporal evolution of G such that the zero level set of G behaves exactly as the flame. Such an expression can be obtained by the consideration that the total velocity of the front consists of two independent contributions: it is advected by the fluid motions at a speed v and it propagates normal to itself with a burning speed s. Since for deflagration waves a velocity jump usually occurs between the pre-front and post-front states, we must explicitly specify which state v and s refer to; traditionally, the values for the unburnt state are chosen. Therefore, one obtains for the total front motion Df = vu + su n .

(2)

The total temporal derivative of G at a point P attached to the front must vanish, since G is, by definition, always 0 at the front: dGP ∂G ∂G = + ∇G · x˙ P = + Df · ∇G = 0 . dt ∂t ∂t

(3)

This leads to the desired differential equation describing the time evolution of G: ∂G = −Df · ∇G. (4) ∂t This equation, however, cannot be applied on the whole computational domain, mainly because using this equation everywhere will in most cases destroy G’s distance function property. Therefore additional measures must be taken in the regions away from the front to ensure a “well-behaved” |∇G|, see [RHN+ 99]. The situation is further complicated by the fact that the quantities vu and su which are needed to determine Df are not readily available in the cells cut by the front. In a finite volume context, these cells contain a mixture of preand post-front states instead. Nevertheless one can assume that the conserved quantities (mass, momentum and total energy) of the mixed state satisfy the following conditions: ρ = αρu

+ (1 − α)ρb ,

ρv = αρu vu + (1 − α)ρb vb , ρe = αρu eu + (1 − α)ρb eb .

(5) (6) (7)

Here α denotes the volume fraction of the cell occupied by the unburnt state. In order to reconstruct the states before and behind the flame, a nonlinear system consisting of the equations above, the Rankine-Hugoniot jump conditions and a burning rate law must be solved. Having obtained the reconstructed pre- and post-front states in the mixed cells, it is not only possible to determine Df , but also to separately calculate

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Fu

Fb

Fb Fu

β

Fb

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Fig. 1. Illustration of the basic principles of the level set method according to [SMK97]: The piecewise linear front cuts the mixed cells into burnt and unburnt parts. α is the unburnt volume fraction of a cell, β is the unburnt area fraction of a cell interface. The ﬂuxes Fu and Fb are calculated from the reconstructed states.

the fluxes of burnt and unburnt material over the cell interfaces. Consequently, the total flux over an interface can be expressed as a linear combination of burnt and unburnt fluxes weighted by the unburnt interface area fraction β (see Fig. 1): ¯ = βFu + (1 − β)Fb . (8) F

3 Implementation For our calculations, the front tracking algorithm was implemented as an additional module for the hydrodynamics code PROMETHEUS [FMA89]. Here we describe a simple implementation of most of the ideas discussed in the previous section which we will call “passive implementation”. It assumes that the G-function is advected by the fluid motions and by burning and is only used to determine the source terms for the reactive Euler equations. It must be noted that there exists no real discontinuity between fuel and ashes in this case; the transition is smeared out over about three grid cells by the hydro-dynamical scheme, and the level set only indicates where the thin flame front should be. However, the numerical flame is still considerably thinner than in the reaction-diffusion approach. A complete implementation contains in-cell-reconstruction and flux-splitting as proposed by [SMK97] and outlined above. Therefore it describes exactly the coupling between the flame and the hydrodynamic flow. This generalized version of the code has been applied to hydrogen combustion in air by [RHN+ 99] and, very recently, also to thermonuclear fusion by [RNH03] (see also Sect. 7.2). 3.1 G-Transport Since the front motion consists of two distinct contributions, it is appropriate to use an operator splitting approach for the time evolution of G. The advection term due to the fluid velocity vF reads in conservation form ∂(ρG) 3 d r+ −vF ρ G df = 0 (9) ∂t ∂V V

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[MOS92]. This equation is identical to the advection equation of a passive scalar, like the concentration of an inert chemical species. Consequently, this contribution to the front propagation can be calculated by PROMETHEUS directly. The additional flame propagation due to burning is calculated at the end of each time step and a re-initialization of G is done in order to keep it a signed distance function (see [RHN+ 99]). 3.2 Source terms After the update of the level set function in each time step, the change of chemical composition and total energy due to burning is calculated in the cells cut by the front. In order to obtain these values, the volume fraction α occupied by the unburnt material is determined in those cells by the following approach: from the value Gij and the two steepest gradients of G towards ˜ of the level set the front in x- and y-direction a first-order approximation G ˜ < 0 can be function is calculated; then the area fraction of cell ij where G found easily. Based on these results, the new concentrations of fuel, ashes and energy are obtained: ′ = max(1 − α, XAshes ) , XAshes ′ ′ XFuel = 1 − XAshes ,

′ e′tot = etot + q(XAshes − XAshes ) .

(10) (11) (12)

In principle this means that all fuel found behind the front is converted to ashes and the appropriate amount of energy is released. The maximum operator in (10) ensures that no “reverse burning” (i.e. conversion from ashes to fuel) takes place in the cases where the average ash concentration is higher than the burnt volume fraction; such a situation can occur in a few rare cases because of unavoidable discretization errors of the numerical scheme.

4 Turbulent nuclear burning The system of equations described so-far can be solved provided the normal velocity of the burning front is known everywhere and at all times. In our computations it is determined according to a flame-brush model of [NH95b], which we will briefly outline for convenience. As was mentioned before, nuclear burning in degenerate dense matter is believed to propagate on microscopic scales as a conductive flame, wrinkled and stretched by local turbulence, but with essentially the laminar velocity. Due to the very high Reynolds numbers macroscopic flows are highly turbulent and they interact with the flame, in principle down to the Kolmogorov scale. This means that all kinds of hydrodynamic instabilities feed energy into a turbulent cascade, including the buoyancy-driven Rayleigh-Taylor instability and the shear-driven Kelvin-Helmholtz instability. Consequently, the picture

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that emerges is more that of a “flame brush” spread over the entire turbulent regime rather than a wrinkled flame surface. For such a flame brush, the relevant minimum length scale is the so-called Gibson scale, defined as the lower bound for the curvature radius of flame wrinkles caused by turbulent stress. Thus, if the thermal diffusion scale is much smaller than the Gibson scale (which is the case for the physical conditions of interest here) small segments of the flame surface are unaffected by large scale turbulence and behave as unperturbed laminar flames (“flamelets”). On the other hand side, since the Gibson scale is, at high densities, several orders of magnitude smaller than the integral scale set by the Rayleigh-Taylor eddies and many orders of magnitude larger than the thermal diffusion scale, both transport and burning times are determined by the eddy turnover times, and the effective velocity of the burning front is independent of the laminar burning velocity. A numerical realization of this general concept is presented in [NH95b]. The basic assumption was that wherever one finds turbulence this turbulence is fully developed and homogeneous, i.e. the turbulent velocity fluctuations on a length scale l are given by the Kolmogorov law v(l) = v(L)(l/L)1/3 , where L is the integral scale, assumed to be equal to the Rayleigh-Taylor scale. Following the ideas outlined above, one can also assume that the thickness of the turbulent flame brush on the scale l is of the order of l itself. With these two assumptions and the definition of the Gibson scale one finds for lgibs l L ≃ λRT 1/3 l v(l) ≃ st (l) ≃ st (lgibs ) lgibs

(13)

and dt (l) ≃ l, where v(lgibs ) = slam defines lgibs , slam is the laminar burning speed and st (l) is the turbulent flame velocity on the scale l. In a second step this model of turbulent combustion is coupled to our finite volume hydro scheme. Since in every finite volume scheme scales smaller than the grid size cannot be resolved, we express lgibs in terms of the grid size ∆, the (unresolved) turbulent kinetic energy per unit mass, ksgs , and the laminar burning velocity: 2 3/2 ul . (14) lgibs = ∆ 2ksgs Here ksgs is determined from a subgrid scale model [Cle93, NH95b] and, finally, the effective turbulent velocity of the flame brush on scale ∆ is given by

with v(∆) = acceleration.

% 2ksgs

st (∆) = max(slam , v(∆), vRT ), (15) √ and vRT ∝ g∆, where g is the local gravitational

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5 Application to the supernova problem Different series of simulations were performed to check the numerical reliability of the employed models and to compare two- and three-dimensional explosions. Resolution study: energy release 4

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Fig. 2. Time evolution of the total energy for identical initial conditions at diﬀerent resolutions. While model c3 2d 128 is clearly under-resolved, the other simulations agree very well, at least in the early and intermediate stages.

5.1 Resolution study A crucial test for the validity of the models for the unresolved scales (in this case the flame and subgrid models) is to check the dependence of integral quantities, like the total energy release of the explosion, on the numerical grid resolution. Ideally, there should be no such dependence, indicating that all effects on unresolved scales are accurately modeled. Figure 2 shows the energy evolution of a centrally ignited white dwarf. The only difference between the simulations is the central grid resolution, which ranges from 2 · 106 cm (model c3 2d 128) down to 2.5 · 105 cm (model c3 2d 1024). Model c3 2d 128 is obviously under-resolved, but the results of the other calculations are in good agreement, with exception of the last stages, where the flame enters strongly non-uniform regions of the grid. So far, this kind of parameter study could only be performed in two dimensions, because of the prohibitive cost of very highly resolved 3D simulations. Nevertheless the results suggest that a resolution of ≈ 106 cm should yield acceptable accuracy also in three dimensions.

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6.0•107 1.0•108 4.0•107 2.0•107 0 0 2.0•107 4.0•107 6.0•107 8.0•107 1.0•108 1.2•108 1.4•108 r [cm] 4.96e+08cm/s

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5.2 Comparison of 2D and 3D simulations In order to investigate the fundamental differences between two- and threedimensional simulations, a 2D and a 3D model with identical initial conditions and resolution was calculated. Figures 3 and 4 show snapshots of the flame geometry at various explosion stages; the energy evolution of both models is compared in Figure 5. It is evident that both simulations evolve nearly identically during the first few tenths of a second, as was expected. This is a strong hint that no errors were introduced into the code during the enhancement of the numerical models to three dimensions. At later times, however, the 3D calculation develops instabilities in the azimuthal direction, which could not form in 2D because of the assumed axial symmetry. As a consequence the total burning surface and the energy generation rate is increased, resulting in a higher overall energy release. 5.3 The effect of different initial conditions In our approach, the initial white dwarf model (composition, central density, and velocity structure), as well as assumptions about the location, size and shape of the flame surface as it first forms fully determine the simulation results. At present, we have not changed the properties of the white dwarf but we concentrate on variations of the latter. In this context, the simultaneous runaway at several different spots in the central region of the progenitor star is of particular interest, a plausible ignition scenario suggested by [GSW95].

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t=0.15s

t=0.45s

t=0.79s

Fig. 4. Burning front geometry evolution of model c3 3d 256. One ring on the axes corresponds to 107 cm.

One simulation was carried out on a grid of 2563 cells with a central resolution of 106 cm and contained five bubbles with a radius of 3·106 cm, which were distributed randomly in the simulated octant within 1.6·107 cm of the star’s center. In an attempt to reduce the initially burned mass as much as possible without sacrificing too much flame surface, a very highly resolved second model was constructed. It contains nine randomly distributed, nonoverlapping bubbles with a radius of 2·106 cm within 1.6·107 cm of the white dwarf’s center. To properly represent these very small bubbles, the cell size was reduced to ∆ =5·105 cm, so that a total grid size of 5123 cells was required. During the first 0.5 seconds, the three models are nearly indistinguishable as far as the total energy is concerned (see Fig. 7), which at first glance appears somewhat surprising, given the quite different initial conditions. A closer look at the energy generation rate actually reveals noticeable differences in the intensity of thermonuclear burning for the simulations, but since the total

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Etot [1050erg]

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-2 -4 -6 0.0

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Fig. 6. Left panel: Snapshots of the ﬂame front for a scenario with 5 ignition spots in 3D. The fast merging between the leading and trailing bubbles and the rising of the entire burning region is clearly visible. One ring on the coordinate axes corresponds to 107 cm. Right panel: The same as before, but for a high resolution model with 9 ignition spots.

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Energy comparison, 3D simulations 10

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Fig. 7. Energy evolution of these three-dimensional explosion models (dashed and dashed-dotted). For comparison we also show the centrally ignited (“three ﬁngers”) model of [RHN02] (solid line) discussed earlier.

flame surface is initially very small, these differences have no visible impact on the integrated curve in the early stages. However, after about 0.5 seconds, when fast energy generation sets in, the nine-bubble model burns more vigorously due to its larger surface and therefore reaches a higher final energy level. Fig. 7 also shows that the centrally ignited model (c3 3d 256) is almost identical to the off-center model b5 3d 256 with regard to the explosion energetics. But, obviously, the scatter in the final energies due to different initial conditions appears to be small. Moreover, all models explode with an explosion energy in the range of what is observed.

6 Predictions for observable quantities In this Section we present a few preliminary results for various quantities which could, in principle, be observed and which therefore can serve as tests for the models. 6.1 Lightcurves The most direct test of explosion models is provided by observed lightcurves and spectra. According to “Arnett’s Law” lightcurves measure mostly the amount and spatial distribution of radioactive 56 Ni in type Ia supernovae, and spectra measure the chemical composition in real and velocity space. [SB03] have used the results of one of our centrally ignited 3D-model, averaged over spherical shells, to compute colour lightcurves in the UBVIbands. Their code assumes LTE radiation transport and loses reliability at later times (about 4 weeks after maximum) when the supernova enters the

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nebular phase. Also, this assumption and the fact that the opacity is not well determined at longer wavelength make I-lightcurves less accurate. Keeping this in mind, the lightcurves shown in Fig. 8 look very promising. The main reason for the good agreement between the model and SN 1994D is the presence of radioactive Ni in outer layers of the supernova model at high velocities which is not predicted by spherical models.

Fig. 8. UBVI-colour lightcurves predicted by a centrally ignited 3D model (solid lines) in comparison with observed data for a bright (98bu), a “normal” (94D), and a subluminous (91bg) supernova.

6.2 Elemental and isotopic abundances A summary of the abundances obtained for all 3D models is given in Table 1. Here “Mg” (as in Fig. 8) stands for intermediate-mass nuclei, and “Ni” for the iron-group. In addition, the total energy liberated by nuclear burning is given. Since the binding energy of the white dwarf was about 5·1050 erg, all models do explode. Typically one expects that around 80% of iron-group nuclei are originally present as 56 Ni bringing our results well into the range of observed Ni-masses. This success of the models was obtained without introducing any non-physical parameters, but just on the basis of a physical and numerical model of subsonic turbulent combustion. We also stress that our models give

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Table 1. Overview over element production and energy release of all discussed supernova simulations. model name mMg [M⊙ ] mNi [M⊙ ] Enuc [1050 erg] c3 3d 256 0.177 0.526 9.76 0.180 0.506 9.47 b5 3d 256 0.190 0.616 11.26 b9 3d 512

clear evidence that the often postulated deflagration-detonation transition is not needed to produce sufficiently powerful explosions. Finally, we have “post-processed” one of our models in order to see whether or not also reasonable isotopic abundances are obtained. The results, shown in Fig. 9, are preliminary and should be considered with care. However, it is obvious that, with a few exceptions, also isotopic abundances are within the expected range and do not differ too much from those computed by means of phenomenological models, i.e. “W7”. Exceptions include the high abundance of (unburned) C and O, and the overproduction of 48,50 Ti, 54 Fe, and 58 Ni.

Fig. 9. Isotopic abundances obtained for oﬀ-center ignited 3D model.

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7 Supplementary studies In order to validate our approach to model the effects on unresolved scales, two studies were carried out. One tests several subgrid scale (SGS) models under the physical conditions encountered in SN Ia explosions and the other concerns the flame stability on small scales, where the turbulent cascade does not dominate the flame evolution.

Fig. 10. Evolution of turbulent thermonuclear burning in a cubic domain subject to periodic boundary conditions. The panels show 2D contour sections of the normalized speciﬁc total energy, etot /c20 . The mean mass density is ρ0 ≈ 2.903 · 108 g cm−3 and the initial sound speed c0 ≈ 6.595 · 108 cm s−1 . Turbulence is produced artiﬁcially by a stochastic solenoidal force ﬁeld. The bright regions of high speciﬁc energy contain burned material. The ﬂame front is indicated by the thin white lines corresponding to the zero level set.

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Fig. 11. Comparison of the evolution of SGS turbulence and thermonuclear burning for LES with three diﬀerent variants of the SGS turbulence energy model. In one simulation, Clement’s law of the wall was used to calculate the closure parameters, the second LES was computed with the constant parameters and for the third simulation, a semi-localized model was applied. In the top panels, the ratio of the mean SGS turbulence velocity to the RMS velocity and the laminar burning speed, respectively, is shown as function of normalized time t/T . In the middle row of panels, the ﬁrst three statistical moments of the mass-weighted SGS turbulence velocity and, in the bottom panels, the time evolution of fuel, burning products and the mean energy generation rate are plotted.

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7.1 Large eddy simulations of turbulent deflagration Subgrid scale models were tested in large eddy simulations (LES) of thermonuclear burning in a turbulent flow. In these simulations, turbulence is driven by an artificial stochastic force field and the computational domain is cubic with periodic boundary conditions. The term large eddy simulation indicates that not all dynamic scales are numerically resolved. Compared to simulations of supernova explosions, there are two additional differences: Firstly, the periodic boundary conditions enforce a constant average mass density. Hence, there is no explosion in this setting. The burning process proceeds in the form of a percolation process as gradually all fuel is consumed and the system approaches a state in which degenerate C+O matter is replaced by 56 Ni and α particles in statistical nuclear equilibrium. Secondly, gravity is not included. Since the integral length scale is of the order 105 cm, this is a sensible approximation. In consequence, we focus on the interaction of the deflagration with purely hydrodynamical turbulence for which the similarity theory of Kolmogorov applies. Whether SGS models which do not account for effects induced by gravity are applicable to thermonuclear supernova explosions is still controversial. However, there are simple scaling arguments in favor of this conjecture. The single most important result emerging from our studies is that the evolution of the burning process is significantly affected by the SGS model in use. This underlines the importance of choosing a faithful SGS model and certainly bears consequences on the simulation of deflagration in SNe Ia. Our approach of determining the turbulent flame speed as outlined in Section 4 is based upon a dynamical equation for the SGS turbulence energy 2 which can be casted into the following equation for the turbulence ksgs = 12 qsgs velocity qsgs :

2 qsgs ℓǫ (16) ∂ D is the Lagrangian time derivative ∂t + v · ∇. In this The operator Dt equation, several heuristic approximations, so-called closures, are incorpo√ rated. √ Associated with these closures √ are the length scales ℓν = Cν ∆eff / 2, ℓǫ = 2 2∆eff /Cǫ and ℓκ = Cκ ∆eff / 2. Cλ accounts for pressure effects in a compressible fluid. The closure parameters Cν , Cǫ , Cκ and Cλ are a priori unknown. In the case of stationary isotropic turbulence in an incompressible fluid, values can be derived from analytic theories of turbulence. Particular examples are Cν ≈ 0.054, Cǫ ≈ 1.0 and Cκ ≈ 0.1 (cf. [Sag01], Section 4.3) and Cλ ≈ −0.2 [FTWG97]. ∆eff is the effective length scale of the finitevolume scheme, in this case, the piece-wise parabolic method. Usually, this scale is set equal to the size ∆ of the grid cells. However, from the energy spectra of direct numerical simulations (DNS) of forced isotropic turbulence, we concluded that it is more appropriate to set ∆eff = β∆, where β is in the

1 D qsgs − ∇ · (ρℓκ qsgs ∇qsgs ) − ℓκ |∇qsgs |2 = ℓν |S ∗ |2 − Dt ρ

1 Cλ + 3 2

qsgs d −

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range 1.6 . . . 1.8 depending on the Mach number. The factor β accounts for the smoothing effect of the numerical scheme on top of the discretization. We have also determined statistical values of the closure parameters from DNS data. For subsonic flows, Cν = 0.06, Cǫ = 0.48 and Cκ = 0.36 appear to be representative values. The outcome of a LES of turbulent deflagration in a cube of 2163 cells with these parameters is illustrated in Figure 10. Burning is ignited in eight small spherical zones at the beginning of the simulation and the fluid is set into motion by stochastic forcing. Turbulence is produced on a time scale T = L/V , where L = 2.16 · 105 cm is the integral length scale and V = 100slam ≈ 9.78 · 107 cm s−1 is the characteristic velocity of the flow. One can think of L being the typical size of the largest vortices generated by the stochastic force field and T is the associated autocorrelation time or, figuratively, the turn-over time. Initially, the burning zones are expanding very slowly as the flame propagation speed mostly equals the laminar speed. After roughly one turn-over time has elapsed, SGS turbulence becomes increasingly space filling and enhances the flame propagation speed. At the same time, the folding and stretching of the flame front by the resolved flow greatly increases the surface and, thus, the rate of fuel consumption. These two effects appreciably accelerate the burning process as one can see from the evolution of the energy shown in Figure 10. At time t 2T , most of the material has already been burnt. Constant closure parameters are a sensible choice for fully developed isotropic turbulence. However, significant deviations are expected for transient, intermittent or anisotropic flows. The deflagration in the cube is certainly transient and, naturally, the physical conditions in the vicinity of the flame front are anisotropic. In the case of a supernova explosion, these qualities are even more pronounced. For this reason, a more sophisticated method of determining the closure parameters is called for. [Cle93] found for stellar structure modeling ad hoc rules to adjust the parameters Cν and Cǫ in order to account for compressibility in stratified media. Since steep density gradients affect turbulence more or less like a wall, we shall refer to the rules proposed by Clement as “law of the wall”. This method has been applied to SGS modeling in the simulations of thermonuclear supernovae as well. In comparison to the SGS turbulence energy model with constant parameters, Clement’s law of the wall produces markedly different results as one can see from the LES statistics plotted in Figure 11. The mean of qsgs increases initially very rapidly and then settles at an almost constant level. Correspondingly, the average rate of energy release per unit volume due to thermonuclear burning, B, increases more gradually as opposed to steep rise and high peak found for the SGS model with constant parameters. An entirely different method utilizes the filtering approach introduced by [Ger92]. Smoothing the resolved velocity field with a filter of characteristic length larger than ∆eff and invoking self-similarity assumptions, equations can be derived which yield closure parameters varying in space and time corresponding to the local structure and evolution of the flow.

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[Pio93, LMK94, GLMA95, MK00]. This so-called dynamical procedure works particularly well for the production parameter Cν . In the case of dissipation and diffusion, however, the method appears to be inadequate because of deficiencies in the respective closures. Therefore, we adopted a semi-localized approach. Results from a LES using this method, are shown in the panels on the very right of Figure 11. The differences compared to the LES with constant parameters are not striking but Clement’s method is clearly dismissed. The prediction of the latter that SGS turbulence is produced right at the beginning in advance of small-scale structure developing in the resolved flow is physically unreasonable. Apart from that, once turbulence has developed in a particular spatial region, it becomes increasingly isotropic on decreasing length scales even in more complex flows. Thus, the remarkable similarity of the evolution of the burning process in the LES with constant and dynamical parameters, respectively. However, the choice of appropriated values for the closure parameters is notoriously difficult, particularly, when it comes to the supernova explosion scenario in which turbulence is generated due to buoyancy effects originating from the density contrast between burnt and unburnt material. In conclusion, the semi-localized model is likely to be the preferable SGS model for the application to SNe Ia simulations after all. 7.2 Investigation of the cellular burning regime Below the Gibson scale, the flame propagation is not dominated by the turbulent cascade, since here the flame burns faster through turbulent eddies than these can deform it. On those scales the flame evolution is determined by the (hydrodynamical) Landau-Darrieus (LD) instability [Lan44] and its counteracting nonlinear stabilization [Zel66]. In case of terrestrial flames this stabilization leads to a cellular steady-state pattern of the flame front giving rise to the cellular burning regime. After [NH95a] have shown by means of hydrodynamical simulation that the LD instability acts under conditions of SNe Ia, we could confirm that also the cellular stabilization of the flame front holds for thermonuclear flames in white dwarfs. This was achieved in a series of 2D simulations. In order to reproduce hydrodynamical effects like the LD instability it turned out to be essential to apply the complete implementation described in Sect. 2 (see [RNH03]). The stability of this cellular flame shape was tested for various fuel densities and for interaction of the flame with vortical flows. Two examples are given in the following. Figure 12 shows the propagation of an initially perturbed flame into quiescent fuel. The initial perturbations grow and in the nonlinear regime the flame exhibits a cellular structure. For the given setup, however, the cellular pattern of the same wavelength as the initial perturbation is not a stable solution. The snapshots at later times (Fig. 12b,c) illustrate the “merging” of cells until the steady-state structure of a single domain-filling cusp has formed. In case of higher numerical resolution, this fundamental flame structure may be superposed by a short-wavelength cellular pattern, which

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Fig. 12. Flame propagation into quiescent fuel at ρu = 5 × 107 g cm−3 , resolution: 200 × 200 cells, snapshots taken at (a) 0, (b) 7.5, (c) 15, and (d) 30 growth times τLD of a perturbation with λ = 3.2 × 104 cm. Color-coded is the vorticity of the ﬂow ﬁeld, see also color ﬁgure, plate 23.

is advected toward the cusp and does not lead to a break-up of the domainfilling cell. This evolution is well in agreement with theoretical predictions. Simulations with different fuel densities ρu led to similar results. In the range of ρu = 1 × 107 g cm−3 . . . 1 × 109 g cm−3 no significant flame destabilization could be observed. Flame interaction with a vortical fuel field is shown in Fig. 13. This is motivated by relic turbulent motions from the eddy cascade around the Gibson scale and from pre-ignition convective motions in the white dwarf. A parameter study with different fuel densities and various strengths of the velocity fluctuations in the fuel flow led to the result that a vortical flow of sufficient strength can break up the cellular pattern of the flame. In this case, however,

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no drastic self-turbulization of the flame but rather a smooth adaptation of the flame structure to the imprinted flow was observed. These results corroborate the assumption of large scale models, that the generation of turbulence is dominated by large-scale effects. In the cellular burning regime, the flame surface is enlarged compared to a planar configuration. This leads to an additional acceleration of the flame. As a result of our study we suggest an increased lower cut-off of the flame velocity (cf. (15)) instead of ulam depending on background turbulence.

8 Conclusion In this project, we have provided a new method to model the physics of thermonuclear combustion in degenerate dense matter of white dwarf stars consisting of carbon and oxygen. Because not all relevant length-scales of this problem can be numerically resolved a numerical model to describe deflagration fronts with a reaction zone much thinner then the cells of the computational grid was presented. This new approach was applied to the simulate thermonuclear supernova explosions of MChan white dwarfs in 2 and 3 dimensions.

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An implicit assumption of our numerical model is that on the resolved scales the flows are turbulent allowing us to describe the physics on the unresolved scales by a subgrid scale model, in the spirit of large-eddy simulations. The results presented here indicate that for supernova simulations this assumption is satisfied if we compute 3D models and use at least a 2563 grid. The reason is simply that more structure on small length scales in better resolved models increases the rate of fuel consumption locally, but the turbulent velocity fluctuations are then smaller on the grid scale, compensating for this gain. All models we have computed (differing only in the ignition conditions and the grid resolution) explode. The explosion energy and the Ni-masses are only moderately dependent on the way the nuclear flame is ignited making the explosions robust. However, since ignition is a stochastic process, the differences we find may even explain some of the spread in observed SN Ia’s. Based on our models we can predict lightcurves, spectra, and abundances, and the first preliminary results look promising. The lightcurves seem to be in very good agreement with observations, and also the nuclear abundances of elements and their isotopes are found to be in the expected range. Of course, the next step is to compute a grid of models, with varying white dwarf properties, and to compare them with the increasing data base of wellobserved type Ia supernovae. The hope is that this will give us a tool to understand their physics and, thus, get confidence in their use for cosmology. Acknowledgement. The numerical computations presented here were in part carried out on the Hitachi SR-8000 at the Leibniz-Rechenzentrum M¨ unchen as a part of the project H007Z. We thank the staﬀ of the HLRB at the LRZ for their continuous help. This work was also supported in part by the Deutsche Forschungsgemeinschaft under Grant Hi 534/3-3.

References [Cle93]

M. J. Clement. Hydrodynamical simulations of rotating stars. I - A model for subgrid-scale ﬂow. ApJ, 406:651, 1993. [FMA89] B. A. Fryxell, E. M¨ uller, and W. D. Arnett. Hydrodynamics and nuclear burning. MPA Preprint, 449, 1989. [FTWG97] C. Fureby, G. Tabor, H. G. Weller, and A. D. Gosman. Diﬀerential subgrid stress models in large eddy simulations. Phys. of Fluids, 9:3578– 3580, November 1997. [Ger92] M. Germano. Turbulence: the ﬁltering approach. J. Fluid Mech., 238:325–336, 1992. [GLMA95] S. Ghosal, T. S. Lund, P. Moin, and K. Akselvoll. A dynamic localization model for large-eddy simulation of turbulent ﬂow. J. Fluid Mech., 286:229–255, 1995. [GSW95] D. Garcia-Senz and S. E. Woosley. Type Ia supernovae: The ﬂame is born. ApJ, 454:895, 1995.

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L. D. Landau. On the theory of slow combustion. Acta Physicochim. URSS, 19:77, 1944. [LMK94] S. Liu, C Meneveau, and J. Katz. On the properties of similarity subgridscale models as deduced from measurements in a turbulent jet. J. Fluid Mech., 275:83–119, 1994. [MK00] C. Meneveau and J. Katz. Scale-Invariance and Turbulence Models for Large-Eddy Simulation. Ann. Rev. Fluid Mech., 32:1–32, 2000. [MOS92] W. Mulder, S. Osher, and J. A. Sethian. Computing interface motion in compressible gas dynamics. J. Comput. Phys., 100:209, 1992. [NH95a] J. C. Niemeyer and W. Hillebrandt. Microscopic instabilities of nuclear ﬂames in type Ia supernovae. ApJ, 452:779, 1995. [NH95b] J. C. Niemeyer and W. Hillebrandt. Turbulent nuclear ﬂames in type Ia supernovae. ApJ, 452:769, 1995. [OS88] S. Osher and J. A. Sethian. Front propagating with curvature-dependent speed - algorithms based on hamilton-jacobi formulations. J. Comput. Phys., 79:12, 1988. [Pio93] U. Piomelli. High Reynolds number calculations using the dynamic subgrid-scale stress model. Phys. of Fluids, 5:1484–1490, June 1993. [RHN+ 99] M. A. Reinecke, W. Hillebrandt, J. C. Niemeyer, R. Klein, and A. Gr¨obl. A new model for deﬂagration fronts in reactive ﬂuids. A&A, 347:724, 1999. [RHN02] M. A. Reinecke, W. Hillebrandt, and J. C. Niemeyer. Reﬁned numerical models for mutidimensional type Ia supernova simulations. A&A, 386:936, 2002. [RNH03] F. K. R¨ opke, J. C. Niemeyer, and W. Hillebrandt. On the small-scale stability of thermonuclear ﬂames in type Ia supernovae. ApJ, 588:952, 2003. [Sag01] P. Sagaut. Large eddy simulation for incompressible flows. Springer, 2001. [SB03] E. Sorokina and S. Blinnikov. Light curves of type Ia supernovae as a probe for an explosion model. In W. Hillebrandt and B. Leibundgut, editors, From Twilight to Highlight: The Physics of Supernovae, page 268, Heidelberg, 2003. Springer. [SMK97] V. Smiljanovski, V. Moser, and R. Klein. A capturing-tracking hybrid scheme for deﬂagration discontinuities. Comb. Theory and Modeling, 1:183, 1997. [SSO94] M. Sussman, P. Smereka, and S. Osher. A level set approach for computing solutions to incompressible two-phase ﬂow. J. Comput. Phys., 114:146, 1994. [Zel66] Ya. B. Zel’dovich. An eﬀect which stabilizes the curved front of a laminar ﬂame. Journal of Appl. Mech. and Tech. Physics, 1:68, 1966.

Hyperbolic GLM Scheme for Elliptic Constraints in Computational Electromagnetics and MHD Y.J. Lee1,2 , R. Schneider3 , C.-D. Munz1 and F. Kemm1 1

2

3

Institut f¨ ur Aerodynamik und Gasdynamik, Universit¨at Stuttgart, Pfaﬀenwaldring 21, D-70550 Stuttgart, Germany lee|kemm|[email protected] ABB Schweiz AG, Corporate Research, CH-5405 Baden-D¨ attwil, Switzerland [email protected] Forschungszentrum Karlsruhe, Institut f¨ ur Hochleistungsimpuls- und Mikrowellentechnik,Postfach 3640, D-76021 Karlsruhe, Germany [email protected]

Summary. The charge conservation laws in general are not strictly obeyed in computational electromagnetics and Magnetohydrodynamics (MHD), due to the presence of various types of numerical errors. In this paper, a ﬁeld theoretical method for the treatment of the often violated charge conservation laws in computational electrodynamics and MHD has been investigated, which reduces to the well-known hyperbolic Generalized Lagrange Multplier (GLM) scheme under particular constraints. The central idea of our divergence correction scheme is the implementation of the physically consistent counter terms to Maxwell and MHD equations, for the restoration of the charge conservation laws. The underlying idea has been veriﬁed by numerical experiments for Maxwell-Vlasov and shallow water MHD systems.

1 Introduction The physics of electromagnetism is described by Maxwell equations, which consist of two hyperbolic evolution equations −∂t E + ∇ × B = je , ∂t B + ∇ × E = 0 ,

(1) (2)

and two additional elliptic constraints ∇ · E = ρe ,

∇ · B = 0,

(3) (4)

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for the electric field E and the magnetic induction B, with external electric charge density ρe and current density je . Here, the unit light velocity c = 1 and Heaviside’s units (with the Coulomb force given by qq ′ /4πr2 ) are taken. The important consequences of Maxwell equations (1)-(4) to which we attend are charge conservation laws. The conservation of the electric charge is described by (5) ∂t ρe + ∇ · je = 0, whereas the conservation of the magnetic charge is described by the elliptic equation (4), due to the permanent absence of magnetic monopoles. These charge conservation constraints must be satisfied by the solutions of hyperbolic subsystem (1) and (2) of Maxwell equations, provided that the initial field configurations satisfy the divergence conditions (3) and (4). However, due to the presence of various types of numerical insufficiencies, the numerical solution of the hyperbolic part of the Maxwell system alone often do not obey the charge conservation constraints (4) and (5) and, consequently, inconsistencies and non-physical field configurations may be generated. In this paper, we approach the problem of the numerical violation of charge conservation laws in computational electrodynamics and MHD from a field theoretical point of view, and present a numerical scheme which is consistent with the symmetries of Maxwell theory, namely the Lorentz-, gauge- and duality symmetries. This scheme under particular constraints will be shown to reduce to the existing hyperbolic GLM method [MSS99, MO00A, MO00B], and thereby defines an extended GLM scheme.

2 Symmetries in Maxwell theory Symmetry plays an important role in understanding characteristic nature of given physical theory and conservation laws. In this regard, we shall review these fundamental symmetries of Maxwell theory in this section. 2.1 Lorentz symmetry The Maxwell theory obeys the physical law of Einstein’s special theory of relativity, which states that the physics is invariant under a special class of space-time coordinate transformations, called Lorentz transformations. This symmetry property of Maxwell theory becomes transparent when the basic equations (1)-(4) are written in term of compact notation for vector and tensor fields which exhibits relativistic covariance. The Lorentz covariant compact form of Maxwell equations is given by ∂µ Feµν = jeν ,

∂µ Fgµν = jgν ,

for the external electric and magnetic four-current densities defined by

(6)

Physical Symmetries and Hyperbolic GLM Scheme

jeµ ≡ (ρe , je ),

jgµ ≡ (ρg , jg ),

387

(7)

where Feµν is the Lorentz covariant antisymmetric rank-2 field strength tensor given in terms of electric and magnetic fields, ⎞ ⎛ 0 −E 1 −E 2 −E 3 ⎜ E 1 0 −B 3 B 2 ⎟ ⎟ Feµν = ⎜ (8) ⎝ E 2 B 3 0 −B 1 ⎠ , E 3 −B 2 B 1 0 and Fgµν the corresponding dual field strength tensor defined as Fgµν ≡

1 µνρσ ǫ Fρσ , 2

(9)

where ǫµνρσ is the Levi-Civita antisymmetric symbol with ǫ0123 = 1. Here, the space-time coordinates in a certain inertial frame are denoted by xµ with the time coordinate x0 = t, where the Greek indices run over 0, 1, 2, 3. The Roman indices stand for the space coordinates. The metric tensor for flat Minkowski space-time is gµν = g µν = diag[ −1, +1, +1, +1 ], and we have adopted the Einstein’s summation convention: the repeated indices in the same term, once upstairs and once downstairs, are summed over (for details, see for instance Refs. [Wei72, Wei95]). For example, equation (6) is equivalent to 3 3 ∂µ Fgµν = jgν , (10) ∂µ Feµν = jeν , µ=0

µ=0

and the metric tensor gµν = g

µν

is used to raising and lowering indices

gµν V ν = Vµ ,

g µν Vν = V µ .

(11)

Using these notations, it is an simple exercise to check that (6) is indeed identical to the genuine set of Maxwell equations (1)-(4). Note that in presenting (6), we have not assumed vanishing magnetic charge and currents for reasons which will become clear later. In the presence of magnetic currents, the Lorentz invariant Lagrangian density for (6) has been given by Zwanziger [Zwa71] LZw = Lfree + Lint. 2 3 1 n ˆµn ˆ κ f µν (A)f˜κν (B) − f µν (B)f˜κν (A) 2 1 ˆµn ˆ κ [f µν (A)fκν (A) + f µν (B)fκν (B)] , + n 2

Lfree =

Lint. = jeµ Aµ + jgµ Bµ , with the definitions

(12)

(13)

(14)

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fµν (A) ≡ ∂µ Aν − ∂ν Aµ , fµν (B) ≡ ∂µ Bν − ∂ν Bµ , 1 1 f˜µν (A) ≡ ǫµν ρσ fρσ (A), f˜µν (B) ≡ ǫµν ρσ fρσ (B), 2 2

(15) (16)

where n ˆ µ is a constant four-vector with n ˆµn ˆ µ = 1, and the two Lorentz covariµ µ ant vector fields A and B describe the interaction with electric and magnetic charges, respectively. (Note that Aµ and B µ are not the conventional vector potentials encountered in classical electrodynamics.) The corresponding expressions for the field-strength tensor Feµν (A, B) and its dual counterpart Fgµν (A, B) are given by Feµν (A, B) = n ˆµn ˆ κ f κν (A) − n ˆν n ˆ κ f κµ (A) 1 ˆρn ˆ κ f κσ (B) − n ˆσ n ˆ κ f κρ (B) − f ρσ (B) ] − ǫµν ρσ [ n 2 ˆµn ˆ κ f κν (B) − n ˆν n ˆ κ f κµ (B) Fgµν (A, B) = n 1 + ǫµν ρσ [ n ˆρn ˆ κ f κσ (A) − n ˆσ n ˆ κ f κρ (A) − f ρσ (A) ] , 2

(17)

(18)

in terms of gauge fields Aµ and B µ . The two seemingly different vector fields Aµ and B µ collectively describe a single photon interacting both with electric and magnetic charges, under the imposition of proper constraints (for details, see Ref. [Zwa71]). Given the full description of the field strengths Feµν (A, B) and Fgµν (A, B), the variation of the action functional for the Lorentz invariant Lagrange density LZw via δSZw = δ dt d3 xLZw = 0 (19) yields the electromagnetic field equations (6). For further details of variational principle, see for example Ref. [Wei95]. 2.2 Gauge symmetry and charge conservation In the absence of the external source terms (jeµ = 0 and jgµ = 0), the action for the Lagrangian density Lfree given by (13) is invariant under local gauge transformations defined by Aµ → Aµ − ∂µ χA ,

Bµ → Bµ − ∂µ χB ,

(20)

where χA and χB are arbitrary differentiable scalar functions. This gauge symmetry is also obeyed by all the experimentally observed electromagnetic phenomena, also for the non-vanishing external sources (jeµ = 0 and jgµ = 0). (Indeed, in the context of the second quantized theory for the electromagnetism, the gauge symmetry is indeed a necessary consequence of the Lorentz symmetry; see Ref. [Wei95].) It is an easy exercise to check that the invariance of the action for LZw under the gauge transformations (20) necessarily implies the conservation of the electric and magnetic currents, which is expressed by

Physical Symmetries and Hyperbolic GLM Scheme

∂µ jeµ = 0,

∂µ jgµ = 0.

389

(21)

(Also refer (5).) That is, the charge conservation laws are the consequence of gauge symmetry of Maxwell theory. 2.3 Duality symmetry It is easy to verify that the extended Maxwell equations (6) for non-vanishing electric and magnetic currents are covariant under global phase rotations represented by (F µν + iF˜ µν ) → eiα (F µν + iF˜ µν ),

(jeµ + i jgµ ) → eiα (jeµ + i jgµ ),

(22)

where α is a real constant. This illustrates that, in the presence of nonvanishing magnetic current density, there is a duality symmetry between the electricity and magnetism. The presence of duality symmetry means that the electricity and the magnetism share the equivalent physical and mathematical structure. The reason why we need this seemingly non-physical duality symmetry in computational electromagnetism will become clear later, in the context.

3 Field theoretical method for divergence correction It is by now clear that the numerical violation of the charge conservation laws is a consequence of the broken gauge symmetry in the discretized spacetime. In this section, we shall derive a Lorentz covariant divergence correction strategy based on field theoretical principles, which aims to restore the once broken gauge symmetry on the grid. This section forms the core of this paper. However, since the present section contains many technical terms, the readers who are not familiar with field theoretical notions can skip the theoretical part, and directly go to the main result presented in (39)-(42). 3.1 Symmetry-based corrections for non-conserving charges We consider the extended Maxwell equations (6) with duality symmetry, and let F¯eµν and F¯gµν be the associated discretized solutions. These numerical solutions F¯eµν and F¯gµν indeed solve (6) only approximately, due to the presence of various types of numerical errors. Let us explicitly define the electric and magnetic numerical errors to be the difference between the discretized divergence µν and the desired conserved electromagnetic of the field strength tensors ∂¯µ F¯e/g µ . Denoting these electric and magnetic numerical errors current densities ˜je/g respectively with Deµ and Dgµ , one notices that the discretized solutions F¯eµν and F¯ µν actually satisfy the equations given by g

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ˆ F¯ µν , xµ ] = ˜j ν , ∂¯µ F¯eµν + Deν [L; e

ˆ F¯ µν , xµ ] = ˜j ν , ∂¯µ F¯gµν + Dgν [L; g

(23)

where ∂¯µ represents discretized partial derivative. The error terms Deµ and Dgµ , in general, depend on the lattice structure collectively parameterized here by ˆ the field strengths F¯eµν and F¯gµν and the space-time coordinate xµ of the L, relevant grid point. Note that the error terms Dµ becomes zero if ∂¯µ F¯ µν e/g

e/g

satisfies the Maxwell equations (6) exactly. Note that, due to the presence of these error terms, the numerical solution µν F¯e/g indeed feels the numerically generated electromagnetic current densities ¯j µ which can be expressed by e/g ¯jeµ ≡ ˜jeµ − Deµ ,

¯jgµ ≡ ˜jgµ − Dgµ .

(24)

It is easy to observe that these numerical currents for non-vanishing numerical errors do not form conserved currents ∂¯µ ¯jeµ = −∂¯µ Deµ = 0,

∂¯µ ¯jgµ = −∂¯µ Dgµ = 0.

(25)

We call this violation of the charge conservation laws the electric and magnetic lattice pseudo-anomalies, respectively. These pseudo-anomalies stem from the artificial generation of the external source terms Deµ and Dgµ to the electric and magnetic current densities, due to numerical errors. The origin of the lattice pseudo-anomalies can be explained as follows. Recall that the charge conservation laws (21) can be deduced from Maxwell equations (6), using the commutativity of differential operators and the antisymmetricity of the field strength tensors Feµν and Fgµν upon interchanging the indices µ and ν. From this, we deduce that the observed numerical violations of the charge conservation laws are mainly due to the fact that the discretized differential operators are not strictly commutative ! (26) ∂¯µ , ∂¯ν ≡ ∂¯µ ∂¯ν − ∂¯ν ∂¯µ = 0

for µ = ν, while the field strengths Feµν and Fgµν by definition remain strictly antisymmetric. For non-commuting discretized differential operators, equations (25) for the lattice pseudo-anomalies take the following forms: ! 1 ∂¯µ Deµ = − ∂¯µ , ∂¯ν F¯eµν = 0, 2

! 1 ∂¯µ Dgµ = − ∂¯µ , ∂¯ν F¯gµν = 0. 2

(27)

Apparently, the presence of the non-vanishing collective error terms Deµ and Dgµ break gauge symmetry, which is crucial for the charge conservation laws. The general strategy to cure these pseudo-anomalies would be to introduce counter terms Lec and Lgc to the Lagrangian density LZw given by (12)-(14), such that the counter terms cancel the anomalous contributions from Deµ and Dgµ , in order to restore gauge symmetry. The Lagrangian density which includes these counter terms reads

Physical Symmetries and Hyperbolic GLM Scheme

L = LZw + Lec (Aµ , Deµ ) + Lgc (B µ , Dgµ ).

391

(28)

The next logical step is to determine Lec and Lgc in terms of known field configurations. However, the analytic determination of these counter Lagrangian density terms is a very difficult task. Instead, we shall look for an indirect method for the determination of these counter terms. Note that the counter terms, upon variation of the corresponding action, should provide the external current densities which cancel the anomalous contributions −Deµ to ¯jeµ and −Dgµ to ¯jgµ ; refer (24). In this regard, the counter Lagrangian densities can be given by Lec = Aµ Deµ ,

Lgc = Bµ Dgµ ,

(29)

treating Deµ and Dgµ as if these are independent degrees of freedom. Explicitly, the introduction of these counter terms modifies the numerical non-conserving electric and magnetic current densities ¯jeµ and ¯jgµ to the conserved ones respectively represented by ˆjeµ and ˆjgµ , ¯jeµ → ˆjeµ ≡ ¯jeµ + Deµ ,

¯jgµ → ˆjgµ ≡ ¯jgµ + Dgµ .

(30)

Now, the corrected electric and magnetic current densities ˆjeµ and ˆjgµ are in principle anomaly free ∂¯µ ˆjeµ = ∂¯µ (¯jeµ + Deµ ) = 0,

∂¯µ ˆjgµ = ∂¯µ (¯jgµ + Dgµ ) = 0,

(31)

which means the restoration of once broken gauge symmetry and thus the charge conservation laws. Still, Deµ and Dgµ have not been determined yet. The idea for the determination of these counter terms is to solve the conservation equations (31) with respect to Deµ and Dgµ . However, in order to solve (31) for Deµ and Dgµ , we do not have sufficient information. That is, we have eight unknowns (Deµ and Dgµ ) for two equations. We overcome this problem by proposing the representations for Deµ and Dgµ respectively in terms of only one unknown. In addition, we require that the representation of Deµ and Dgµ is Lorentz covariant four-vectors, in order to keep Lorentz symmetry. The simplest representations which satisfy these two conditions can be given by Deµ (Φ) = λe ∂¯µ Φ,

Dgµ (Ψ ) = λg ∂¯µ Ψ

(32)

where λe and λg are real constants and Φ and Ψ real scalar fields. Note that, in the context of variational principle, λe and λg are Lagrange multipliers; refer (29) and (32). We can now determine the scalar fields Φ and Ψ from the conservation equations (31) which now become ∂¯µ (¯jeµ + λe ∂¯µ Φ) = 0,

∂¯µ (¯jgµ + λg ∂¯µ Ψ ) = 0.

(33)

Once Φ and Ψ are computed from (33) with the knowledge of ¯jeµ and ¯jgµ , one can algebraically solve the modified Maxwell equations,

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∂¯µ Fˆeµν − λe ∂ ν Φ = ¯jeν ,

∂¯µ Fˆgµν − λg ∂ ν Ψ = ¯jgν ,

(34)

for the corrected electric and magnetic field strengths Fˆeµν and Fˆgµν , for each time step. This in principle completes the derivation of field theoretical divergence correction scheme. The counter terms indeed can be more conveniently implemented by directly solving modified Maxwell equations, ∂µ Feµν − λe ∂ ν Φ = 0,

∂µ Fgµν − λg ∂ ν Ψ = 0,

(35)

which is the analytic counterpart to (34). The direct application of the counter terms to the Maxwell equations reduces considerable amount of computational effort, and such simplification can be justified for reasons stated below. Recall that the conventional Maxwell equations (1)-(4) are comprised of eight equations for six unknowns, and the numerical violation of the charge conservation laws is a result of solving only six hyperbolic equations (1) and (2), while leaving the two divergence constraints (3) and (4) not directly imposed. On the other hand, note that the modified Maxwell equations are comprised of eight equations for eight unknowns, thanks to the appearance of additional two scalar fields Φ and Ψ , and the corresponding numerical solver therefore should attain a good control over the two divergence constraints at each time step. Furthermore, the divergence constraints in this case naturally lead to inhomogeneous Klein-Gordon wave equations (33) which serve as modified charge conservation laws. This guarantees the fact that the Lagrange multiplier terms Deµ = λe ∂ µ Φ and Dgµ = λg ∂ µ Ψ indeed provide the counter terms to the non-vanishing numerical pseudo-anomalies ∂µ jeµ = 0 and ∂µ jgµ = 0, locally in time. As a consequence, the numerical solutions of (34) preserve the symmetries of Maxwell theory, and therefore the charge conservation laws. 3.2 A field theoretical interpretation of Φ and Ψ fields We attend to the fact that (33) is mathematically the inhomogeneous KleinGordon wave equations with the source terms provided by ∂µ jeµ and ∂µ jgµ , which can be obtained by varying the action for the Lagrangian densities given by LΦ =

λe ∂µ Φ∂ µ Φ + Φ∂µ jeµ , 2

LΨ =

λg ∂µ Ψ ∂ µ Ψ + Ψ ∂µ jgµ . 2

(36)

These Lagrangian densities describe the conventional massless scalar field theories, where the scalar fields Φ and Ψ respectively couples to the non-vanishing divergence errors, ∂µ jeµ = 0 and ∂µ jgµ = 0; see Ref. [Wei95]. This fact physically implies that the Φ and Ψ wavelets (generated by the error source terms ∂µ jeµ and ∂µ jgµ ) transport the divergence errors away from the computational domain, via the wave equations (33). In this context, in order to properly implement our divergence correction scheme to computational electromagnetism

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for initial divergence free configurations, it is logical to set transmissive boundary conditions for Φ and Ψ at the computational boundaries, with the initial conditions given by Ψ |t=0 = Φ|t=0 = 0. 3.3 Broken Lorentz symmetry and extended GLM scheme In the derivation of the counter terms to the electric and magnetic lattice pseudo-anomalies, we have emphasized the preservation of the Lorentz symmetry of Maxwell theory. However, the Lorentz symmetry is only an approximate symmetry in discretized space-time, which should be recovered in the continuum limit; refer for instance Ref. [Rot92]. Therefore, one may break (at least slightly) the Lorentz covariance of the counter term representations of electric and magnetic pseudo-anomalies in the discretized space-time, for further extension of the presented symmetry-based divergence correction scheme. An intentional breaking of Lorentz symmetry is achieved by taking following representations of the anomaly canceling counter currents Deµ (Φ) = Ceµν ∂ ν Φ,

Dgµ (Ψ ) = Cgµν ∂ µ Ψ,

(37)

where Ceµν = Ce µν and Cgµν = Cg µν are Lorentz non-covariant real matrices which are to be chosen for the optimization of the numerical performance of the divergence clearance. Upon implementing the modified counter terms (37), the divergence correction scheme discussed so far becomes equivalent to numerically solving the following set of modified Maxwell equations: ∂µ F µν − Ceµν ∂ ν Φ = jeν ,

∂µ F˜ µν − Cgµν ∂ ν Ψ = 0.

(38)

If the matrices Ceµν and Cgµν are set to be diagonal such that Ceµν = cµe δ µν and Cgµν = cµg δ µν for cµe ∈ R+ and cµg ∈ R+ , then (38) becomes equivalent to −∂t E + ∇ × B − diag[c1e , c2e , c3e ] · ∇Φ = je , ∂t B + ∇ × E + diag[c1g , c2g , c3g ] · ∇Ψ = 0,

c0e ∂t Φ + ∇ · E = ρe , c0g ∂t Ψ + ∇ · B = 0,

(39) (40) (41) (42)

in conventional terms, for vanishing external magnetic charge and currents. These modified Maxwell equations (39)-(42) keep the hyperbolicity of genuine Maxwell equations (1)-(4), and reduces to GLM-Maxwell equations of Munz et. al. [MSS99, MO00A, MO00B] upon choosing particular matrix elements cke = ckg = 1 for k = 1, 2, 3. 3.4 Counter Term Approach and Standard Correction Schemes Many of the standard correction schemes are recovered by setting various types of counter terms to the lattice pseudo-anomalies,

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Deµ = [−De (t), ∇Φ] ,

Dgµ = [−Dg (t), ∇Ψ ] ,

(43)

where De/g are linear differential operator: De/g ≡ αe/g ∂t + βe/g for real αe/g and βe/g . The choice D(t) ≡ 0 (α = β = 0) yields the hyperbolicelliptic constrained formulation of Assous et al. [ADH93] which couples the Gauss law divergence constraint with the hyperbolic evolution equations for the electric field and magnetic induction. This enlarged system can be solved numerically within a FE framework by applying a penalization technique or within the context of FD by using the projection method proposed by Boris [Bor70]. For D(t) ≡ I (α = 0, β = 1), the hyperbolic-parabolic formulation introduced by Marder [Mar87] is obtained, which can be approximated by the pseudo-current approach discussed for instance in Refs. [Mar87, Lan92]. As discussed in the previous subsection, the hyperbolic-hyperbolic form of the GLM Maxwell model system is recovered by setting D(t) ≡ ∂/∂t (α = 1, β = 0) [MO00B, MO00A]. This hyperbolic-hyperbolic GLM method can be most efficiently implemented into the finite-volume scheme. In this case, the numerical errors are transported out of the computational domain with the light velocities. Very recently, Dedner et al. [DKK02] adapted the mixed parabolic- hyperbolic operator D(t) (α = 0, β = 0) for divergence cleaning in the context of ideal MHD. The resulting mechanism underlying this mixed correction approach is the combination of error advection with additional damping due to diffusion.

4 Numerical Experiment I: Maxwell-Vlasov Equations We consider a simple but very typical situation occurring in the context of particle-in-cell (PIC) simulations [HEa81] which is based on the MaxwellVlasov equations. The simulation model consists of a plane diode with anode and cathode situated respectively on the left and right sides, with the domain of computation Ω = [0, 1]×[0, 1]. Here, electrons are injected at the left side of the device and accelerated throughout the diode by an external electric field. Under this physical setting, we have calculated the electric field configuration in the computational domain, once without correction to the divergence constraints and then with the presented Lorentz covariant hyperbolic GLM correction scheme. The snapshots of the E1 after a certain time and plots of the L2 norm of ∇ · E − ρe as a function of time are illustrated in Fig. 1. Indeed, if one neglects the transverse variations, the Gauss law d E1 /dx = ρe predicts that the E1 configuration monotonically decreases with x, since there are only negatively charged electrons in the device. In this regard, we observe from Fig. 1 that the non-corrected case runs into an unrealistic regime, whereas the corrected computation shows physical reliability. Also, we observe the clear violation of the charge conservation law without GLM correction. On the other hand, we observe the clear suppression of the L2 norm of ∇ · E − ρe to zero and thus

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the restoration of the electric charge conservation law, after implementing our GLM correction recipe.

5 Numerical Experiment II: Shallow Water MHD Equations 5.1 GLM Based Shallow Water MHD Equations The symmetry-based divergence cleaning method can also be applied to MHD equations. A toy (but realistic) MHD model upon which our extended GLM concept can be implemented is “shallow water” MHD (sw-MHD) [Gil00]. The corresponding sw-MHD equations are given by ⎡ ⎡ ⎤ ⎡ ⎤ ⎤ 0 h hv ∂ ⎣ hv ⎦ + ∇ · ⎣ hvv − hBB + I(gh2 /2) ⎦ = − ⎣ B ⎦ ∇ · (hB), (44) ∂t hB hvB − hBv v

where h represents the height of the shallow water system, v the two dimensional velocity and the B the two dimensional magnetic field which must

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satisfy the divergence constraint ∇ · (hB) = 0. The source term appearing on the right hand side is the Powell’s term [Pow94]. Here and hereafter, we shall simplify the notation by setting µ0 = 1, as has been conventionally adopted in many MHD literatures. Especially for the sw-MHD case, it √ is convenient to include the factor (4πρ)1/2 in the magnetic field, B = B/ 4πρ, where ρ is the fluid density. (Note that sw-MHD deals with incompressible fluids, and therefore ρ can be set to be constant.) Then, the physical quantities are ˆ and unit time Tˆ. It is easy to check the described in terms of unit length L ˆ 1 Tˆ−1 . The graviB-field in sw-MHD now takes the dimension of the velocity, L tational constant shall be fixed to unity (g = 1) for the numerical experiments presented below. The GLM concept can be applied to the Maxwell subsystem described by the last row of (44) and the divergence constraints: ∂ B + ∇ × E = 0, ∂t ∇ · B = 0,

(45) (46)

for B ≡ hB, E ≡ −v × B = Eˆ z. The GLM concept now can be realized by implementing counter term Ansatz via operator splitting ∂ B Cg Ψ = 0, (47) +∇· χ2 B ∂t Ψ where the degree of broken Lorentz symmetry has been parameterized by θ via 2 cos θ 0 Cgij = ζ 2 . (48) 0 sin2 θ Here, note that ∇ · B = 0 is the stronger condition than the magnetic charge conservation law. Therefore, the error transport via Ψ waves of the hyperbolic GLM plays an crucial role here for the divergence cleaning. 5.2 Numerical Experiments In this section, we present numerical results for the GLM-based shallow water MHD model. In the spirit of operator splitting, the genuine sw-MHD equations (44) are solved with a second order HLL-based FV solver, and the linear GLM subsystem (47) is solved with a second order Godunov solver. The numerical systems considered are a one-dimensional Riemann problem (RP) and a stationary two-dimensional RP which can be traced back to a one-dimensional standard problem for the conserved variables. One-Dimensional Riemann Problem We consider the sw-MHD equation (44) in the one dimensional computational domain Ω = [−1, +1] which is discretized by 1000 grid points. For the numerical experiments, we consider an initial value data configuration which explicitly violates the the divergence constraints in 1D (i.e. ∂x (hBx ) = 0),

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IC1 : U(x, 0) =

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(49)

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for U ≡ [h, hv, hB] . The resulting RP is investigated with various numerical approaches, which are: (i) the scheme without correction, (ii) the Powell source term Ansatz, (iii) the GLM correction without Powell sources (with one or ten sub-cycles per each time step) and (iv) the GLM correction combined with the Powell Ansatz (also with one or ten sub-cycles). The linear GLM subsystem has been implemented via operator splitting, adopting second order Godunov method. To avoid interaction between the Ψ waves, the CFL number √ has been fixed to σCF L = 1/2 = c1 χ∆t/∆x. For typical values like θ = 0, χ = 10 and the spatial increment ∆x = 2 · 10−3 , the time step size is given by ∆t = ∆x/20 = 10−4 . 0

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The RP of the GLM subsystem can be solved exactly [LMS02]. Simply quoting the result, the numerical solution started with the “wrong” initial data (49) should approach to the configuration with the “good” initial configuration given by 0 T UL = (1.00, 0.00, 0.00, 1.00, 0.00) for x < 0 , (50) IC2 : U(x, 0) = T UR = (2.00, 0.00, 0.00, 1.00, 2.00) for x ≥ 0 where hB1 is now a constant quantity. This has been confirmed by the numerical solutions at t = 0.3 illustrated in Fig. 2. From these plots, we can

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Fig. 3. Transport of the ∇ · (h B) error computed with no correction and with the GLM technique with one and ten sub-cycles (from left to right). These plots are recorded for 0 ≤ t ≤ 0.025 and −0.2 ≤ x ≤ 0.2.

summarize the following observations: If one solves the RP (with IC1 ) without correction, then the initial divergence error stays at the original site with respect to time. The Powell Ansatz based scheme, on the other hand, transports error parallel with the background fluid velocity u1 , as has been analyzed by de Sterck [DS01A]. Interestingly, we observe that the velocity component in y-direction is also affected by Powell’s error transport (cf. Fig. 2 second row), leading to a wrong wave structure. This suggests that for certain cases the Powell’s source term approach may not be reliable. Clearly, Fig. 2 indicates that GLM approach works very properly, and the solution agrees to the one obtained from the good initial condition IC2 . Also, note that the Ψ error waves do not affect the wave structure of hB2 . Though not shown in Fig. 2, we have also observed that the combined GLM-Powell technique performs slightly better for divergence cleaning than the GLM approach alone. The reason for this is that Powell’s source term also transports the error with the background velocity u1 and, hence, enhance the cleaning efficiency. The error transport of the GLM-based scheme is once again depicted in Fig. 3. There, we recognize that the ten sub-cycles in each time step results in roughly the ten-times faster divergence error cleaning than the case of only one sub-cycling. Stationary Two-Dimensional Riemann Problem We consider the two-dimensional sw-MHD system on the computational domain Ω = [−1, +1] × [−1, +1]. We numerically solve the time-dependent sw-MHD equations until it reaches the steady state, in order to obtain stationary configuration. We set the divergence free initial configuration given by ⎡ ⎤ ⎡ ⎤ 1.0 2.0 ⎢ 4.5 ⎥ ⎢ 11.0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ U(x, y, 0)|y≥0 = Qu = ⎢ 0.0 ⎥ and U(x, y, 0)|y<0 = Qd = ⎢ ⎢ 0.0 ⎥ (51) ⎣ 2.0 ⎦ ⎣ 1.0 ⎦ 0.0 0.0

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while keeping the boundary conditions U(x, y, t)|x=−1, y≥0 = Qu and

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(52)

at the computational boundary x = −1 for t ≥ 0. For all other borders of the domain Ω, we impose transmissive boundary conditions (see, e.g. Ref. [Tor97]). For the divergence-free initial configurations, we require the initial condition Ψ (x, y, 0) = 0 for the GLM-Ψ waves. Since the effective transport of the divergence errors out of the computational domain via Ψ waves is desired, it is in principle logical to impose transmissive boundary conditions for Ψ waves. However, it is often necessary to enhance the efficiency of the transmissive characteristics of the Ψ waves at the boundaries, particularly in the presence of shocks. For this purpose, we apply the so-called sponge layer technique [Hes97] in setting boundary conditions for Ψ . This is equivalent to applying parabolic-hyperbolic GLM correction scheme for the ghost cells, with the damping exponentially increasing from zero at the computational boundaries to infinity at the outermost ghost cell boundaries. The application of the sponge layer procedure absorbs the partial reflection of the Ψ waves at the boundaries, and massively enhances the transmitting property at the boundaries. The parameterization of the GLM sub-system has been achieved by setting ζ = 1 in (48) and determining χ from χ=

∆x 0.5 max{sin θ, cos θ} ∆t

for given ∆x and ∆t and fixed CFL number σCFL = 0.5. Now, only the parameter for the isometry-breaking, θ remains to be a free parameter. The computed wave configurations at t = 1.0 of the stationary RP obtained from various time-dependent, two-dimensional numerical methods are depicted in Fig. 4 for selected GLM parameterizations. The steady state configuration consists of two shock waves and one rarefaction wave. In essence, we notice from these plots a qualitative improvement of the wave resolution if the GLM sub-system correction is switched on either with or without the presence of the Powell’s source term (for further comparisons we refer to Ref [DS01B]). The wave patterns obtained only from the Powell source term approach do not appear in Fig. 4 because the code crashes due to the always increasing of the divergence error with time. The reason for this observation is the ill-defined divergence operator in the FV sense at the vicinity of the shock: The locally evaluated, inaccurate divergence sources has been used as right-side inputs of equation (44) for each time step which leads to a numerical fluctuation of h such that it locally enters into the “non-physical” negative region. The quantitative improvement of the wave structure as a consequence of the GLM sub-system correction is illustrated in Fig. 5 by the left plot. There, the temporal evolution of the discrete L1 -norm of the global divergence error is compared for different situations. It is vividly seen that the FV-based

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Fig. 5. Discrete L1 -norm of the divergence error for the two-dimensional, steadystate RP as a function of time (left) as well as a function of the isometry parameter θ (right) at t = 1.0. The time-dependent results are computed with θ = 0.05 π and θ = 0.375 π for the GLM sub-system and the combined GLM-Powell correction, respectively. Here, the computational domain consists of 100 × 100 grid cells.

divergence error is effectively carried out of the domain Ω by the Ψ waves if the simple GLM sub-system correction is used (dashed-dotted line). This effectiveness is slightly improved if the additional Powell transport is taken into account (dash-dot-dot). Obviously, performing no correction (solid line) as well as taking the divergence cleaning which solely based on Powell’s source term is not recommended by these results. As already discussed above, the observed behavior of the pure Powell correction can lead back to the strong accumulation of the divergence errors at (−1, 0) where two shocks collide, which is inherent in the FV context. The curves seen in the right part of Fig. 5 indicate the dependence of the L1 -norm of the divergence error on the degree of broken isometry. The best performance has been achieved with θ ≃ 0.05 π and θ ≃ 0.38 π for the GLM and combined GLM-Powell method, respectively. These optimal values of θ have been also identified for various parameters χ and for several tested grid cell sizes. The local divergence error distribution has been illustrated in Fig. 6. It is clearly seen that the GLM subsystem performs the error correction also on the local level. For comparison, the right most figure shows how the Ψ field “feels” the divergence error after the full configuration reached the numerical steady state, which has been calculated from (47). Different from the FV approximation of ∇ · (hB), the Ψ wave virtually does not detect the error for t ≥ 1.

6 Discussion In this paper, we have presented a field theoretical derivation of the GLM Maxwell equations. In particular, we have extensively explored the basic sym-

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Fig. 6. Divergence error distributions at = 1 0 obtained without correction (left) and with the combined GLM-Powell correction approach for = 0 375 (middle and right). The ﬁrst two plots show the distribution of ( B) in FV sense while the right plot depicts the distribution computed from t Ψ .

metry features of Maxwell theory for the derivation of an extended version of the genuine hyperbolic GLM Maxwell system. We have then combined these extended GLM Maxwell equations with the shallow water magnetohydrodynamics (sw MHD) via operator splitting. A new extended feature of the presented GLM scheme is that the direction of divergence error transport can, in principle, be controlled by breaking the isometry artificially. In some practical applications for instance, if the divergence perturbation occurs near a corner of the computational domain such a speed-up of the information transport into a certain direction would be an advantage. The underlying theoretical framework of the extended GLM scheme has been validated by the presented numerical experiments for the Maxwell-Vlasov test system. A further justiﬁcation of the extended GLM idea has been provided by the numerical solutions for the Riemann problems in sw MHD, where the GLM subsystem has been implemented via operator splitting. This means that the updating of the conserved sw MHD variables is decoupled from the subsequent divergence cleaning step in such a way that it can be combined with any available standard sw MHD solver. This basic plot can be easily extended to the full GLM solvers. Since the derivation of the extended GLM scheme has been based on fundamental symmetries of Maxwell theory, the underlying concept can be applied to certain class of numerical solvers for physical problems which are described by hyperbolic partial diﬀerential equations with elliptic constraints, where underlying physical symmetry plays an important role. These include relativistic MHD, general relativity, non-abelian gauge ﬁeld theory in the context of non-perturbative quantum ﬁeld theory, etc.

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References [ADH93]

Assous, F., Degond, P., Heintze, E., Raviart, P., Segr´e, J.: On a ﬁniteelement method for solving the three-dimensional Maxwell equations, J. Comput. Phys., 109, 222–237 (1993) [Bor70] Boris, J. P.: Relativistic plasma simulations – Optimization of a hybrid code, in Proc. 4th Conf. on Num. Sim. of Plasmas. NRL Washington. Washington DC, 3–67 (1970) [DKK02] Dedner, A., Kemm, F., Kr¨oner, D., Munz, C.-D., Schnitzer, T., Wesenberg, M.: Hyperbolic Divergence Cleaning for the MHD Equations. J. Comput. Phys., 645–673 (2002) [DS01A] DeSterck, H.: Hyperbolic theory of the shallow water magnetohydrodynamics equation, Phys. Plasmas, 8 3293–3304 (2001) [DS01B] DeSterck, H. : Multi-dimensional upwind constrained transport on unstructured grids for ’shallow water’ magnetohydrodynamics. AIAA Paper, 2001–2623 (2001) [Gil00] Gilman, P.A.: Magnetohydrodynamic “Shallow Water” Equations for the Solar Tacholine. The Astrophys. J., 544, L79-L82 (2000) [Hes97] Hesthaven, J.: The analysis and construction of perfectly matched layers for the linearized Euler equations. ICASE Report no. 97-49 (1997) [HEa81] Hockney, R. W., Eastwood, J. W.: Computer Simulation using Particles. McGraw-Hill, New York (1981) [Lan92] Langdon, A. B.: On enforcing Gauss’ law in electromagnetic particle-incell codes, Comput. Phys. Commun., 70, 447–450 (1992) [LMS02] Lee, Y.J., Munz, C.-D., Schneider, R.: Lagrangian and Symmetry Structure of the Diveregnce Cleaning Model Based on Generalized Lagrange Multipliers. to appear in Int. J. of Mod. Phys. C., 15 (2004) [Mar87] B. Marder: A method incorporating Gauß’ law into electromagnetic PIC codes, J. Comput. Phys., 68, 48–55 (1987) [MO00A] Munz, C.-D., Omnes, P., Schneider, R.: A three-dimensional ﬁnitevolume solver for Maxwell equations with divergence cleaning on unstructured meshes, Computer Physics Communications, 130, 83–117 (2000) [MO00B] Munz, C.-D., Omnes, P., Schneider, R., Sonnendr¨ ucker, E., Voss, U.: Divergence correction techniques for Maxwell solvers based on a hyperbolic model. J. Comput. Phys., 161, 484–511 (2000) [MSS99] Munz, C.-D., Schneider, R., Sonnendr¨ ucker, E., Voss, U.: Maxwell equations when the charge conservations is not satisﬁed. C. R. Acad. Sci. Paris, t. 328, S´erie I, 431–436 (1999) [Pow94] Powell, K.: An approximate Riemann solver for magnetohydrodynamics (that works in more then one dimension), ICASE-Report 94-24 (NASA CR-194902), NASA Langley Research Center, Hampton, VA 23681 (1994) [Rot92] Rothe, H.J.: Lattice Gauge Theories. World Scientiﬁc, Singapore (1992) [Tor97] Toro, E.: Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, Berlin – Heidelberg (1997) [Wei72] Weinberg, S.: Gravity and Cosmology. John Wiley & Sons, New York (1971) [Wei95] Weinberg, S.: The Quantum Theory of Fields. Cambridge University Press, Cambridge (1995)

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Flexible Flame Structure Modelling in a Flame Front Tracking Scheme Heiko Schmidt1 and Rupert Klein2 1

2

´ Laboratoire EM2C, Ecole Centrale Paris et C.N.R.S., Grande Voie des Vignes, 92295 Chˆ atenay-Malabry Cedex, France, [email protected] Fachbereich Mathematik and Informatik, Freie Universit¨ at Berlin, Arnimallee 2-6, 14195 Berlin, Germany, [email protected]

Summary. A numerical technique for the simulation of accelerating turbulent premixed ﬂames in large scale geometries is presented. It is based on a hybrid capturing/tracking method. It resembles a tracking scheme in that the front geometry is explicitly computed and propagated using a level set method. The basic ﬂow properties are provided by solving the reactive Euler equations. The ﬂame-ﬂow-coupling is achieved by an in-cell-reconstruction technique, i.e., in cells cut by the ﬂame the discontinuous solution is reconstructed from given cell-averages by applying RankineHugoniot type jump conditions. Then the reconstructed states and again the front geometry are used to deﬁne accurate eﬀective numerical ﬂuxes across grid cell interfaces intersected by the front during the time step considered. Hence the scheme also resembles a capturing scheme in that only cell averages of conserved quantities are updated. To enable the modelling of inherently unsteady eﬀects, like quenching, reignition, etc., during ﬂame acceleration, the new key idea is to provide a local, quasi-onedimensional ﬂame structure model and to extend the Rankine-Hugoniot conditions so as to allow for inherently unsteady ﬂame structure evolution. A source term appearing in the modiﬁed jump conditions is computed by evaluating a suitable functional on the basis of a onedimensional ﬂame structure module, that is attached in normal direction to the ﬂame front. This module additionally yields quantities like the net mass burning rate, necessary for the propagation of the level set, and the speciﬁc heat release important for the energy release due to the consumption of fuel. Generally the local ﬂame structure calculation takes into account internal (small scale) physical eﬀects which are not active in the (large scale) outer ﬂow but essential for the front motion and its feedback on the surrounding ﬂuid. If a suitable set of diﬀerent (turbulent) combustion models to compute the ﬂame structure is provided, the new numerical technique allows us to consistently represent laminar deﬂagrations, fast turbulent deﬂagrations as well as detonation waves. Supplemented with suitable criteria that capture the essence of a Deﬂagration-to-Detonation-Transition (DDT), the complete evolution of such an event can be implemented in principle.

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1 Introduction Turbulent premixed flames have been theoretically studied since many years. In spite of the increasing computational power numerical simulation of turbulent premixed flames remains in these days a challenging task. Concerning the physics of combustion processes one has to cope with different length and time scales. Flame structures may be of the order of millimeters while the characteristic extension of explosions for, e.g., astrophysical applications is larger than a kilometer. An additional problem is the unsteady behaviour of combustion. Due to the feedback between turbulence, gas expansion, and flame front dynamics a continuous acceleration of premixed flames – ranging from laminar speed of the order of cm/s up to speeds of several km/s – can occur. This process typically occurs, e.g., in large scale gas explosions and astrophysical nova- and supernova explosions. In the context of flame accelerations and the Deflagration-to-Detonation-Transition (DDT) one is faced with rapidly changing thermodynamic, mean flow, and turbulence conditions. One consequence is that the internal structure of the propagating combustion front will become inherently time dependent. In addition, the turbulence intensities associated with the accelerating flow will increase and grow rapidly beyond the characteristic burning velocity of a laminar flame. While turbulence intensities are still low, quasi laminar combustion takes place in thin “flamelets”. Turbulent combustion modelling will in this case aim at a description of the net flame surface area and of the mean quasi-laminar burning velocity in order to arrive at the net rate of unburnt gas consumption. If, on the other hand, turbulence intensities increase dramatically, then the turbulence-induced strains will locally distort the flamelet structures or even quench them completely, and a more stochastic interaction between reaction, turbulent transport, and diffusion becomes significant. As a consequence, in these “thin-reaction zone”, “broken-reaction zone” and “well-stirred reactor” regimes, very different effective turbulent combustion models must be employed, [Pet86, Pet99, Pet00]. The different regimes are illustrated in the famous “Borghi-Diagram” (Sketched qualitatively in Figure 1). In the flamelet regime the flame thickness is much smaller than the smallest turbulent (Kolmogorov) scale, and a similar estimate holds for the time scale ratio. The flamelet regime occupies the region Ka < 1. Above Ka = 1 one enters the “thin-reaction zone regime” according to Peters [Pet99]. In this regime, the deflagration preheat zones are larger than the Kolmogorov scale, so that the turbulence non-trivially distorts these regions, but the actual reaction zones are still thin and compact. When the Kolmogorov eddy size becomes comparable even to the reaction zone thickness lδ , turbulence may disrupt the reaction layers and we enter the regime of “broken reaction zones”. This is a regime where the Damk¨ ohler number is still large, i.e., reactions are fast compared to fluid mechanical effects. Below the threshold Da ≈ 1, however, the assumption of rapid chemistry and the strong coupling between reaction and diffusion, present in the two previous regimes, is destroyed. This is called the “well-stirred-reactor regime”.

Flexible Flame Structure Modelling in a Flame Front Tracking Scheme 4

v s

, 10

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Fig. 1. Borghi-diagram. v ′ : turbulent ﬂuctuation velocity; slam : laminar burning velocity; l: integral scale of turbulence; lF : laminar ﬂame thickness; Ret : turb. Reynolds number v ′ l/ν; Ka: Karlovitz number = (lF /η)2 ; Kaδ : mod. Karlovitz number = ohler number. (lδ /η)2 ; lδ : Reaction zone thickness; η: Kolmogorov length; Da: Damk¨

Thus a numerical code that is supposed to cover the full range of combustion phenomena during flame acceleration and DDT must be capable of dynamically accessing the correct combustion model for all the regimes the solution passes through. Here we present a new numerical technique which— given such a set of (turbulent) combustion models—allows us to consistently represent laminar deflagrations, fast turbulent deflagrations as well as detonation waves. Supplemented with suitable DDT criteria, the complete evolution of a DDT process can be implemented in principle. A slightly modified version of the original algorithm of Smiljanovski et al. [SMK97], which we started from, is briefly described in the next section. It implements a version of the flamelet model of turbulent premixed combustion combining level-set techniques to propagate interfaces with conservative finite volume methods for compressible flows. The flame physics are condensed into a burning rate law. Standard Rankine-Hugoniot jump conditions are applied to achieve a coupling between flame and flow. Our base scheme for the present developments differs from that in [SMK97] in that it removes a formal inconsistency between level set propagation and the fuel consumption source terms. This is a very important modification to end up with a consistent and stable numerical scheme. Nevertheless we refer the reader to [SK03] for a detailed description, since in this paper we restrict to the basic ideas of modelling flames with different unsteady structures. This even improved base scheme still has the following limitations: • It assumes a quasi stationary flame structure, and • depends on the availability of an algebraic burning rate law.

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Since these two points in most cases are not fulfilled, our goal is to develop a modular numerical approach for general unsteady flame structures, while retaining the assumption of a thin flame in comparison with the overall flow geometries. Concerning the notion of the required scale separation, the applicability of the method will even be extended to a “numerical flamelet regime”, i.e., flames can be computed as long as the maximum thickness of all locally appearing flame structures is smaller than the numerical grid resolution. Overcoming this restriction by allowing flame structures larger than the grid size is imaginable but not within the scope of this paper. The modular extensions of the “standard method” are presented in section three. Exemplary results are shown in section four. Sample implementations incorporate a laminar combustion model and an unconventional multi-regime turbulent combustion model based on the linear eddy model (LEM) by Kerstein [Ker89]. The implementation of a turbulent combustion model developed by Thibault and Zhang, [ZT98] based on a classical PDF (Probability Density Function) approach by Pope [Pop81, Pop85] and using a 8-reaction H2 –O2 scheme according to Maas and Warnatz [MW88] has also been performed by the authors and is described in detail in [Sch02, SK03]. In the latter work the occurring problem of coupling an overall binary species system to a local multispecies system has been discussed. Finally, results for three dimensional flame propagation over a single obstacle are shown. The detailed flame physics in that case are provided by a turbulent flame structure module based on a LEM ansatz, simplified for the flamelet regime.

2 Basic Level-set/in-cell-reconstruction Scheme Smiljanovski et al. [SMK97] introduce a deflagration capturing/tracking hybrid scheme which allows a robust representation of turbulent high speed combustion. A deflagration is considered as a reactive discontinuity, which is embedded in a compressible surrounding flow. The flame surface is represented as the level set of a dynamically evolving scalar function. Flame-flow-coupling is realized by explicitly invoking the Rankine-Hugoniot jump conditions at the discontinous front. An explicit burning rate law expresses the net unburnt gas mass consumption as a function of the unburnt gas conditions. We emphasize that this approach is strongly limited to conditions where such burning rate laws are available and where the standard Rankine-Hugoniot conditions apply. 2.1 Governing equations The present developments are based on the reactive Euler equations in conservation form for an ideal gas mixture. Wherever possible we describe our approach for the general case of a mixture of nspec chemical species. However,

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the concrete implementations in this contribution restrict to a binary system with nspec = 2. Applications for more complex chemical systems are beyond the scope of this paper, but scheduled to be addressed in future work. Here the following notation is used: ρ is the density, v the flow velocity vector, p the pressure, Yi , Qi the mass fraction and formation enthalpy of the ith chemical species, respectively, E the total energy, I the unit matrix and γ is the isentropic exponent of the mixture. Throughout this paper we use the spec simplifying assumption that γ = const., and we will abbreviate Y = {Yi }ni=1 . With these conventions the governing equations read ∂ ρ + ∇ · (ρv) = 0 ∂t ∂ (ρv) + ∇ · (ρv ◦ v + pI) = 0 ∂t ∂ (ρE) + ∇ · ([ρE + p]v) = 0 ∂t

(1)

with the equation of state spec

n p 1 + ρv · v + ρ ρE = Qi Yi . γ−1 2 i=1

(2)

The chemical reactions are described by the balance laws ∂ (ρY ) + ∇ · (ρvY ) = −ρω ∂t

(3)

where ω is the vector of effective reaction rates. The detailed formulation for the species source term ω depends on the combustion mode considered, i.e., it depends on whether we consider instantaneous reactions within a flame discontinuity or distributed finite rate chemistry. Notice that for resolved calculations (local flame structures concerning our present work) diffusion terms have to be added to (1) and (3), while we neglected them for unresolved (large scale) ones, since in such cases their physical effect is smaller than the artificial diffusion of the numerical scheme. In our concrete implementations we will restrict to a simple two-species chemical system, i.e., nspec = 2, and let Y = Y1 denote the mass fraction of the relevant energy carrying fuel species. For resolved computations with such a binary mixture we presently use a standard Arrhenius rate law Ea

ω = BY e− RT

(4)

with the temperature T given by the ideal gas law T =

p . ρR

(5)

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Here B is a preexponential factor, Ea the activation energy and R the gas constant. Notice that our general set-up allows for more sophisticated schemes when needed. In particular in the context of flame tracking via our level set technique the reaction source terms are extracted from the level set propagation. Concerning this important point we refer the reader to [SK03]. In addition, the effect of turbulence can be taken into account through a standard k − ǫ turbulence model as has been demonstrated by Smiljanovski et al.[SMK97]. Notice, that we currently transfer our overall scheme to a more sophisticated Large Eddy Simulation (LES) type turbulence model approach [Sch03]. 2.2 Level set representation of flame front geometries The appropriate formulation of the ith species’ reaction source term for a reactive discontinuity is a delta distribution, ωi = [ Yi ] s δ (G) |∇G|, where [ Yi ] is the jump of the ith species’ mass fraction across the discontinuity, s is the flame speed, G(x, t) is a scalar function whose level set G = 0 coincides with the flame surface and δ(·) is the dirac delta distribution. The evolution of the level set function is described by the well-known “G-equation” [Wil85, Pet86] ∂ G + (v + sn) · ∇G = 0 . ∂t

(6)

Here n denotes the front normal and v the local flow velocity. We note that the expression (v + sn) · n is invariant across the flame surface, provided that s always denotes the local relative velocity between the flame and the surrounding gas. That is, s has a nonzero jump across the flame surface. The meaning of G and s in (6) depends on the flame surface considered. In Figure 2 Damk¨ ohler’s ansatz for thin, locally laminar flames is illustrated. It states that the ratio of the turbulent to the laminar flame speed is equal to the ratio of the increment dA of the turbulent flame area A to the increment of the averaged flame surface dA. Thus we have dA sl = dA st . In our case s ≡ st is a turbulent burning velocity and G indicates the mean turbulent flame surface. Only a single level set, here G = 0, has a physical meaning. As a consequence the distribution of G away from the mean flame front is not unique. It has turned out to be numerically convenient, (see, e.g., [FAX99, Set99, SMK97]), to assign G(x, t) to be a signed distance function satisfying |∇G| = 1 .

(7)

Modern numerical methods for level set equations are described, e.g., by Sethian in [Set99]. In our formulation we use a propagation scheme that combines equation (6) and (7) to ∂ |G| G + (v + sn) · ∇G = sign(G) (1 − |∇G|) · . ∂t ζ

(8)

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Fig. 2. Flamefront described by a level set function.

The source term on the right hand side is designed to “drag” the level set function towards a signed distance function wherever it tends to deviate from it. In our code (8) is solved only within a layer of thickness ζ surrounding the flame front. Values of G at larger distances are set to a constant ±ζ (positive in the burnt and negative in the unburnt gas). A more detailed discussion of the level set formalism and the used discrete numerical implementation are given in [Sch02, SK03]. 2.3 Flame-flow coupling for reactive fronts Tracking the time evolution of a flame surface is only one part of a complete description of gasdynamic discontinuities. The second equally important issue is the description of flame-flow coupling. Fedkiw et al. [FAX99] perform this by applying a (non-conservative) ghost fluid method. Smiljanovski et al. [SMK97] introduce a coupling scheme that relies on “in-cell reconstruction” and maintains the grid cell-by-grid cell conservation properties of the underlying flow solver. The idea is to construct a finite volume scheme which, between time steps, updates only the averages of all conserved quantities for each grid cell. As described shortly, there is a way to reconstruct the two separate preand post-front substates within each “mixed-cell” from the grid cell averages by invoking the averaging properties and the Rankine-Hugoniot type jump conditions. Given such a reconstruction, at each intersected cell interface there are two pairs of adjacent states available (see Figure 3b): One pair of burnt gas conditions (index b) and one pair of unburnt gas conditions (index u). Modern higher order shock capturing schemes use approximate Riemann solvers to derive effective numerical fluxes, F , for mass, momentum, and energy from pairs of adjacent states U at grid cell interfaces [LeV92]. Having two appropriate pairs of left and right states available a straight-forward application of a standard Godunov type numerical flux functions yields separate flux densities for burnt and unburnt conditions.

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F ν = F God. (U lν , U rν ) with

(ν ∈ {u, b})

(9)

An area-weighted superposition of these fluxes (see Figure 3a) then allows an accurate numerical approximation of the net flux across the grid cell interface F = β F u + (1 − β) F b − δβ[[F ] upw ,

(10)

where β is the unburnt area fraction of the interface, δβ is the change of β in one time step (see Figure 3c), and the term [ F ] upw is an upwind-evaluated jump of the flux vector across the flame surface (for details see [SMK97]). An implementation using directional operator splitting is described in [SMK97], and, say, the first step of the scheme updates all conserved quantities via ' ∆t & n+ 12 n+ 12 n F = − − U , (11) U n+1 − F i,j,k i,j,k i+ 21 ,j,k i− 21 ,j,k ∆x

where n is the discrete time level tn , ∆t the discrete time step, ∆x the grid n+ 1 size, and i, j, k are indices of a grid cell center and the fluxes F i± 12,j,k are area 2 weighted averages of burnt and unburnt fluxes according to (10). Note that equation (11) only holds for the conserved quantities. For the species equation, the source term has to be evaluated additionally. Concerning this point we refer to [SK03]. The remaining task in summarizing the base scheme on a)

y b

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Fig. 3. Calculation of net ﬂuxes in “mixed cells” (a); adjacent “mixed cells” in x-direction (b); decomposition of the cell interface in a y-t diagram into fractions facing burnt (1 − β) and unburnt gas (β − δβ), and one fraction that is swept by the front during the time step (δβ) (c).

which the present developments are founded is to describe the in-cell reconstruction procedure. The deflagration is considered as a reactive discontinuity embedded in a compressible surrounding flow. Thus, it should satisfy the standard Rankine-Hugoniot jump conditions, which are nothing but an expression for the conservation of mass, momentum and energy across the front. These conditions can be written in a compact form as D [ U C ] − n · [ F C (U )]] = 0

(12)

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U C = (ρ, ρv, ρE) vector of conserved quantities U = (U C , ρY ) total state vector FC flux tensor of conserved quantities s burning velocity D = (s + v · n) normal front propagation speed [ Φ]] = (Φb − Φu ) jump of Φ across the flame front The total state vector includes the vector of chemical species mass densities ρY . These conservation conditions must be supplemented with suitable jump conditions for the chemical species mass fractions. For the chemical species, the relevant chemical kinetic reaction models yield the appropriate conditions. Generally, one either may assume complete reaction, in which case only reaction products are left and the mass fractions of the deficient reactant species are zero in the burnt gases, or one finds chemical equilibrium in the burnt, in which case one obtains a set of algebraic equilibrium equations. Together with the atom conservation laws, these relations will provide a number of additional constraints between the pre- and post-front states. Another set of relations connects the pre- and post-front states with the integral cell averages within all “mixed-cells” in a numerical computation. The averaging conditions state that the computed cell averages should be equivalent to volume weighted averages of the burnt and unburnt states U i = (αU u + (1 − α)U b ),

(13)

where α is the unburnt volume fraction of the i-th mixed cell. Both the level set and the Rankine-Hugoniot jump conditions depend on the burning velocity s. For deflagration discontinuities this burning velocity must be supplemented, e.g, through a flame speed function, [Cho80], [SMK97], s = s(p, ρ, Y ).

(14)

Within each mixed cell we may consider the cell averages U as given and the burnt and unburnt states U b and U u as unknowns for the in-cell reconstruction procedure. It turns out, [SK03], that not only does the number of equations match that of the unknowns, but the resulting equations are also independent, so that the equation system can be solved using standard nonlinear equation solvers. Thus, burnt and unburnt gas conditions can be reconstructed from the mixed cell averages as conjectured above. This completes the description of the base scheme for deflagration tracking.

3 Tracking Flames with Unsteady Internal Structure 3.1 Motivation Deflagration-to-Detonation Transition (DDT) can principally occur through at least two different mechanisms. The first, which we call here “Mode A”DDT, is characterized by pressure wave accumulation at considerable distances away from the (turbulent) deflagration, local autoignition and a purely

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gasdynamics-induced runaway to detonation. In contrast, “Mode B”-DDT occurs through physical processes that are intimately related to the turbulent combustion within the “flame brush”. It can be argued that excessive strain may locally quench quasi-laminar flamelets, then re-mix hot burnt gas with cold fresh mixture, and in this fashion establish an ignitable mixture with variable auto-ignition delay. A sequence of auto-ignitions may then lead to DDT via SWACER-type mechanisms [LKV78, ZLMS70, Zel90]. A detailed numerical description of this sequence is not possible when the interface between burnt and unburnt is represented as a sharp gasdynamic discontinuity by a standard tracking scheme. Thus, our numerical technique must be supplemented with additional capabilities allowing a dynamic description of the internal unsteady structure of a reactive front. Since flame quenching by highintensity turbulence is associated with a disruption of laminar flamelets (Peters [Pet99]), and a stochastic interaction between reaction, turbulent transport, and diffusion, we have opted to extend the flame front tracking algorithm by including turbulent combustion models based on the PDF-[Pop81, Pop85] and LEM-approaches, [Ker88, Ker89], where the former developments can be found in [SK03]. The general strategy for extending the scheme is, however, more general, so that the final result is a modular algorithmic structure that allows the incorporation of a variety of internal flame structure descriptions. This will ultimately enable us to capture turbulent combustion processes within any of the regimes in the “Borghi-Diagram” (see Figure 1). 3.2 Generalized formulation of a tracking scheme In the “well-stirred-reactor regime”, and also in the “thin-reaction-zone regime”, the assumption of a limitingly thin flame sheet is not applicable and, even though flame fronts will still tend to be compact, their internal dynamics must be accounted for. In order to systematically derive the necessary modifications to the existing numerical technique we reconsider the original derivation of the standard Rankine-Hugoniot conditions in a single space dimension first. Conservation of mass, momentum and energy is required for a space-time control volume that encloses the discontinuity (see Figure 4). The general conservation law for the vector of conserved quantities, U C = (ρ, ρv, ρE), ∂ C ∂ U + (F C (U )) = 0 , (15) ∂t ∂x where F C is the flux tensor, is first transformed to a moving coordinate system with ξ = x − Dmod t and τ = t. The transformed equation then reads ∂ C ∂ ∂ U − Dmod U C + (F C (U )) = 0. ∂τ ∂ξ ∂ξ

(16)

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δ << L Temperature

τ

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ξ dx =D box dt

x

Fig. 4. Travelling wave with unsteady internal structure.

Here Dmod denotes the velocity of an observer or a moving coordinate system, later called “module”, which encloses the onedimensional flame structure. The thickness δ of the structure is still assumed small compared to the characteristic geometrical scale L of the system considered. When the structure is a stationary travelling wave, then for a suitable choice of D the time derivative C ∂ vanishes identically and straight forward integration w.r.t. ξ yields the ∂τ U standard Rankine-Hugoniot conditions, with [Φ] [ ] = (Φb − Φu ) denoting a jump of a quantity Φ and D = (s + v · n) as the normal flame velocity −D[[U C ] + [ F C ] = 0.

(17)

This, however, is a special situation and the general result of such an integration is ξ1 ∂ C C C U dξ, (18) −Dmod [ U ] + [ F ] = ∂τ ξ0 where ξ0 , ξ1 are the boundaries of the moving coordinate system. Our extension of the flame front algorithm may be summarized as follows (see Figure 5 for a better understanding): 1. Unsteady jump conditions: For every cell cut by the flame (mixedcell), a reconstruction of the solution U across the flame discontinuity is performed (see section 2). But the jump conditions from (17) are replaced with (18), where the right hand side is computed in a onedimensional coordinate system, now called flame structure module. 2. Flame structure modules: For each grid cell intersected by the flame, a separate quasi-onedimensional flame structure module is running parallel to the overall flow solver (the “main module”) to provide the needed flame structure information. The entire flame structure is considered to evolve continuously in time and to travel with points on the flame front,

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i.e., with trajectories of dx/dt = (v + sn) (see (6)). This also provides a rule for distributing the flame structure information available in the current “flame cells” into cells that are newly intersected during a time step: The flame structure in the upwind cell with respect to (v + sn) is used to initiate a new flame structure computation. When the flame leaves a grid cell, the associated flame structure module is eliminated. The choice of the flame structure module depends on the combustion regime described by turbulent and chemical time and length scales (see Figure 1). During our studies we implemented four different flame structure modules, three of which are briefly described now. The fourth module that computes turbulent flame structures in the ”well-stirred-reactor-regime” via a PDF type model is explained in [SK03], where one also finds additional information concerning the structure modules and their coupling to the ”main module”. Note that these are only sample implementations and actually this is the point where the “user” of our algorithm may insert his preferred module. Module A: • Flow Model – “compressible”, onedimensional Navier-Stokes solver – explicit “shock capturing” technique • Reaction Model – One-step Arrhenius kinetics • Combustion regime validity – laminar flames; region of Ret < 1 in Figure 1 Module B: • Flow Model – 1D zero Mach number solver (projection method) – turbulent transport with Linear Eddy Model, Kerstein [Ker88] • Reaction Model – One-step Arrhenius kinetics • Combustion regime validity – Valid for all combustion regimes, in principle Module C: • Flow Model – GLEM-Method according to Menon and Kerstein with flamelets described by local level set functions [MK92] – turbulent transport by Linear Eddy Model applied to the associated G-Field (gridless implementation) [Wun01] • Reaction Model – Each Flamelet propagates with a prescribed laminar burning speed • Combustion regime validity – Model only valid in Flamelet Regime 3. Quantities provided by the flame structure module: After the flame structure computation between two global time levels the right hand side of (18) is computed by a suitable discrete approximation

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The net mass burning rate also is obtained from the structure module computations again by an integral evaluation. Since convective and diffusive effects only redistribute the chemical species, but do not consume or produce them, the net burning rate is determined solely by the integral of the reaction source term. Thus, for a binary mixture ρs =

ξ1

ξ0

ρωY dξ =

ξ1

ξ0

∂ (ρY )dξ + ∂τ

ξ1

∂ (ρuY )dξ. ∂x

(19)

ξ0

A further quantity that must be extracted from the flame structure module in a multi-species case is the specific heat release ∆Q = Qi (Yb,i − Yu,i ). i

For a binary mixture and assumed complete reaction this reduces to ∆Q = Y1 Q1 , i.e., to the specific heat release of the energy carrying species. 4. Quantities needed by the module: The flame structure module receives its input from the “outer flow computation”. The input can be of two different types: • Explicit boundary conditions: This means that the 1D-module receives time and space interpolated states from the main grid. These states may be needed on the boundary of the onedimensional domain, or in “ghost-cells” depending on the method used. • Integral mean values: These are values defined for the whole 1Dmodule, i.e., turbulence quantities or the temporal change of the thermodynamic background pressure, say dP/dt, needed when the incompressible flow equations are solved in the 1D-domain. Note that, until now we neglected any turbulence effects, but a turbulent flame structure computation will at least need turbulent length and time scales as input parameters. For details concerning the inclusion of a k − ǫtype turbulence model for the burnt and unburnt gas flows the reader is referred to [SMK97]. Extension of our method to LES type turbulent flow modelling is work in progress [Sch03]. 5. How to keep the flame structure in the module’s computational domain: A further important step is the determination of the effective observer velocity Dmod . A criterion for it is obtained by requiring that in the moving frame of reference the reaction front is nearly stationary or may oscillate only mildly about a given center. The general rule is to identify a characteristic species whose spatial variation locates the reaction zone. For a Hydrogen-Air-Steam mixture, this might be the Hydrogen mass fraction Y . In an attached normal frame with the coordinate ξ, the reaction zone can be localized approximately by requiring the expression ξ1 1 Y dξ, (20) ξ∗ = Ymax ξ0 to be nearly constant in time, where ξ0 , ξ1 are the boundaries of the internal structure computation. The observer velocity should then be adjusted

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in time such that ξ ∗ (t) ≈ ξ0∗ with ξ0∗ = 12 (ξ1 + ξ0 ). Notice that the net effect of modifying Dmod is a Galilei-transform of the system states within the structure module, which is easily done by adding a suitable constant correction to all flow velocities and to accordingly correct kinetic energies. For a practical implementation we refer to [Sch02, SK03]. Our extended algorithm is schematically summarized in Figure 5. It is based on the ansatz (see upper box) of Smiljanovski et al. that combines the solution of (1) and (3) with a level set approach in which (8) is solved. The combination is achieved by an in-cell-reconstruction scheme, see section 2. Our extended general coupling procedure (see box in the middle) that makes it possible to track flames with unsteady internal flame structure was described in this section. Within the in-cell-reconstruction procedure, a flame structure module is “called” for each “mixed-cell”. The module runs for the time of one discrete global timestep ∆t and, at the end, provides quantities like the burning velocity, the specific heat release and source terms in the

Conservation Laws

Level Set Approach Ansatz of Coupling

Smiljanovski,

via

Moser and Klein

In Cell Reconstruction strategy

burning velocity s Boundary Conditions

R.H. Integrals specific heat release ∆Q

generalized modular coupling procedure

Interface to Flame Structure Module Compressible Flow

Incompressible Flow

Equations

Equations ?

G LEM Ansatz (Kerstein, Menon)

Description of quasi onedimensional laminar Flames

Our choice of

pdf Ansatz

ODT Ansatz

LEM Ansatz

(Pope)

(Kerstein)

(Kerstein)

stochastic Turbulence Models

Fig. 5. A schematic illustration of the extended algorithm.

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jump conditions appearing in (18). During its run the module needs explicit and/or integral boundary conditions extracted from the “outer flow” solution. In the lower box of Figure 5 the components of such flame structure modules are exemplarily shown. To describe quasi onedimensional laminar flames one can solve the compressible or incompressible Navier-Stokes equations. Since turbulence is a threedimensional phenomenon the neglected effects might be added via stochastic turbulence models. In the wide range of available models, one chooses one that produces good results for a certain combustion regime (see Figure 1). What kind of combustion regime we have to consider is extracted by local length and time scales of chemistry and flow.

4 Results 4.1 Ignition of a laminar flame First we consider a laminar flame in one space dimension. The internal structure of the flame is explicitly computed using “Module A”. The flame structure is represented on a sub-mesh of about 100 grid points, yielding sufficient spatial resolution for a detailed unsteady flame computation. No emphasis is given here to efficiency, as this is merely a demonstration of the technology. Thus, the internal structure module uses explicit time stepping and time steps corresponding to a Courant number based on the local sound speed. The internal structure module is coupled to our one-dimensional flame front tracking-capturing hybrid scheme as described before. The internal and outer flows are coupled in the present simulation through the modified Rankine Hugoniot conditions from (18) and explicit boundary conditions for the internal module that are interpolated from the outer flow data. The specific heat release Q as well as the burning speed s are computed from the structure module. The initial data represent a discontinuity separating hot burnt and cold unburnt gases. The left plot in Figure 6 shows the time history of the fuel mass fraction in the internal structure computation. Molecular mixing and heat conduction lead to a sudden ignition of the unburnt. The process induces a continuous smoothing of the fuel mass fraction and the asymptotic establishment of a laminar flame profile. Two pressure pulses emerge from the ignition location. The pressure evolution is displayed in the right plot of Figure 6. In the right plot of Figure 7 the time evolution of the net combustion rate within the flame structure module is shown. This quantity enters the outer flow flame tracking module as the relevant burning speed needed in the reconstruction of burnt and unburnt states within a “mixed-cell”. The increasing burning speed leads to a thermal gas expansion and to the generation of outward propagating pressure pulses as can be seen in the left picture in Figure 7. With the same type of model the extinction of a laminar flame by pressure pulses has been calculated in [SK03].

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0 * dt 50 * dt 100 * dt 1000 * dt

0.8

1.04

p (pressure)

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(fuel mass fraction )

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10 * dt 50 * dt 100 * dt

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0

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0.1

Fig. 6. Mass fraction Y (left) and pressure p (right) in the ﬂame structure module at diﬀerent times. 1.05

Y 10*dt 30*dt 100*dt 220*dt

p (pressure)

1.03

0.01

s (laminar flame speed)

1.01

0.99

0 0

2

4

x

6

8

10

0

0.2

0.4

t

0.6

0.8

Fig. 7. Pressure evolution in ”outer ﬂow ﬁeld” (left) and time history of burning velocity s (right).

4.2 Linear Eddy Model (LEM) In this section we present results obtained with “Module B” as flame structure module. In this module stochastic rearrangement events occur during deterministically solving the one-dimensional incompressible Navier-Stokes equations (see Figure 8). First we want to consider what happens after an ignition of a laminar flame, e.g. in an obstacle configuration (for an example, we refer to the following section). Due to the combustion process pressure waves are emitted by the flame front which induce a turbulent flow field nearby the obstacles due to shear. In a simplified way, we want to study how an initially laminar flame structure, entering these regions, behaves under such changing turbulence conditions. In our test cases we have frozen the integral length scale as well as the chemical time and length scales. Only the turbulent Reynolds number Ret is modified. This means we move on a line parallel to the y-axis of the “Borghidiagram” (see Figure 1). For increasing Ret the Karlovitz number becomes smaller and smaller. So we can observe how the flame structure changes under these conditions.

1

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421

Scalar field (1D, normal to the front) Triplet map

l

Eddy of size l

before eddy

l

after eddy Flame Shape

LEM Line (Flame structure normal to the front)

Fig. 8. Stochastic rearrangement of a scalar ﬁeld caused by an eddy of size l in the 1D LEM structure module (above) and geometrical illustration of an eddy acting on a ﬂame front with LEM module in normal direction to the average front (below).

In the left picture in Figure 9 a typical flame structure in the flamelet regime is visible. The entire flame structure is an agglomeration of laminar flames (flamelets). Each individual Flamelet (see Figure 9 right) naturally is a laminar reaction-diffusion profile. Further increase of Ret leads to smaller Kolmogorov scales η and increases the likelihood that the smallest eddies disturb the flamelets diffusion profiles. For this case Figure 10 (left) shows a typical flame structure, which adjusts after some time on the prescribed turbulence conditions. It becomes obvious that the diffusion profiles of the Flamelets are partly disturbed by eddies, while the reaction zones still tend to be compact. A further increase of turbulence intensity reduces the length scale of the smallest turbulent eddies so strongly that they even can disturb the reaction zone of the flame. This leads to a flame profile as shown in Figure 10 (right), which is qualitatively comparable with the results obtained in [SK03] from a PDF type flame structure calculation. 4.3 G-LEM In order to show that the presented procedure is applicable also in three space dimensions, a simplified G-LEM flame structure module (“Module C”) which was explained in section 3.2 is used. Here the flame propagation over an obstacle in 3D is simulated. For each cell cut by the flame the flame speed is calculated by local G-LEM modules. The computational domain is composed of six cubes of length two, each one discretized by 32x32x32 cells. Turbulent

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1 0.8 0.6

Y

0.5

(fuel mass fraction)

0.4 0.2 0 5.37

0 4.6

4.8

5

5.2

5.4

5.38

ξ

5.39

ξ

Fig. 9. Fuel mass fraction distribution in the ﬂame structure module obtained with the linear eddy model for Ret = 3 (left) and zoom into the most right ﬂamelet (right). 1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

Y (fuel mass fraction)

0 4.7

0 4.8

4.9

ξ

5

5.1

5.2

3

4

5

ξ

6

7

Fig. 10. Fuel mass fraction distribution in ﬂame structure module for diﬀerent turbulent Reynoldsnumbers. Left:Ret = 10; right: Ret = 100.

length and time scales are provided by a k − ǫ turbulence model. The temporal evolution of the flame geometry is illustrated in Figure 11. In Figure 12 a cut plane in x-y direction through the domain for the turbulent kinetic energy is shown. It becomes obvious that the highest turbulence intensities are observed in the wake of the obstacle. In these regions the computed turbulent flame speeds st are orders higher than the laminar burning speed. In fact a more physical flame structure module, i.e. “Module B”, should be applied adaptively in this flow section, whereas in the other regions the flame propagates more or less with slam , and “Module C” is a sufficient and very efficient flame structure module.

5 Summary and conclusions Hybrid Capturing/Tracking schemes which use In-Cell-Reconstruction techniques are able to model accelerating premixed flames with the following features [SMK97]:

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• flame is thin compared to the large geometrical scales • Flamelet ansatz; flame physics condensed into a burning rate law • standard Rankine-Hugoniot jump conditions apply They have the following limitations: • inherent assumption of a quasi stationary flame structure • depend on availability of burning rate law In this paper we present a generalized numerical In-Cell-Reconstruction strategy that • extends the level set technique to complex internal front structures, while retaining the assumption of scale separation between flame structure and global scales • flexibly models a wide range of different combustion regimes by handling non-stationary flame structures in a modular fashion • provides a basis for the modelling of Deflagration-to-Detonation-Transition (DDT) in large scale systems

6 Outlook Currently the following modifications of our method are investigated in DFGProject Schm 1682/1-1: • Consistent extension to Large Eddy Simulation (LES) type turbulent flow and flame modeling • Improved coupling between outer flow solution and local flame structure computation Acknowledgement. This research was supported by the German Science Foundation (DFG) through grant no KL-611/5 and KL-611/7 (within the priority research Programme on “Analysis and Numerics of Conservation Laws” (ANumE)). The authors gratefully acknowledge fruitful, stimulating discussions with A. Kerstein and S. Wunsch (SANDIA Nat. Lab. Livermore). We also thank W. Hillebrandt, J. Niemeyer, M. Reinecke, and A. Groebl (Max Planck Inst. for Astrophysics (Garching)) for continuous constructive discussions. Finally, we would like to thank G. Warnecke for the coordination of the ANumE Programme.

References [Cho80]

Chorin, A.J.: Flame Advection and Propagation Algorithms. Journal of Computational Physics, 35, 1–11 (1980) [FAX99] Fedkiw, R., Aslam, T., Xu, S.: The Ghost Fluid Method for Deﬂagration and Detonation Discontinuities. J. of Comp. Phys., 154, 393–427 (1999)

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Fig. 11. to be continued

Flexible Flame Structure Modelling in a Flame Front Tracking Scheme sequel to Fig. 11.

Fig. 11. to be continued

425

426

H. Schmidt, R. Klein sequel to Fig. 11.

Fig. 11. Three-dimensional propagation of a premixed ﬂame over an obstacle: Representation of the isoline G=0 as well as the velocity ﬁeld (vector arrows); local burning speeds are modeled over G-LEM ﬂame modules. (See also color ﬁgure, Plate 26.)

Fig. 12. Two-dimensional cut plane (x-y-direction) through the ﬂow ﬁeld: The velocity ﬁeld (vector arrows) and the turbulent kinetic energy are shown. (See also color ﬁgure, Plate 27.)

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Kerstein, A.R.: Linear-eddy model of turbulent transport and mixing. Combustion Sci. and Tech., 60, 391–421 (1988) [Ker89] Kerstein, A.R.: Linear-eddy model of turbulent transport II. Combustion and Flame, 75, 397–413 (1989) [LKV78] Lee, J.H.S., Knystautas, R., Yoshikawa, N.: Photochemical initiation and gaseous detonations Regime. Astronautica Acta, Vol. 5, 971–972 (1978) [LeV92] LeVeque, R.: Numerical Methods for Conservation Laws. Birkh¨auser Basel (1992) [MW88] Maas, U., Warnatz, J.: Ignition processes in hydrogen-oxygen mixtures. Combustion and Flame, 74, 53–69 (1988) [MK92] Menon, S., Kerstein, A.R.: Stochastic Simulation of the structure and propagation rate of turbulent premixed ﬂames. Proc. of the Comb. Institute, Twenty-Fourth Symp. (International) on Comb., 443–450 (1992) [Pet86] Peters, N.: Laminar ﬂamelet concepts in turbulent combustion. The Combustion Institute, Pittsburgh PA USA, Volume 21, 1291–1301 (1986) [Pet99] Peters, N.: The turbulent burning velocity for large scale and small scale turbulence. Journal of Fluid Mechanics, 384, 107-132 (1999) [Pet00] Peters, N.: Turbulent Combustion. Cambridge University Press (2000) [Pop81] Pope, S. B.: A Monte Carlo method for the PDF equations of turbulent reactive ﬂow. Combustion Science and Technology, 25, 159–174 (1981) [Pop85] Pope, S.B.: PDF methods for turbulent reactive ﬂows. Prog. Energy Combustion Science, 11, 119–192 (1985) [Set99] Sethian, J. A.: Level Set Methods and Fast Marching Methods. Cambridge Univ. Press (1999) [Sch02] Schmidt, H.: Ein numerisches Verfahren zur Verfolgung von Vormischﬂammen unter Ber¨ ucksichtigung instation¨ arer Frontstrukturen. Dissertation, Universit¨ at Duisburg Deutschland (2002) [Sch03] Schmidt, H.: Ein hybrides LES-Tracking-Verfahren f¨ ur Vormischﬂammen unter Ber¨ ucksichtigung instation¨ arer Frontstrukturen. DFG-Forschungs´ stipendium an der Ecole Centrale Paris, Frankreich (2003-2004) [SK03] Schmidt, H. and Klein, R.: A generalized level-set/in-cell-reconstruction approach for accelerating turbulent premixed ﬂames. Combustion Theory and Modelling, 7, 243–267 (2003) [SMK97] Smiljanovski, V., Moser, V., Klein, R.: A Capturing-Tracking Hybrid Scheme for Deﬂagration Discontinuities. Combustion Theory and Modelling, 2(1), 183–215 (1997) [Wil85] Williams, F.A.: Turbulent Combustion. In: Buckmaster, J. (ed) The Mathematics of Combustion. 97–131 (1985) [Wun01] Wunsch, S.: Private Communication. 2001 [ZLMS70] Zeldovic, Y.B., Librovic, V.B., Makhviladze, G.M., Sivashinsky, G.I.: On the development of detonation in a non-uniformly preheated gas. Astronautica Acta, 15, 313–321 (1970) [Zel90] Zeldovic, Y.B.: Regime classiﬁcation of an exothermic reaction with nonuniform initial conditions. Combustion and Flame 39, 211–214 (1990) [ZT98] Zhang, F., Thibault, P.: A PDF-Module for Turbulent Flame Structure Computation. Final Report on subcontracted work for EC-Project FI4SCT96-0025, Combustion Dynamics Ltd. Halifax Canada (1998)

Riemann-Solver Free Schemes Tim Kr¨ oger and Sebastian Noelle Institut f¨ ur Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, D-52056 Aachen, Germany, [email protected]

Summary. In this article, we use the recently developed framework of state and ﬂux decompositions to point out some interesting connections and diﬀerences between several Riemann-solver free numerical approaches for systems of hyperbolic conservation laws. We include a numerical comparison of Fey’s Method of Transport with a second order version of the HLL scheme and prove an interesting property of the former scheme for linear waves contained in the equations of ideal gas dynamics. Key words: systems of conservation laws, Fey’s Method of Transport, Euler equations, kinetic schemes, bicharacteristic theory, state decompositions, ﬂux decompositions, exact and approximate evolution operators, quadrature rules, numerical comparison, HLL scheme.

1 Introduction Many physical problems, for example the behaviour of a compressible fluid, can be modelled as systems of hyperbolic conservation laws, ∂t U + ∇x · F (U) = 0,

(1)

if certain effects (for instance viscosity) are neglected. Here, x ∈ Rd , F : Ω → Rm×d , and U : Rd × [0, ∞) → Ω ⊂ Rm . Most classical numerical methods for systems of hyperbolic conservation laws are based on solving one-dimensional Riemann problems. Schemes for multi-dimensional systems are then obtained by either performing a dimensional splitting or by a finite volume approach. See for example the textbooks of LeVeque [Lev90], Kr¨ oner [Kr¨ o97], Godlewski and Raviart [GR96], and Toro [Tor99]. Since about 15 years, however, there is an ongoing discussion (see for example Roe et al [Roe86, DRS93]) whether one-dimensional Riemann solvers do justice to the multi-dimensional effects arising in such systems. As discussed e. g. in the introduction of [Noe00], there were a number of approaches which therefore purposely dispensed with Riemann solvers. The following three of them form the main subject of the current paper:

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• the Method of Transport (MoT), originally developed by Fey [Fey93, Fey98a, Fey98b] and later modified by Noelle [Noe00], • the Evolution Galerkin (EG) approach of Butler [But60], Morton et al [LMS97] (exploiting the transport collapse operator of Brenier [Bre84]), Ostkamp [Ost95, Ost97], Luk´ aˇcov´a, Morton, Warnecke [LMW00] as well as (based on this) the Finite Volume Evolution Galerkin (FVEG) approach of Luk´ aˇcov´a, Morton, Saibertov´ a and Warnecke [LSW02, LMW02], and • the kinetic approach of Deshpande [Des86] and Perthame [Per90, Per92]. In this paper, we present the framework of so-called state decompositions and flux decompositions, which will be shown to be able to cover both the MoT and the EG approach, as well as the standard finite volume approach. In this framework, a fundamental difference between the MoT and the EG schemes will be pointed out. On the other hand, the flux decompositions used in the MoT can be derived systematically from the kinetic theory (for this result, we refer to [KNZ04]). The main tool for this are suitable quadrature rules applied to the moment integral used in the gaskinetic derivation of the Euler equations from the Boltzmann equation. Together with some quadrature in space and time (see Subsect. 3.2), this forms a new derivation of the second author’s [Noe00] version of the MoT, called MoT-ICE, which is mainly based on quadrature. All important connections within and between the mentioned approaches are schematically shown in the diagram in Fig. 1. The unifying themes of this paper are state decompositions, flux decompositions and the corresponding exact and approximate evolution operators: In Sect. 2, we define state and flux decompositions and give some simple examples. In Sect. 3, we derive exact and approximate evolution operators from state and flux decompositions, respectively. In Sect. 4, we state the so-called characteristic state decomposition for an arbitrary system. It can be derived from the classical bicharacteristic theory and is in general not flux-consistent. It is necessary to see that our concept includes the EG approach. In Sect. 5, we present a simple but interesting numerical comparison of the Method of Transport in its ICE version and the Riemann solver of Harten, Lax, van Leer [HLL83] (briefly called HLL). The HLL scheme is known to be a very simple and at the same time very robust scheme. Also, it has the property that for initially constant velocity and pressure, it preserves these components exactly for all time, which we prove in Sect. 6. However, the HLL scheme suffers from being very dissipative: Contact discontinuities or, more generally, density variations are smeared out quite heavily. On the other hand, Quirk [Qui94] already pointed out that the damping effect of the HLL scheme prevents it from producing instabilities or unphysical solutions. Moreover, about ten years later, Pandolfi and D’Ambrosio [PA01] analyzed the development of two significant examples Quirk had observed, namely the carbuncle phenomenon and the odd-even-decoupling phenomenon (the latter also known as crossflow-instability). They precisely came to the conclusion that a scheme

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kinetic theory

R em

ar k ray 8 vel o ci tie s

class. charact. theory

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FD

eig env ect ors ,

std. integral form

Lemma 11

EG operator

derivation

[Noe00]

inconsist. MoT-CCE [Ost95, Ost97]

equal for linear equations

Legend:

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[Fey98b]

Ostkamp’s consistent scheme

Fey

[Fey93]

Ostkamp’s incons. sch.

Fey’s continuous FD

(theory only)

00] [LMW

[Ost95, Ost97]

[Ost95, Ost97]

EG 3 scheme

characteristic SD

original MoT

differ only by some factor d

discretization

specialization

MoTCCE

MoTICE

gas dynam. only

Fig. 1. Overview of state decompositions (SDs), ﬂux decompositions (FDs), the classical bicharacteristic and the kinetic theories, the Evolution Galerkin (EG) schemes and Fey’s Method of Transport (MoT).

which damps out density variations gives correct results for this type of test problems. In fact, they classify the HLL scheme as a ‘carbuncle-free scheme’. In opposite, especially the HLLC scheme, which is just a simple modification of HLL specially designed for a better resolution of contact discontinuities (see Toro, Spruce, Speares [TSS94]), is classified as a ‘strong carbuncle prone scheme’. Summarizing the last paragraph, we can say that a scheme that claims to be better than Riemann solver based schemes should firstly be less dissipative

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than HLL, and then be carbuncle-free. The result of our test, however, shows that the Method of Transport does not meet the first requirement. In fact, it is approximately as dissipative as HLL. Recalling that we mentioned a common root (see [KNZ04] for details) of the Method of Transport and Perthame’s kinetic schemes [Per90, Per92], we can say that this numerical result confirms this close relationship because it is well-known that the kinetic schemes also tend to damp out contact discontinuities and purely advective parts quite heavily. We would also like to recall a similar result by Gressier et al [GVM99], who show that a positivity preserving flux vector splitting scheme cannot keep stationary contacts sharp. Further numerical test examples, which include multi-dimensional computations and shocks and confirm the result presented here, can be found in [KN04]. ˇ a and GerWe would like to thank Bill Morton, Maria Luk´ aˇcov´a-Medvidov´ ald Warnecke for interesting discussions on Evolution Galerkin Schemes, and Michael Fey and Rolf Jeltsch for sharing their insights into the Method of Transport.

2 Decompositions of Hyperbolic Systems In this section, we will define state decompositions and flux-consistent state decompositions for arbitrary systems of hyperbolic conservation laws (1) and give some simple examples. Definition 1. Let L ∈ N, and for every l = 1, . . . , L let Sl be some Rm -valued and al some Rd -valued continuously differentiable functions on the set Ω of physical states U. The family (Sl , al )l=1,...,L is called a state decomposition for (1) if L

Sl (U) = U

(2)

l=1

holds.

Definition 2. A state decomposition is called flux-consistent if additionally L l=1

holds.

Sl (U) · al (U) = F (U)

(3)

A flux-consistent state decompositions will sometimes simply be called a flux decomposition. In section 4, we will also work with a continuous state decomposition where the sum over l is replaced with an integral. We would like to emphasize that this is different from the Fluctuation Splitting approach of Deconinck, Roe and Struijs [DRS93] where the divergence ∇ · F (U) is decomposed rather than the flux matrix F itself.

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Example 3. For any system, choosing L = 1, S1 = U, a1 = 0 forms a very simple state decomposition. We call it the trivial state decomposition. It is not flux-consistent except in the trivial case F ≡ 0. In remark 8, we will recover the classical integral form of conservation laws from the trivial state decomposition. Example 4. If we consider a one-dimensional system of conservation laws, i. e. d = 1, and further assume that F is homogeneous, i. e. F(U) = F′ (U) · U, then we can set L = m, Sl = rl and al = λl where rl are the eigenvectors of F′ (U) (normalized such that they sum up to U) and λl are the corresponding eigenvalues. The resulting state decomposition is flux-consistent. This decomposition has been used by Steger and Warming in their well-known flux-vector splitting scheme [SW81]. Example 5. The previous example can be generalized to the multi-dimensional case if one assumes the Jacobian matrices Fs to commute (i. e. to be simultaneously diagonalizable): If rl are the common eigenvectors and λls is the corresponding l-th eigenvalue of the s-th Jacobian matrix F′s (U), set Sl = rl as before and al = (λl1 , . . . , λld ). This again results in a flux-consistent state decomposition. For the general case in which the Jacobian matrices are not simultaneously diagonalizable, there does not seem to be such a simple mechanism to construct a flux-consistent state decomposition. However, we have found a systematic approach to derive a whole class of flux decompositions for Euler’s equations from the gaskinetic theory. This class especially includes the (discrete) flux decomposition suggested by Fey [Fey98b]. For a description of this derivation, we refer to [KNZ04] and [Kr¨ o04].

3 Exact and Approximate Evolution Operators for Smooth Solutions Assume now that a state decomposition for (1) is given. Based on this, we will derive an exact evolution operator for (1). If further more the state decomposition is flux-consistent, we will explain how the use of quadrature rules in time and space leads to approximate evolution operators which form the basic building block of the MoT-ICE. 3.1 Exact Evolution Operator If a system (1) and a state decomposition (2) are given, we can write ∂t Sl (U) + ∇ · (Sl (U)al ) = −S′l (U)∇ · F (U) + ∇ · (Sl (U)al ) =: Tl (x, t), (4)

434

T. Kr¨ oger, S. Noelle K

tn+1

x˙ =

al

al ′

= x˙

Ωl

′

Ωl

n ′

K l (tn )

K l (tn )

tn

Fig. 2. The cell K is transported backwards in time along advection curves.

where al (x, t) := al (U(x, t)). The representation (4) can be considered as an advection equation for each Sl (U) with a right hand side. Lemma 6. For a flux-consistent state decomposition, we have that l Tl = 0. Proof.

L l=1

Tl = −

L

L

l=1

l=1

∂ (Sl (U)al ) Sl (U)∇ · F (U) + ∇ · ∂U

∂U ∇ · F (U) + ∇ · F (U) = 0. =− ∂U

⊓ ⊔ Define the l-th advection curve ξ lx through the point (x, tn+1 ) by ∂τ ξ lx (τ ) = al (ξ lx (τ ), τ ),

ξ lx (tn+1 ) = x,

and for any K ⊂ Rd (with sufficiently smooth boundary) define K l (τ ) = {ξ lx (τ ) : x ∈ K} and Ω l = {(ξ lx (τ ), τ ) : x ∈ K, tn ≤ τ ≤ tn+1 } ⊂ Rd × R, see Fig. 2. Then, integration of (4) over Ω l yields , ∂t Sl (U) + ∇ · (Sl (U)al ) dxdτ Tl (x, τ )dxdτ = l Ωl Ω Sl (U)(al , 1) · nds. = ∂Ω l

(5)

Riemann-Solver Free Schemes

435

The boundary ∂Ω l of the tube Ω l consists of the top K × {tn+1 }, the bottom K l (tn ) × {tn } and the surface S := {(x, t) ∈ ∂Ω l : tn < t < tn+1 }. At each point (x, τ ) ∈ S, we have that n is perpendicular to the advection curve (ξ lx,τ (τ ), τ ). Due to (5), this means that the integral over S vanishes. Thus, we have Tl (x, τ )dxdτ = (6) Sl (U(x, tn+1 ))dx − Sl (U(x, tn ))dx. Ωl

K l (tn )

K

Summing over all l yields L

Tl (x, τ )dxdτ =

Ωl

l=1

L K l=1

>

Sl (U(x, tn+1 )) dx − ?@

U(x,tn+1 )

A

L

Sl (U(x, tn ))dx

K l (tn )

l=1

and thus gives rise to the following representation of the solution U: Lemma 7 (Exact evolution operator). Let U be a smooth solution of (1) and (Sl , al )l be a (not necessarily flux-consistent) state decomposition. Then,

U(x, tn+1 )dx =

K

L l=1

Sl (U(x, tn ))dx +

K l (tn )

L l=1

Tl (x, τ )dxdτ.

(7)

Ωl

Remark 8. Choosing the trivial state decomposition (compare Example 3) leads to the standard integral form of the conservation law, tn+1 U(x, tn+1 )dx = ∇ · F (U)dxdτ, U(x, tn )dx − K

tn

K

K

which is the basis of classical finite volume discretizations. In section 4, we will show that lemma 7 also includes the EG operator, which is derived from the classical characteristic theory. 3.2 Approximate Evolution Operators The exact evolution operator cannot be used in practice because it contains a time integral over [tn , tn+1 ], and in a practical scheme, one does not know anything about the solution at those intermediate time levels. Application of the trapezoidal rule to the time integral in the left hand side of (6) yields ∆t 2

∆t Tl (x, tn+1 )dx + Tl (x, tn )dx 2 K l (tn ) K = Sl (U(x, tn+1 ))dx − Sl (U(x, tn ))dx + O(∆t3 |K|), K

K l (tn )

436

T. Kr¨ oger, S. Noelle

K

K l (tn )

ˆl K

Fig. 3. The cell edges are transported backwards in time and approximated by ˆ l which consist of a union of ﬁnitely cellwise axiparallel lines. These deﬁne the cells K many rectangles.

or & K

Sl (U) −

∆t ' Tl (x, tn+1 )dx 2 & ∆t ' Sl (U) + Tl (x, tn )dx + O(∆t3 |K|). = 2 K l (tn )

If we again sum over all l and now assume the state decomposition to be flux-consistent, the term Tl on the left hand side cancels and we get: Lemma 9 (Approximate evolution operator I). For a smooth solution U and a flux-consistent state decomposition of (1),

U(x, tn+1 )dx =

K

L l=1

K l (tn )

&

Sl (U) +

∆t ' Tl (x, tn )dx + O(∆t3 |K|). (8) 2

In general, K l (tn ) has a curvilinear boundary and cannot be determined exactly. In [Noe00, section 3], the second author approximated the cell by a ˆ l which is a union of finitely many rectangles1 , see Fig. 3. For this K ˆ l, set K the following result holds: Lemma 10 (Approximate evolution operator II). Let & 1 ∆t ' ˆ Tl (x, tn )dx. Sl := Sl (U) + |K| Kˆ l 2

(9)

Then,

1 |K|

K

U(x, tn+1 )dx =

L

ˆ l + O(∆t3 ) S

(10)

l=1

for a smooth solution U and a flux-consistent state decomposition of (1). 1 Similar approximations were developed by M. Fey and coworkers (private communication).

Riemann-Solver Free Schemes

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4 Comparison with the EG Evolution Operator In this section, we will show that the evolution operator derived in lemma 7 is closely related to the one developed by Ostkamp [Ost95, Ost97] and later used by Luk´ aˇcov´a, Morton, Saibertov´ a, and Warnecke [LMW00, LSW02, LMW02] for the derivation of several schemes. This evolution operator, which we call EG operator in the following, is based on the classical characteristic theory, see [But60, CH62, Pra01]. Ostkamp already showed that there is a close connection between her characteristic Galerkin scheme and Fey’s version of the Method of Transport. We will now show that there is a canonical continuous state decomposition (derived from the classical characteristic theory) for which our exact evolution operator (7) becomes identical to the EG operator. In other words, the connection between the two approaches is lifted up from the level of schemes to the level of exact evolution operators, and Lemma 11 below can be considered as the fundamental interface between the two ideas. Note that in [Ost95, Ost97, LMW00, LSW02, LMW02] the EG operator has only been used for linear or linearized systems. Thus, to compare both evolution operators, we will restrict ourselves to the linear, constant coefficient case. Hence, let ∂t U +

d

As ∂xs U = 0

(11)

s=1

be a linear system of hyperbolic conservation laws. Here, As are m × m matrices. The consistency condition of a state decomposition, for a linear system, reads L

Sl (U) = U,

(12)

l=1

and flux-consistency would additionally require L l=1

Sl (U) · al,s (U) = As U,

(13)

where al,s , s = 1, . . . , d denote the single components of the speeds al . It seems to be sensible for linear systems that al,s (U) should not depend on U (i. e. only on l and s). As a consequence, the advection curves ξ lx are straight lines, ξ lx (τ ) = x + (τ − tn+1 )al . In this case, by considering the limit of a one-point set K, i. e. K → {x}, the exact evolution operator (7) can be stated pointwise as L L tn+1 Sl (U(x − ∆t · al , tn )) + Tl (x + (τ − tn+1 )al , τ )dτ. U(x, tn+1 ) = l=1

l=1

tn

(14)

438

T. Kr¨ oger, S. Noelle

ne la

n. ve l.

lp

no rm al

di re ct

tia en ng

io n

ta

ray velocity

Fig. 4. Example for a Friedrichs diagram for the linearized equations of magnetohydrodynamics.

Here the state decomposition still has to be specified. One can construct a continuous state decomposition (i. e. the sum in (2) is replaced with an integral or, more general, a sum of integrals) in which Sl (U) are the eigenvectors of the Jacobian of F (U) (when a certain normal direction in the x space is given) and the velocities al are the associated ray velocities arising from the characteristic theory (see Courant and Hilbert [CH62] or Jeffry and Taniuti [JT64]): Consider a characteristic surface of the linearized system in the (x, t) space which at a given time level (say, at t = t0 ) concentrates in one point. Then construct the intersection of this surface with the t = t0 + 1 plane, see Fig. 4 for an example (such a diagram is called Friedrichs diagram). If a normal direction is given, the ray velocity is the position vector of the point on the Friedrichs diagram in which the tangential plane is orthogonal to the given normal direction. It must not be confused with the normal velocity. The normal velocity always points into the normal direction, the ray velocity doesn’t, but the projection of the ray velocity onto the normal direction equals the normal velocity.

Riemann-Solver Free Schemes

439

As this state decomposition is derived from the characteristic theory, we call it the characteristic state decomposition. It is defined for an arbitrary system but in general not flux-consistent; for details see [Kr¨ o04]. In symbols, the characteristic state decomposition is defined as follows: Let p ∈ Rd be the normal direction, p = 0. From the hyperbolicity of the system we know that the matrix A(p) :=

d

p s As

s=1

is diagonalizable with real eigenvalues. Let rkp be the right (column) eigenvectors, lkp the left (row) eigenvectors and λkp the eigenvalues—where k = 1, . . . , m. We only consider these terms for |p| = 1, but it is important to understand that they are defined as well for other values of p so that derivatives of λkp with respect to a component ps are defined. The eigenvectors are where Rp and Lp are the assumed to be normalized such that Rp = L−1 p matrices whose columns are rkp or whose rows are lkp , respectively. Now, set Sl (U) := Skp (U) :=

1 rk lk U, |S d−1 | p p

(15)

where S d−1 ⊂ Rd is the unit sphere. Then set al := akp := ∇p λkp ,

(16)

this is the gradient of λkp with respect to p. These are the ray velocities that arise from the Friedrichs diagrams. Note that the normal velocities in our notation are p · λkp . Although this may seem somewhat artificial, we would like to emphasize that it is in fact the most natural choice of the velocities: Recall that the ray velocities are precisely those velocities with which a point disturbance propagates (see Fig. 4). Equations (15) and (16) define a continuous state decomposition. In fact, (12) is satisfied:

m

S d−1 k=1

Skp (U)dp =

1 |S d−1 |

= U.

m

S d−1 k=1

rkp lkp dp U =

1 |S d−1 |

Rp Lp dp U

S d−1

Lemma 11. For a smooth solution U of the linear system (11) of hyperbolic conservation laws, we have that our evolution operator (14) where Skp and akp are given by (15) and (16) is identical with the EG operator. For a proof, see [KNZ04] or [Kr¨ o04]. We would like to recall that an overview of these connections has been given in Fig. 1.

440

T. Kr¨ oger, S. Noelle

Remark 12. 1. The characteristic state decomposition can also be introduced for the non-linear case. Also, the exact evolution operator (7) remains valid in that case. We have therefore found a generalization of the EG operator to non-linear systems. 2. Since the characteristic state decomposition is in general not flux-consistent, a discretization as in lemmas 9 and 10 would not lead to a consistent scheme for (1). Indeed, we are convinced that this scheme (for the linear, constant coefficient case) would coincide with Ostkamp’s ‘inconsistent’ scheme. 3. Perhaps the techniques of Luk´ aˇcov´a, Morton and Warnecke [LMW00] can be used to construct yet another EG scheme out of this generalized evolution operator in the nonlinear case. Unfortunately, for the non-linear case the coupling terms Tl = Tkp would become extremely complicated due to the fact that rkp , lkp and λkp in general also depend on U. For details see [KNZ04]. As Ostkamp pointed out, her consistent scheme coincides with Fey’s original MoT [Fey93] for the linearized, constant coefficient Euler equations. Note that for Euler’s equations, however, we have that m = d + 2 and we get the very special situation that (for appropriate numbering) ap1 = am −p and all other akp (k = 2, . . . , m − 1) are equal and do not depend on p. Using these facts the continuous state decomposition (15)–(16) can be simplified by combining k S1p and Sm −p to one component and integrating all the Sp components (for d−1 ) to one single component. If this is done and k = 2, . . . , m − 1 and p ∈ S further more the p-dependent part of S1p = Sm −p is multiplied by the space dimension d, one gets exactly Fey’s continuous state decomposition used in [Fey93], and therefore, one get’s Ostkamp’s consistent scheme. This state decomposition is flux-consistent, but only because some special part of Skp has been multiplied by a factor d in some places. Ostkamp [Ost95] derived this factor d systematically, but this derivation relies on a number of rather restrictive assumptions which seem to be tailored to the equations of gas dynamics. For a general system of conservation laws, it does not seem to be able to make the continuous state decomposition flux-consistent this way. Even though Ostkamp was able to identify her EG scheme and Fey’s original MoT, we would like to suggest that the two approaches are fundamentally different.2 As we see it, the main difference between the MoT and the EG approach is that the state decomposition which leads to the EG operator is not flux-consistent. The MoT approximates the evolution operator in lemma 7 exploiting the flux-consistency of a given state decomposition (see especially the approximate evolution operator of lemma 9), while the EG schemes use an approximation technique which does not rely on the flux-consistency. How2

Quoting D. Adams’ ‘The Hitch Hiker’s Guide to the Galaxy’ (Random House Value Publishing Inc., 1979), we are tempted to claim that the MoT is ‘almost, but not quite, entirely unlike’ the EG schemes.

Riemann-Solver Free Schemes

441

ever, this has the consequence that the coupling terms Tl become more essential in the EG approach. In the MoT-ICE, if the Tl are neglected, one still gets a first order scheme; this can be seen in (8) by recognizing that the sets;K l (tn ) differ from K only by sets of Lebesque measure O(∆t|K|), and l K Tl dx = 0 by lemma 6. So, the error made if Tl is neglected in (8) is of order O(∆t2 |K|), thus the scheme would still be of first order. In distinction to that, if the coupling terms are neglected in the EG operator, one gets an inconsistent scheme. In fact, this is just what Ostkamp [Ost95, Ost97] did in her ‘inconsistent’ scheme (which would be first order consistent in Ostkamp’s terminology). Neglecting the coupling terms in the first order MoT seems to be closely related to dropping the interaction terms Q(f, f ) in Boltzmann’s equation during a time step, which leads to the first order version of the kinetic schemes. One could suggest that the second order MoT corresponds to approximating Q(f, f ) in a suitable way.

5 Numerical Results and Comparison In this section, we present a simple numerical test example. It consists of a static one-dimensional Riemann problem for Euler’s equations of gas dynamics using the equation of state for an ideal gas. We set p = 1, u = 0 and ρ = ρl = 3 (for x < 0) or ρ = ρr = 1 (for x > 0). The exact solution obviously remains constant for all time. We computed this example using both the MoT-ICE and the HLL scheme in either second order versions. MoT-ICE was coded according to the description in [Noe00], except that the monotonized centered slope limiter was used instead of the WENO limiter, and for the HLL scheme, we used Einfeldt’s [Ein88] choice of the signal velocities and lifted the scheme to second order using a piecewise linear reconstruction (again using the monotonized centered slope limiter) and a Runge–Kutta time step (more precisely, the method of Heun). Results for the HLL scheme at selected time levels are shown in Figs. 5 to 7 (one figure for each component). Here, we used a resolution of ∆x = 1/256 and a computational domain which is large enough such that any effects originating at the contact at x = 0 are not able to reach the boundary within the time up to which the computation is carried out. (We show only the domain [−0.5, 0.5], but the real computational domain is larger.) This prevents effects which are due to the handling of the boundary conditions. We see that the HLL scheme conserves the u and p components exactly while the ρ component is smeared out quite heavily. (The ρ plot is a split plot where each side is cut off at the position where the smearing begins. So the width of the gap between the two sides is a measure for the intensity of the smearing.)

442

T. Kr¨ oger, S. Noelle t ρ

0.4 0 4 3.015 3.01 3.005

ρ

1.005

0 25 0.25 2.995 2.99

0.995 0.99 0.985

0.12 0 12

one step 1 2

− 12

Fig. 5. ρ component of the static contact problem, using second-order HLL. t 0.4 u +0.002 +0.001

0.25 −0.001 −0.002 −0.003

0.12

one step − 12

1 2

Fig. 6. u component of the static contact problem, using second-order HLL.

In fact, it is true, that the HLL scheme leaves pressure and velocity absolutely constant (up to machine precision) for all time (for an ideal gas) if they are constant in the initial data. This statement applies for first and second order versions, for arbitrarily many space dimensions, for any choice of the signal velocities and for any slope limiter out of the class of limiters that were discussed by Sweby [Swe84]. For a proof, see Sect. 6. Figures 8 to 10 show the result of the same problem, but now using MoTICE. First, we see that there are now small waves in all components, originating at the contact and moving to both sides with the respective speed of sound. These pertubations arise in all three components, even in the first

Riemann-Solver Free Schemes

443

t p

0.4 1.01 1.005

0.25 0.995 0.99

0.12

one step 1 2

− 12

Fig. 7. p component of the static contact problem, using second-order HLL. t ρ

0.4 0 4 3.015 3.01 3.005

ρ

1.005

0 0.25 25 2.995 2.99

0.995 0.99 0.985

0.12 0 12

one step − 12

1 2

Fig. 8. ρ component of the static contact problem, using second-order MoT-ICE.

step, although the initial pertubation in the p component is so small that it can hardly be seen in the plot (but it exists). Anyway, we want to point out that the size of the emerged pressure waves is extremely small—don’t forget to look at the scales, and for the ρ component have a look at figure 11 for an unsplit plot. The second and more important observation to mention is that MoT-ICE smears out the contact approximately as much as HLL (remember that this is indicated by the width of the gap in the ρ plot, or compare the unsplit plots in figures 11 (MoT-ICE) and 12 (HLL)). As pointed out in the introduction, this principally already suffices to demonstrate that the results of the MoT are not satisfactory. However, a single, one-dimensional test problem

444

T. Kr¨ oger, S. Noelle t 0.4 u +0.002 +0.001

0.25 −0.001 −0.002 −0.003

0.12

one step 1 2

− 12

Fig. 9. u component of the static contact problem, using second-order MoT-ICE. t p

0.4 1.01 1.005

0.25 0.995 0.99

0.12

one step − 12

1 2

Fig. 10. p component of the static contact problem, using second-order MoT-ICE.

Riemann-Solver Free Schemes

445

3 2.75 2.5

rho

2.25 2

1.75 1.5 1.25 1 -0.5

-0.25

0

0.25

0.5

x Fig. 11. ρ component of the static contact problem unsplit at t = 0.25, using second-order MoT-ICE.

might be considered to be not a fair comparison, especially when comparing a Riemann-solver based scheme to a scheme which was designed to capture multi-dimensional effects particularly well. For a more extensive numerical comparison which includes multi-dimensional examples as well as calculation of shocks, we refer to [KN04] and [Kr¨ o04]. The result that MoT-ICE does not meet the claim of being better than Riemann-solver based schemes is confirmed by those examples.

6 A Property of the HLL Scheme In Sect. 5, we saw that for our test example, the HLL scheme preserved the constant velocity and pressure exactly. In fact, the following result holds: Lemma 13. For Euler’s equations for an ideal gas, if the initial data has constant pressure and velocity over the whole computational domain, the first

446

T. Kr¨ oger, S. Noelle

3 2.75 2.5

rho

2.25 2

1.75 1.5 1.25 1 -0.5

-0.25

0

0.25

0.5

x Fig. 12. ρ component of the static contact problem unsplit at t = 0.25, using second-order HLL.

and second order HLL schemes on a Cartesian grid keep these variables constant (up to machine precision) for all time. This statement applies for all choices of the signal velocities and for any slope limiter of the class discussed by Sweby [Swe84]. Proof. Under the made assumptions, we have that U = V + Wρ in the initial data where ⎞ ⎛ 0 ⎜ T⎟ ⎟ V=⎜ ⎝0 ⎠ p γ−1

and

(17) ⎛

1

⎞

⎜ T ⎟ ⎟ W=⎜ ⎝ u ⎠ 1 2 2 |u|

are constant vectors and only ρ varies between the cells. We have to show that (17) still holds after one time step with the same constants V and W.

Riemann-Solver Free Schemes

447

The first part of a time step of the second order HLL scheme is the piecewise linear reconstruction. If Ui−1 , Ui , and Ui+1 are the states in three (in some coordinate direction) neighboring cells and upper indices denote components, the slope (in this direction) of the k-th component in the i-th cell is (component-wise) given by k Si,i−1,i+1

k k k Ui − Ui−1 − Uik Ui+1 ·φ = , k − Uk h Ui+1 i

(18)

where φ is a function depending on the slope limiter (for first order HLL we can just set φ ≡ 0). If we state (17) component- and cell-wise, we get Uik = V k + W k ρi (note that V and W do not have spatial indices, because V and W are constant vectors, and ρ does not have a component index, because it is a scalar variable). Using this, we can simplify (18) to yield ρi − ρi−1 Wk k · (ρi+1 − ρi ) · φ Si,i−1,i+1 = . h ρi+1 − ρi Especially, the argument of φ is the same for all components. It follows that the reconstructed, piecewise linear function U(x) still satisfies (17) all over the domain. Following Einfeldt [Ein88], we get that the numerical flux over the interface between the i-th and (i + 1)-th cell equals G=

b+ F(Ui,+ ) − b− F(Ui+1,− ) + b+ b− (Ui+1,− − Ui,+ ) b+ − b −

where we skip the index i + 12 of G, b+ and b− , and we denote F = p · F where p is the normal vector to the cell interface, and Ui,+ and Ui+1,− are the one-sided limits of the piecewise linear function U at the midpoint of the interface. We now note that for Euler’s equations for an ideal gas, we have F (U) = Uu + Q where ⎛

0

⎞

⎜ ⎟ ⎟ Q=⎜ ⎝p1⎠ pu

is (under our assumptions) a constant matrix. Using this and the shortcut Q = Q · p, we get

448

T. Kr¨ oger, S. Noelle

b+ (Ui,+ (u · p) + Q) − b− (Ui+1,− (u · p) + Q) + b+ b− (Ui+1,− − Ui,+ ) b+ − b − + − b Ui,+ − b Ui+1,− b+ b− (Ui+1,− − Ui,+ ) = · p) + Q + (u , b+ − b− b+ − b −

G=

and inserting (17) yields b+ ρi,+ − b− ρi+1,− b+ b− (ρi+1,− − ρi,+ ) G= V+W · p) + Q + W (u . b+ − b− b+ − b − Now, the state Ui in the i-th cell is increased by λGi− 21 and decreased by λGi+ 21 (where λ = ∆t/h is constant). As the lower indices only apply to b+ and b− , all terms containing V or Q cancel in the flux difference, and we get that the cell update is just a scalar multiple of W. Thus, (17) still holds for the new states. For a second order scheme, this procedure is embedded into a Runge– Kutta time step. This just consists of performing multiple time steps and then taking some (possibly weighted) average of the results. But averaging of functions U which all satisfy (17) with the same constants V and W results in a function U which still satisfies this equation. ⊓ ⊔

7 Conclusion The concept of state decompositions, flux decompositions and evolution operators, which is the natural framework for the Method of Transport, was shown to include also the Evolution Galerkin approach and the standard finite volume approach. This etablishes a connection between these approaches and, at the same time, allows to point out an essential difference between the MoT and the EG schemes, namely the flux-consistency. On the other hand, there is an important connection between the MoT and the kinetic schemes, because both are based on the kinetic theory. Therefore, it is not surprising that the MoT-ICE is a very diffusive scheme. In our numerical experiments, MoT-ICE smears out contact discontinuities approximately equally much as the HLL scheme. Recall that the HLL scheme does not suffer from any of the known multidimensional instabilities, which have been a prime motivation for the development of Riemann-solver free schemes, and that HLL’s most significant drawback is its high dissipativity. Moreover, we have proved in this paper that HLL produces no pressure or velocity oscillations for initial data with pure density variation, a property which does not hold for the MoT. Note furthermore that MoT-ICE, while being significantly faster and easier to implement than the original MoT-CCE [Fey98b], is still slower and more complicated to implement than HLL. Together with more numerical results (see [KN04] and [Kr¨ o04]), this should be enough reason to rethink the future development of the MoT, at least the ICE version proposed

Riemann-Solver Free Schemes

449

by the second author. We believe that only a considerable less dissipative, and yet carbuncle free, version of the MoT would justify further research into ideas like a positivity preserving slope limiter [Zim98] or non-reflecting boundary conditions. It would be interesting to see how other Riemann-solver free schemes perform for the test problems discussed in the present paper.

References [Bre84]

Brenier, Y.: Average multivalued solutions for scalar conservation laws. SIAM J. Numer. Anal., 21, 1013–1037 (1984) [But60] Butler, D.S:: The numerical solution of hyperbolic systems of partial diﬀerential equations in three independent variables. Proc. Roy. Soc., 255A, 232–252 (1960) [CH62] Courant, R., Hilbert, D.: Methods of Mathematical Physics. Volume II: Partial Diﬀerential Equations. Interscience Publishers, New York (1962) [DRS93] Deconinck, H., Roe, P.L., Struijs, R.: A multidimensional generalization of Roe’s ﬂux diﬀerence splitter for the Euler equations. Comput. & Fluids, 22 no. 2–3, 215–222 (1993) [Des86] Deshpande, S.M.: A Second-Order Accurate Kinetic-Theory-Based Method for Inviscid Compressible Flows. NASA Technical Paper, 2613 (1986) [Ein88] Einfeldt, B.: On Godunov-type methods for gas dynamics. SIAM J. Numer. Anal., 25 no. 2, 294–318 (1988) [Fey93] Fey, M.: Ein echt mehrdimensionales Verfahren zur L¨osung der Eulergleichungen. PhD Thesis, ETH Z¨ urich (1993) [Fey98a] Fey, M.: Multidimensional upwinding. I. The method of transport for solving the Euler equations. J. Comput. Phys., 143, 159–180, (1998) [Fey98b] Fey, M.: Multidimensional upwinding. II. Decomposition of the Euler equations into advection equations. J. Comput. Phys., 143, 181–199 (1998) [FNT01] Fey, M., Noelle, S., von T¨orne, C.: The MoT-ICE: a new multidimensional wave-propagation-algorithm based on Fey’s method of transport. With application to the Euler- and MHD-equations. Internat. Ser. Numer. Math., 140, 141, 373–380 (2001) [GR96] Godlewski, E., Raviart, P.-A.: Numerical Approximation of Hyperbolic Systems of Conservation Laws. Applies Mathematical Sciences, 118. Springer, New York (1996) [GVM99] Gressier, J., Villedieu, P., Moschetta, J.-M.: Positivity of ﬂux vector splitting schemes. J. Comput. Phys., 155 no. 1, 199–220 (1999) [HLL83] Harten, A., Lax, P.D., van Leer, B.: On upstream diﬀerencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev., 25 no. 1, 35–61 (1983) [JT64] Jeﬀrey, A., Taniuti, T.: Non-Linear Wave Propagation. Academic Press, New York (1964) [Kr¨ o04] Kr¨ oger, T.: Multidimensional systems of hyperbolic conservation laws, numerical schemes, and characteristic theory—Connections, diﬀerences, and numerical comparison. PhD Thesis, RWTH Aachen (2004)

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Kr¨ oger, T., Noelle, S., Zimmermann, S.: On the Connection between some Riemann-Solver Free Approaches to the Approximation of MultiDimensional Systems of Hyperbolic Conservation Laws. Accepted for publication in M2 AN, (2004) [KN04] Kr¨ oger, T., Noelle, S.: Numerical Comparison of the Method of Transport to a Standard Scheme. Accepted for publication in Comp. & Fluids., (2004) [Kr¨ o97] Kr¨ oner, D.: Numerical Schemes for Conservation Laws. John Wiley & Son Ltd., Stuttgart (1997) [Lev90] LeVeque, R.J.: Numerical Methods for Conservation Laws. Lectures in Mathematics. Birkh¨ auser, Basel (1990) [LMS97] Lin, P., Morton, K.W., S¨ uli, E.: Characteristic Galerkin Schemes for Scalar Conservation Laws in Two and Three Space Dimensions. SIAM J. Numer. Anal., 34 no. 2, 779–796 (1997) ˇ a, M., Morton, K.W., Warnecke, G.: Evolution [LMW00] Luk´ aˇcov´ a-Medvidov´ Galerkin methods for hyperbolic systems in two space dimensions. Math. Comp., 69, 1355–1384 (2000) ˇ a, M., Morton, K.W., Warnecke, G.: Finite Vol[LMW02] Luk´ aˇcov´ a-Medvidov´ ume Evolution Galerkin (FVEG) Methods Hyperbolic Sytems. Preprint, (2002) ˇ a, M., Saibertov´ [LSW02] Luk´ aˇcov´ a-Medvidov´ a, J., Warnecke, G.: Finite volume evolution Galerkin methods for nonlinear hyperbolic sytems. J. Comput. Phys. 183 no. 2, 533–562 (2002) [Noe00] Noelle, S.: The MoT-ICE: A New High-Resolution Wave-Propagation Algorithm for Multidimensional Systems of Conservation Laws Based on Fey’s Method of Transport. J. Comput. Phys., 164 no. 2, 283–334 (2000) [Ost95] Ostkamp, S.: Multidimensional Characteristic Galerkin Schemes and Evolution Operators for Hyperbolic Systems. PhD Thesis, University of Hannover (1995) [Ost97] Ostkamp, S.: Multidimensional Characteristic Galerkin Methods for Hyperbolic Systems. Math. Meth. Appl. Sci., 20, 1111 (1997) [PA01] Pandolﬁ, M., D’Ambrosio, D.: Numerical Instabilities in Upwind Methods: Analysis and Cures for the “Carbuncle” Phenomenon. J. Comp. Phys., 166 no. 2, 271–301 (2001) [Per90] Perthame, B.: Boltzmann Type Schemes for Gas Dynamics and the Entropy Property. SIAM J. Numer. Anal., 27 no. 6, 1405–1421 (1990) [Per92] Perthame, B.: Second-Order Boltzmann Schemes for Compressible Euler Equations in One and Two Space Dimensions. SIAM J. Numer. Anal., 29 no. 1, 1–19 (1992) [Pra01] P. Prasad, Nonlinear hyperbolic waves in multi-dimensions. Chapman & Hall/CRC, New York (2001) [Qui94] Quirk, J.J.: A contribution of the great Riemann solver debate. Int. J. Numer. Methods Fluids, 18, 555-574 (1994) [Roe86] Roe, P.: Discrete models for the numerical analysis of time-dependent multidimensional gas dynamics. J. Comput. Phys., 63 458–476 (1986) [SW81] Steger, J.L., Warming, R.F.: Flux vector splitting of the inviscid gasdynamic equations with application to ﬁnite-diﬀerence methods. J. Comput. Phys., 40 no. 2, 263–293 (1981)

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[Tor99] [TSS94] [Zim98]

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Sweby, P.K.: High resolution schemes using ﬂux limiters for hyperbolic conservation laws. SIAM J. Num. Anal., 21 no. 2, 995–1011 (1984) von T¨ orne, C.: MOTICE – Adaptive, Parallel Numerical Solution of Hyperbolic Conservation Laws. PhD Thesis, Bonner Mathematische Schriften, Nr. 334 (2000) Toro, E.: Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction. Second edition. Springer, Berlin (1999) Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL-Riemann solver. Shock Waves, 4 25–34 (1994) Zimmermann, S.: The Method of Transport for the Euler Equations Written as a Kinetic Scheme. Internat. Ser. Numer. Math., 141 999–1008 (2001)

Relaxation Dynamics, Scaling Limits and Convergence of Relaxation Schemes Hailiang Liu Department of Mathematics, Iowa State University, Ames, IA 50011 [email protected]

Summary. Relaxation dynamics, scaling limits, and relaxation schemes are three main topics on hyperbolic relaxation problems that, remarkably, can be well understood with one model equation. The criterion that leads to desired results for the three problems is the so called “sub-characteristic condition”. The criterion of this nature is also pivotal in the study of general hyperbolic relaxation problems. In this article we review the recent research development in hyperbolic relaxation problems. The emphasis is on contributions associated with our own project within ANumE priority research program. We will ﬁrst review some basic properties and notions for hyperbolic relaxation problems, and then focus our investigation on three main topics associated with the underlying relaxation model: relaxation dynamics, scaling limits as well as convergence theory of relaxation schemes.

1 Introduction This article is written on the occasion that a special book on the ANumE program (a priority research program on Analysis and Numerics for Conservation Laws) is being published, and serves as a review article on a few selected topics on hyperbolic relaxation problems. The emphasis is mainly on our research contributions in these topics associated with the ANumE research project, entitled “Stability in Hyperbolic Systems with Relaxation”, DFG grant Wa 633/11-1. Following the main theme of this article, we restrict the discussion to our results on basic models which contribute to an understanding of relaxation dynamics, scaling limits as well as the relaxation schemes. We also include some earlier complementary results of the author in hopes to provide the reader with a richer picture. However we make no attempt to summarize the extensive literature to the vast amount of special and intrinsic analysis for various hyperbolic relaxation problems in recent years. Historically the program on hyperbolic relaxation problems was pioneered by Whitham from examining interactions of hyperbolic waves of different orders in one single equation

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ut + cux + ǫ(utt − a2 uxx ) = 0,

(1)

in which first and second order waves are present simultaneously, see [85, Chapter 10]. Indeed if the lower order terms were absent (ǫ = ∞), the general solution would be u = u1 (x − at) + u2 (x + at). Conversely, if the higher order waves were absent (ǫ = 0), the solution would be u = u0 (x − ct). It turns out that both kinds of wave play important roles, and there are important interaction effects between the two. The higher order waves carry the “first signal” with speed a, but the “main disturbance” travels with the lower order waves at speed c. One important observation by Whitham is the following stability criterion −a < c < a,

(2)

which ties in nicely with the ideas on propagation. The relevant ideas can be formally taken over to the nonlinear situation ( c → f ′ (u)), and the condition (2) becomes (3) −a < f ′ (u) < a. The equation (1) with c replaced by f ′ (u) gives

ut + f (u)x + ǫ(utt − a2 uxx ) = 0,

(4)

which can be compared to the widely accepted viscous approximation ut + f (u)x = ǫuxx . Also the equation (4) when written into a system of first order PDEs leads to

ut + vx = 0, (5) vt + a2 ux = f (u)−v . ǫ Nonlinear analysis of hyperbolic relaxation problems began with Liu [38] for a more general 2 × 2 relaxation system (than (5)), in which the condition of the type (3) is identified as the “sub-characteristic condition”. The system analyzed in [38] belongs to the following class d 0 u Fj (u, v)xj = + q(u, v) v t

(6)

j=1

with the unknown function U = (u, v) ∈ Rm , and the given flux functions Fi , q(u, v) is a given vector valued smooth function. Systems of form (6) occur in a large number of applications involving various non-equilibrium processes.

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They also occur as approximations to systems in various applications, see e.g. [6, 23, 24, 32, 69, 76]. As the simplest model in the class of (6), the system (5) has become quite popular after Jin and Xin proposed to use it as a numerical device to compute the entropy solution for hyperbolic conservation laws, see [24]. In particular this model possesses the key features of a more general hyperbolic relaxation system, thus serves as an ideal model problem to understand the more general ones. The requirement (3) is shown to be essential for relaxation model (5) to enjoy the stability property. Actually Leveque and Wang [48] showed that if (3) is violated, the instability may, and actually does occur even for linear flux f = cu. For more general hyperbolic relaxation systems first successful attempts to identify necessary stability conditions started with Chen, Liu and Levermore in [12], and independently with Yong in [90]. A more recent account on basic structures of hyperbolic relaxation systems can be found in [91]. For better understanding of conditions of such nature more contributions would be certainly desirable. Among others along this line, let me mention Boillat & Ruggeri [9], Zeng [94], Bouchut [6], Yong [91] and Hanouzet and Natalini [19]. Also consult [15, 14] for stability conditions in some balance laws. The study of problems in this area suggests the whole range of the problem from the microscopic kinetic equations, say Boltzmann equation, to the macroscopic systems, say Euler/Navier-Stokes equations. Therefore it is not surprising that, in recent years, hyperbolic relaxation problems have received a considerable attention, see e.g., [4, 12, 20, 24, 32, 38, 45, 70, 76]. Our research project on hyperbolic relaxation problems has been conducted in developing stability and convergence theory for a class of hyperbolic relaxation systems, see e.g. [34, 35, 36, 37, 42, 51, 52, 53, 60]. The main theme for underlying relaxation system is to examine the stability of solutions in terms of the delicate balance of the nonlinear convection and the relaxation forcing. Our main emphasis in this project is to understand the relaxation dynamics and see how this mechanism affects the stability of the hyperbolic systems. Thus, we seek both the justification of stable relaxation wave patterns and the convergence analysis of approximate solutions in terms of the competition of relaxation and the nonlinear convection. Relaxation dynamics, scaling limits, and relaxation schemes are three main topics for hyperbolic relaxation problems that, remarkably, can be well understood with the model equation (5). The “sub-characteristic condition” (3) plays a pivotal role in the study of above three topics. Therefore we shall review our research results mainly associated with the model (5), while knowing that some of these underlying issues for more general relaxation systems remain to be unveiled. The rest of this article is organized as follows. In the second section, some fundamental notions and properties about the relaxation will be discussed. We will mostly focus on the model (5) though some of what we present can be easily carried over to some more general hyperbolic relaxation systems. Sect. 3

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is devoted to the study of long time solution behaviors driven by the relaxation dynamics. Depending on the pattern of the initial data we will meet the large time diffusion waves, asymptotic towards refraction waves, nonlinear stability of relaxation shocks as well as the stability of relaxation boundary layers. Scaling limits will be addressed in Sect. 4, where results on zero relaxation limit and diffusive limit are reviewed, respectively. Finally convergence results of relaxation schemes will be described in Sect. 5.

2 An Overview of Relaxation Basics The goal of this section is to remind the reader of some basic structure properties of hyperbolic system with relaxation, which will be needed in following sections. We start with a hyperbolic balance law of the form

ut + vx = 0, (x, t) ∈ IR × IR+ (7) vt + a2 ux = f (u) − v, with f = f (u) being a given smooth function. The variables u and v are the unknowns, a > 0 is a given constant. Equilibrium Manifold We are interested in the large time behavior of solutions developed by the relaxation dynamics starting with a certain class of initial data. Intuitively if some stable criteria are met the solution should be attracted to the equilibrium states or to wave patterns connecting equilibrium states. The set of the equilibrium states of a relaxation system for which the source term vanishes is named as the equilibrium manifold. For (7) such manifold can be described as Γ (u) := {(u, v); v = f (u)}. (8) A simple check shows that Γ (u) is a stable manifold in the sense that for any constant initial data (u, v) = (α, β) ∈ IR2 , the corresponding solution (u, v) = (α,

βe−t + f (α)(1 − e−t )),

will approach Γ (α) exponentially as t goes to infinity, independent of the choice of β. Scaling Laws In the relaxation system the solution behavior clearly relies upon the competition of the convection and the driving force from the source term. Under

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457

different scalings such competition could drive the solution towards different states. Two typical scalings are: the hyperbolic scaling and the parabolic scaling. (a) Hyperbolic scaling. Under the hyperbolic scaling t x , (t, x) → ǫ ǫ the system (7) takes the form (5) in Sect. 1, i.e., ut + vx = 0, vt + aux = 1ǫ (f (u) − v).

(9)

Here ǫ is a positive constant representing the rate of relaxation. Relaxation effect is known to provide a subtle dissipative mechanism for discontinuities against the destabilizing effect of nonlinear response (see, for example, Liu [38]). The relaxation model (9) can be loosely interpreted as a discrete velocity kinetic equation. The relaxation parameter, ǫ, plays the role of the mean free path and the limiting system (as ǫ ↓ 0) models the macroscopic conservation laws. (b) Parabolic scaling. Under the parabolic scaling t x , (t, x) → , (u, v) → (ǫu, ǫ2 v) ǫ2 ǫ the system (7) takes the form ut + vx = 0, ǫ2 vt + a2 ux = f (u) − v,

(10)

where we have formally replaced the scaled function f (ǫu) by ǫ2 f (u), see [21, 22, 42]. One of important tasks in the study of relaxation problems is to justify the scaling limits in various regimes, depending on the relative size of physical scales. The limiting process under the hyperbolic scaling is usually called the zero relaxation limit, which provides a passage in parallel to that from the Boltzmann equation to the Euler equation. In contrast, the limit under parabolic scaling is connected to the diffusive kinetic limit of the Boltzmann equation to the Navier-Stokes equation. Further insight about this statement may be obtained via the ChapmanEnskog expansion, an expansion on the differential operator in terms of the small relaxation parameter. With this procedure one can also discover some necessary structure conditions for both scaling limits. Sub-characteristic condition The leading order of the relaxation system (9) gives

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ut + f (u)x = 0,

(11)

which shows the role of (9) as a new way of regularizing the hyperbolic conservation law (11). The Chapman-Enskog expansion on (9), up to O(ǫ) order, yields ut + f (u)x = ǫ[(a2 − f ′ (u)2 )ux ]x .

(12)

Clearly, in order to guarantee the dissipative nature of this convectiondiffusion equation it is necessary to require a2 − f ′ (u)2 > 0, i.e., (3). Note that the sub-characteristic condition (3) plays a similar role to the CFL condition for numerical approximations. The condition of this nature requires that the limit of domain of dependence for the approximation system contains the domain of dependence for original system. Indeed, under this condition, the rigorous passage from (9) to (11) has been justified, see e.g. [68]. Diffusive sub-characteristic condition. Using the Chapman-Enskog expansion on the diffusive scaled model (10), one obtains ut + f (u)x = [(a2 − ǫ2 f ′2 )ux ]x + a2 ǫ2 [f (u)xx + f ′ (u)uxx ]x − a2 ǫ2 uxxxx , (13) and the formal limit system reads ut + f (u)x = a2 uxx .

(14)

Obviously, to ensure the dissipative nature of (13), one just needs a weakened characteristic condition −a < ǫf ′ < a. (15)

In [22] this condition is called the diffusive sub-characteristic condition, under which we rigorously justified that (14) is the limit of (10) as ǫ → 0, see Theorem 4.3 to be presented in Sect. 4.2. In the study of the solution behavior to hyperbolic relaxation systems there are two different kinds of asymptotic limits. The first one is the large time behavior of solutions if we look at the long–time effects of relaxation, called the relaxation dynamics. A different behavior is found when dealing with scaling limits in various regimes, which are essentially driven by equilibrium equations (11) or (14). We summarize this discussion in stating that large time dynamics and the scaling limits are related issues where a key role is played by the subcharacteristic condition (3), under which we establish rigorous theorems for various settings. Moreover these basic properties for the relaxation system (7) are shared by most of the general relaxation systems even though structure conditions appear more complicated in general contexts, see e.g. [12]. This leads us first to the discussion on relaxation dynamics as described in the next section.

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459

3 Relaxation Dynamics This section is devoted to the time-asymptotic behavior of solutions of (7) subject to various class of initial (and boundary if applicable) data. According to Whitham’s observation for linear problem (1), the striking feature of the relaxation dynamics would be the interaction of the waves of the equilibrium equation and the waves of the full relaxation system. Actually in hyperbolic relaxation systems, initial disturbances often propagate along the characteristics of the full system, whose waves will have important roles to play. Yet the equilibrium system is expected to determine the large time relaxation dynamics if some structure conditions are met. The large time solution behavior will certainly rely on the basic pattern of the initial data, which will be detailed in following subsections when the model (7) is being used. The key in dealing with different cases is to clarify the roles of each wave set, and to see how each set is modified by the presence of the other. 3.1 Long-time diffusive behavior Consider the relaxation system (7) with f (u) = αu2 /2, α > 0, subject to the initial data (u, v) = (u0 , v0 ) in IR × {0}, (16) where

u0 , v0 ∈ L1 (IR) ∩ L∞ (IR), u0 ≥ a|v0 | 1

a.e. x ∈ IR.

(17)

Solutions of (7), (16) with L initial data satisfy the following property u(x, t)dx = u0 (x)dx = m, ∀ t ≥ 0. (18) IR

IR

We assume that m > 0 and a is large enough such that the sub-characteristic condition (10) holds for u ∈ I, where I is a bounded interval of range of the component u, see [42]. Since u0 (x) → 0 as |x| → ∞ and the mass m > 0, it follows from Sect. 2 that the sub-characteristic condition may strengthen the diffusive effect hidden in the relaxation dynamics. Thus, in the long time behavior, the hyperbolic system (7) is expected to approach the convection-diffusion equation (14). With Roberto Natalini in [42], we showed that the large time attractor of the solution to (7) subject to the above initial data is indeed a diffusion wave profile to (14). More precisely, let θm : IR → IR be the fundamental solution for the convection diffusion equation (14), starting with a Dirac mass mδ(x). Our result in [42] can be summarized in Theorem 3.1 [42] Assume the sub-characteristic condition (3) be satisfied. 2 Let (u, v) be a solution of (7), (16), with f = α u2 , for bounded initial data

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; (u0 , v0 ) ∈ L1 (IR)2 carrying a finite mass m = IR u0 dx > 0. Assume that there exists a constant K > 0 such that |v0 | ≤ au0 and au0x ± v0x ≤ K for x ∈ IR. Then for every p ∈ [1, ∞) lim t

t→∞

p−1 2p

u(·, t) − θm (·, t)Lp (IR) = 0.

(19)

Some remarks are in order. 1. The diffusive phenomena in the relaxation system was first observed by Liu in [38], via the Chapman-Enskog expansion for a class of 2 × 2 hyperbolic relaxation system. Later, Chern [10] showed that the global smooth solution of relaxation system approaches a diffusive wave θ(x, t) in the Lp –norm at a certain rate if the initial data are sufficiently small. Respect to Chern’s result, the main achievement of Theorem 3.1 consists in removing the smallness on initial data by using a quite different approach. This seems remarkable for hyperbolic relaxation problems since the dissipation from the relaxation effect is known to be weaker than that in the viscous approximation. 2. Our approach is due to a parabolic scaling which has led to (10). With this argument the investigation of the asymptotic behavior of the solution {u, v} can be reduced to studying the convergence of the family {uǫ , v ǫ }, as ǫ → 0. The main difficulty is to obtain the appropriate a priori bounds on the family {uǫ }. One of our main tools for doing that is the following entropy-type estimates C ∂u m ≤ , uL∞ (IR) ≤ C √ , t > 0. ∂x t t The scaling arguments of such nature have been widely used in studying the large time behavior of solutions for equations of parabolic type, see for instance [17, 25]. We want to stress here the main phenomena that, in the long time behavior, the hyperbolic system (7) is driven by an equation which is different in type, e.g.: parabolic. 3. The asymptotic rate obtained in Theorem 3.1 is sharp, and it implies p−1 p−1 that t 2p θ is the Lp attractor of t 2p u. 4. For more general flux functions similar results can be established by exploiting the same scaling argument. Actually it is enough to assume that f (u) behaves like |u|q−1 u for q ≥ 2 and for u small enough. For q > 2 the asymptotic behavior will be driven by a simple heat equation, see [42] for details. 3.2 Asymptotic towards rarefaction waves In this section and in the following one, our discussion about large time relaxation dynamics will be continued. Here we assume that the initial data are asymptotically approaching equilibrium states i.e, (u, v)(x, 0) = (u0 , v0 )(x) → Γ (u± )

as

x → ±∞,

(20)

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461

where u± are constants satisfying u− < u+ . The complementary situation, when u− > u+ , and the problem (7), (20) admits relaxation shock profiles, will be dealt with in Sect. 3.3. Again we assume the sub-characteristic condition (3) holds in a bounded interval u ∈ I such that the global existence of the solution (u, v), as well as the L1 contraction property are secured, see [35]. Our goal is again to show the large time behavior of solutions and to measure their asymptotic rates to the large time wave profiles. The time asymptotic behavior of the solutions to (7), (20) is related to that of the Riemann problem for the equilibrium conservation law: rt + f (r)x = 0, r(x, 0) = u± ,

(x, t) ∈ IR × IR+ ,

(21)

for ± x > 0.

Its entropy solution r(x, t) is called the rarefaction wave, given explicitly by ⎧ x < f ′ (u− )t ⎨ u− , ′ −1 x r(x, t) = (f ) ( t ), f ′ (u− )t ≤ x ≤ f ′ (u+ )t. (22) ⎩ u+ , x > f ′ (u+ )t

Compared to the relaxation shock waves to be discussed in Sect. 3.3, the rarefaction wave is time-varying. One does not know what the exact large time wave profile is when trying to prove it stable. The rarefaction stability results depend strongly on how well one defines the approximate profile. Therefore in [35] we proposed a notion of the relaxation rarefaction profile in the following way: for any constant γ > 0, there exists t0 > 0 such that 0 ≤ r(x, t0 )x ≤ γ.

The u-component of the solution to (7) with initial data Γ (r(x, t0 )) is called the relaxation rarefaction profile. In fact this solution is shown to be monotone non-decreasing in space and flatten out in a way as the rarefaction wave does. Such relaxation wave profile is shown to behave like the usual rarefaction wave for the equilibrium conservation laws, and serves as an Lp (p > 1) attractor for a large class of initial data in L∞ + L1 ∩ H 1 . Theorem 3.2 [35] Let (u0 , v0 ) ∈ Γ (r(x, t0 )) + L1 (IR) ∩ H 1 (IR) be the initial data, and (¯ u, v¯) be the relaxation rarefaction profile as described above. Suppose that the stability condition (3) holds. Then there exists C > 0 such that u(t) − u ¯(t)Lp + v(t) − v¯(t)Lp ≤ C(1 + t)−(1/2)+(1/2p) ,

∀t ≥ 0.

An immediate consequence of the above theorem for the case u+ = u− is (u, v)(t) − Γ (u− )Lp ≤ C(1 + t)−(1/2)+(1/2p) ,

∀t ≥ 0.

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One may also ask how fast the solution converges to the rarefaction wave (22). The answer provided in [35] is the sharp estimate u(·, t) − r(·, t + t0 )Lp ≤ CN (t)(1/2)+(1/2p) (t + t0 )−(1/2)+(1/2p) ,

(23)

where N (t) := ¯ u(·, t) − r(x, t + t0 )L1 ,

t ≥ 0.

If N (t) ∼ 1 + ln(t + t0 ) holds, then the above estimate (23) is consistent with the result obtained in [40] for viscous case. We use both the L1 -contraction and the time-weighted L2 energy approach in proving the above results, for which the regularity estimate for u-component plays essential role and with such regularity we were able to investigate the global solution behavior of the relaxation system starting with the general large data, as described in Theorem 3.2. Consult [38, 39] for stability results of rarefaction wave profiles of some hyperbolic relaxation systems. 3.3 Nonlinear stability of relaxation shocks As pointed out earlier, if u− > u+ , the problem (7), (20) is expected to approach a travelling shock profile. Actually under the stability condition (3), the relaxation system (7) admits a smooth travelling wave solution of the form (u, v)(x, t) = (φ, ψ)(x − st),

(φ, ψ)(±∞) = Γ (u± )

(24)

where the shock speed s = [f (u+ ) − f (u− )]/(u+ − u− ), see e.g. [50]. In [37] the travelling wave (φ, ψ) is named as the relaxation shock profile. Our main interest is to investigate the asymptotic stability of relaxation shock profiles. •

Stability of relaxation shocks for 1-D scalar law.

In a series of works [50, 55, 56] joint with Wang, Yang and Woo, we studied the stability of relaxation shock profiles for some 2 × 2 hyperbolic relaxation models. In these papers the stability of strong shock profiles with possibly nonconvex flux were established via the weighted energy method. The selection of the weight depends on the underlying shock profile and the flux function f . We refer to these papers for the full story, a decay rate result from [55] is summarized in the following ′ Theorem 3.3 [55]. Let (φ, ψ) be a non-degenerate (f ′ (u ; + ) < s < f (u− )) relaxation shock profile connecting Γ (u± ), determined by IR (u0 (x) − φ(x))dx = 0. Let (u, v) be a global solution to (7) subject to initial data, having asymptotic rate |x|−α/2 to (φ, ψ) as |x| → ∞ in the following sense #2 # x # # 2 α # (25) (u0 − φ)dx, u0 − φ, v0 − ψ(x))## dx ≤ C. (1 + x ) 2 #( IR

−∞

Relaxation Dynamics and Scaling Limits

Then

463

α

sup |(u, v)(x, t) − (φ, ψ)(x − st)| ≤ Const(1 + t)− 2 .

x∈IR

1. In [55] the decay rate for non-integer α was obtained as t−α/2+ǫ . Actually this rate can be improved by removing ǫ, see the discrete version to be reviewed in Sect. 5.3. If the relaxation shock is degenerate (s = f ′ (u+ ) or s = f ′ (u− )), the decay rate obtained in [55] is t−α/4 provided that an additional spatial decay rate on the degenerate side is imposed. 2. These results reveal that the information on the decay rate can be transferred from space to time. 3. The L1 stability of the relaxation shocks for (7) was first obtained in [64], which when combined with our result in [42] stated in Sect. 3.1 may lead to a definitive statement: the underlying relaxation shock stands as an L1 large time attractor; Consult [79] for an independent argument. 4. These decay rates are shown to be the same as those for the viscous conservation laws obtained by Matsumura and Nishihara in [63]. •

The equilibrium equation is a n × n system

If the limit system (11) is a n × n system, the stability issue becomes quite subtle. In such situation there are more wave fields for equilibrium system as well as for the full relaxation system, much more effort needs to be involved to handle the interaction of various wave modes. Let λk ∈ σ(Df (u)), spectrum of Df , be a simple eigenvalue in a neighborhood of some reference state u∗ , u± are close to u∗ . For each k ∈ {1, · · · , n}, we assume that the shock speed s satisfies the strict Liu’s entropy condition s = s(ρ+ ) < s(ρ)

(26)

for ρ between 0 and ρ+ , where the parameter ρ = lk (u− ) · (u(ρ) − u− ) parameterizes the k-th Hugoniot curve u(ρ) = u(ρ, u− ) passing through u− and u+ = u(ρ+ ). When restricting ourselves to the non-degenerate shocks, i.e., λk (u+ ) < s < λk (u− ), we recorded here a main result obtained in [37]. Theorem 3.4 [37] Assume the sub-characteristic condition for λ ∈ σ(Df (u)) |λ| < a,

|u − u∗ | < ǫ0 .

(27)

There are numbers ǫ0 , β0 > 0 such that if |u± − u∗ | < ǫ0 and (φ, ψ)(x − st) is a relaxation shock profile connecting Γ (u± ) u(x, 0) = φ(x) + Ux (x, 0), with

IR

|∂xα U (x, 0)|2 dx +

IR

v(x, 0) = ψ(x) − W (x, 0)

|∂xα−1 W (x, 0)|2 dx ≤ β0 ,

α = 0, 1, 2,

(28)

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then lim sup |(u, v)(x, t) − (φ, ψ)(x − st)| = 0.

t→∞ x∈IR

1. Note that the present result restricts to perturbations with zero total mass in u-component, so there are no diffusion waves. We refer to [10, 42] for the decay estimate of diffusion waves when the equilibrium system is a scalar law. 2. The stability of relaxation shocks for the 2 × 2 system has been well understood in previous studies. The results for 3 × 3 Broadwell model can be found in [13, 29, 81]. It would be interesting to extend our stability result to more general relaxation systems, at least to those the existence of relaxation shocks has been established by Yong and Zumbrun [93]. 3. We refer to [67] for an elegant pointwise analysis of relaxation shocks and references therein for the recent development along this line. •

2-D relaxation shock fronts.

The stability issue of relaxation shock fronts in multi-D case is much more delicate. First the wave front will not only translate in the heading direction but is forced to reshape in the transverse direction. In [36] we prove nonlinear stability of planar shock fronts for certain relaxation system in two spatial dimensions. If the sub-characteristic condition is assumed and the initial perturbation is sufficiently small, even though the mass carried by initial perturbations is not necessarily finite, then the solution converges to a shifted planar shock front solution as time t ↑ ∞. The asymptotic phase shift of shock fronts is in general non-zero and governed by a similarity solution to heat equation. The asymptotic decay rate to the shock front is proved to be t−1/4 in L∞ (IR2 ) without imposing extra decay rate in space for initial perturbations. To make things more precise we follow [36] and consider a simple example of Jin-Xin’s 2-D relaxation system ut + v1x + v2y = 0,

(t, x, y) ∈ IR+ × IR2 ,

v1t + a2 ux = f (u) − v1 ,

(29)

2

v2t + b uy = −v2 .

The unknowns u, v1 , v2 belong to IR, the function f = f (u) is in C 2 , and a, b > 0 are fixed constants satisfying the sub-characteristic condition (3). The equilibrium manifold reads Γ2 (u) = {(u, v1 , v2 ),

v1 = f (u),

v2 = 0}.

The initial data are asymptotically constants as x → ±∞, i.e., (u, v1 , v2 )(0, x, y) = (u0 , v10 , v20 )(x, y) → Γ2 (u± ), with u± being two given constants such that u− > u+ .

x → ±∞

(30)

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A planar relaxation shock front is a travelling wave solution to (29) of the form (u, v1 , v2 ) = (U, V1 , V2 )(z), z = x − st, connecting Γ2 (u± ). Its existence is ensured by the sub-characteristic condition (3) combined with the Oleinik entropy condition i.e., f (u) − f (u± ) < s(u − u± ) for u+ < u < u− . We focus on the stability of the relaxation shock front with perturbation carrying infinite mass, i.e., (u0 (x, y) − U (x))dxdy = ∞, IR2

in such a situation the asymptotic state is shown to be (U, V1 , V2 )(x − st + d(t, y)),

and the effective phase shift d(t, y) in the shock front may have different values at y = ±∞. In fact the location of the shock front may be determined by solving a wave equation of d(t, y), which satisfies 1 [u(t, x, y) − U (x)]dx, d(t, y) = u+ − u− IR for all y, t. The effective phase shift d(t, y) is time-asymptotically governed by a solution to the heat equation dt = b2 dyy . The asymptotic ansatz can be actually given by √y +∞ b 4π 2 2 e−πz dz + d+ e−πz dz → d± ρ(y) = d− b

y √ 4π

−∞

as

y → ±∞,

(31)

; m1 := IR dt (0, y)dy and θ(y) is a smooth function with compact support and unit integral.

Theorem 3.5 [36] Let (u, v1 , v2 ) be a global solution to (29) subject to initial data being ; x a small perturbation of the relaxation shock front (U, V1 , V2 )(x, y), with ( −∞ (u0 − U )dx, u0 − U, v10 − V1 , v20 ) having decay rate |x|−α/2 as |x| → ∞ in the sense of (25). Then the following convergence rate estimate holds # # # α 1 y + y0 ## # √ sup #(u, v1 , v2 )(t, x, y) − (U, V1 , V2 )(x − st + ρ( ))# ≤ C(1+t)−min{1, 2 + 4 } . 2 t + 1 IR 1. This result indicates that, as t → ∞, the wave equation has a parabolic structure through the effective phase shift d. 2. Theorem 3.5 suggests that the decay rate of perturbations could not be faster than t−1 even if a stronger localization of perturbation may be imposed. However the decay rate is always not slower than t−1/4 . This is in sharp contrast to the one dimensional theory which we have seen in Theorem 3.3. 3. We refer to [59] for a stability result on weak shock front to (29) in the presence of transverse force g(u), i.e., with g(u) − v2 on the right of the third equation in (29). In order to control the effect from g(u), the convexity of f plays essential role in the stability analysis of [59].

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3.4 Stability of relaxation boundary layers We now return to the 1-D model (7) and discuss boundary layers under relaxation. Sub-characteristic condition (3) has been proved to be necessary and also sufficient for the global-wellposedness of the initial value problem for (7). One would expect some additional criterion to ensure stable boundary layers. Such requirement is complicated by the interaction of two sets of waves at the boundary. With Wen-An Yong in [60] we proved the time-asymptotic boundary layer under a dissipative boundary condition. We consider (7) on the quarter-plane x, t ≥ 0 subject to initial data (u0 , v0 ) for x ≥ 0. Since the coefficient matrix of the vector (ux , vx )⊤ in (7) 0 1 A= a2 0 has only one positive eigenvalue at x = 0, it is well known, [26], that one relation of boundary data: B(u(0, t), v(0, t)) = 0

(32)

should be given with B = B(u, v) satisfying Bu + aBv = 0,

(33)

where (1, a)⊤ is the right eigenvector associated with the positive eigenvalue of the coefficient matrix of (7). The asymptotic behavior has been studied for some special cases of boundary data, say Bv = 0, by several authors, for instance [44, 66, 74, 75]. Despite the known results concerning IBVP, one may still wonder what is a natural condition on B in order that IBVP can have globally stable boundary layer. Our answer provided in [60] is the dissipative stability condition Bu = 0 and Bu Bv ≥ 0,

(34)

with which we came up with the following Theorem 3.6 [60] Let (U, V )(x) be a bounded steady solution to IBVP and f ′ (u+ ) < 0. Assume (3) and (34) hold. Then there exists a positive constant δ > 0 such that if +∞ (u0 − U )(x)dxH 3 (IR+ ) + (v0 − V )(x)H 2 (IR+ ) + |B(u+ , f (u+ ))| < δ, x

then the IBVP (7), (32) has a unique global solution (u, v)(x, t) satisfying lim sup |(u, v)(x, t) − (U, V )(x)| = 0.

t→∞ x∈IR+

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467

See [60] for details and we conclude this section by two remarks. 1. Note that different from the initial value problem(IVP), for the IBVP the total mass in u-component is not preserved, but changing with the boundary flow. 2. The condition of the type (34) seems also necessary, as evidenced by previous studies on zero relaxation limit problems with boundaries, see e.g. [27, 73, 86, 92]. For boundary layer analysis associated with Broadwell model see [58]; for more general Jin-Xin model see [88, 89]. 3. Consult [16] for kinetic approximation of a boundary problem for conservation laws.

4 Scaling Limits Scaling limits represent sharper physical modelling of macroscopic equations, the prototypical example being the hydrodynamic limit problem for the Boltzmann equation. So one of the central issues in the hyperbolic relaxation problems is to justify the limiting process in various scaling regimes. 4.1 The zero relaxation limit In a relaxation limit process the primary system describes the physical dynamics on a finer scale than the system of conservation laws. As we pass to the relaxation limit fine scale features are often lost and we recover the hyperbolic conservation laws. The relaxation approximation to conservation laws is in spirit close to the description of the hydrodynamic equations by the detailed microscopic evolution of gases in kinetic theory. The rigorous theory of kinetic approximation for solutions with shocks is well developed when the limit equation is scalar. For works using the continuous velocity kinetic approximation, see Giga and Miyakawa [18], Lions, Perthame and Tadmor [45] and Perthame and Tadmor [76], for discrete velocity approximation of entropy solutions to multidimensional scalar conservation laws see Natalini [69], Katsoulakis and Tzavaras [32]. For the scaled relaxation system (9), as ǫ ↓ 0, the u-component is expected to converge to the entropy solution of scalar conservation laws (11). We make the following assumptions: v0ǫ = f (u0 (x)) + K(x)ω(ǫ) (H1 ) uǫ0 (x) := u0 (x), ∞ 1 where K ∈ L ∩ L (IR) ∩ BV (IR), ω : [0, ∞[→ [0, ∞[ is continuous and ω(0) = 0. (H2 ) the flux function f is a C 1 function with f (0) = f ′ (0) = 0; (H3 ) the initial data satisfy (uǫ0 , v0ǫ ) ∈ L1 (IR) ∩ L∞ (IR) ∩ BV (IR). Equipped with assumptions made in (H1 )-(H3 ), it has been proved that, as ǫ → 0+ , the solution sequence to (9) converges strongly to the unique entropy

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solution of (11), see Natalini [68], also [46] when the equilibrium equation is a n × n system. Our interest is more than convergence, and we want also detect the convergence rates. With Gerald Warnecke in [49], we have obtained sharp estimates. Theorem 4.1 [49] Consider the relaxation system (9), subject to L∞ (IR) ∩ BV (IR)-perturbed initial data satisfying (H1 )-(H3 ). Then the global solution (uǫ , v ǫ ) converges to (u, f (u)) as ǫ ↓ 0 and the following error estimates hold, √ uǫ (·, t) − u(·, t)1 ≤ CT ǫ, (35) t

t

v ǫ (·, t) − f (uǫ (·, t))1 ≤ CT [e− ǫ ω(ǫ) + ǫ(1 − e− ǫ )],

0 ≤ t ≤ T.

(36)

Some remarks are in order. 1. Our estimates are sharp and (36) reflects two sources of error: the initial error of size ω(ǫ) and the relaxation error of order ǫ. 2. It is striking that the effect of initial error persists only for a short time of order ǫ! Beyond this time the non-equilibrium solution approaches the equilibrium state at an exponential rate. 3. We would like to mention that an analogous result for a class of relaxation systems was obtained by Kurganov and Tadmor [31] by using the Lip′ framework initiated by Nessyahu and Tadmor [72]. But their argument uses the convexity of the flux function. For the case of a possibly nonconvex flux function f , our work uses Kuznetzov-type error estimates, see [8, 33]. We also refer to [82, 83] for first order convergence rates when piecewise smooth solutions with finitely many discontinuities are to be computed with the assumption of convex fluxes f (u). 4.2 The diffusive scaling limit We now turn to discuss the diffusive scaling limit. Consider the system (10), which is obtained from (7) under a parabolic scaling as stated in Sect. 2. The limit equation is expected not to be the scalar conservation law (11) but the convection diffusion equation (14). With Shi Jin in [21], we rigorously justify the diffusive limit even the relaxation shocks are present in the solution. To state our result more precisely we make the following assumptions on initial data: u± a travelling wave (U ǫ , V ǫ ) of (10). (B1 ) ; xgenerate ǫ (B2 ) −∞ (u − U ǫ )(y)dyH 3 + ǫ(v0ǫ − V ǫ )(x)H 2 ≤ c1 , (B3 ) (v0ǫ − V ǫ − s(uǫ0 − U ǫ )H 2 ≤ c2 and 1 ǫ v − f (uǫ0 ) + a∂x uǫ0 H 1 ≤ c3 , ǫ 0 (B4 )

;

R

(uǫ0 − u0 )(x)dx = 0 and, as ǫ → 0,

uǫ0 − U ǫ → u0 − U 0 ,

v0ǫ − V ǫ → v0 − V 0 ,

in H 2

as

ǫ → 0.

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469

In these assumptions, c1 , c2 , c3 are positive constants independent of ǫ. Under these conditions, we have Theorem 4.2 [21] Assume 0 < ǫ ≤ 1, f is a smooth convex function satisfying the sub-characteristic condition (3) for u ∈ (u+ , u− ) and (B1 ) − (B4 ) hold, then there exists an u ∈ C 1 ([0, T ] × IR) for any T ≥ 0 such that 2−δ1 uǫ − U ǫ → u(x, t) − U 0 in C 0 ([0, T ]; Hloc )

v ǫ − V ǫ → v(x, t) − V 0

in

for any

δ1 > 0,

2−δ1 ), C 0 ([0, T ]; Hloc

and the limit function (u, v) solves the relaxed parabolic problem: ut + f (u)x = a2 uxx ,

u|t=0 = u0 (x)

with v = f (u) − aux . From the discussion in Sect. 2 we know that the requirement of (3) for diffusive limit is too strong. Using a refined energy argument we were finally able to prove the same result but under a weaker requirement (15). Theorem 4.3 [22] Assume that there exists a constant ǫ0 > 0 such that 0 < ǫ ≤ ǫ0 ,

a > ǫ|f ′ |

for u ∈ (u+ , u− ) ,

(37)

and (B1 ) − (B4 ) hold, then the convergence result in Theorem 4.2 still holds if |u+ − u− | ≤ β for some β > 0. 1. Our arguments also provide a clear picture of the large time behavior of solutions for both original system and the reduced equation. 2. The diffusive scaling introduced in (10) is very typical in many important physical problems, for example, in transport equation in diffusive regime [5, 41, 77], in kinetic equations near incompressible Navier-Stokes regimes [3, 11], in hyperbolic balance laws [62] and in nonlinear parabolic equation [7]. 3. For similar diffusive limits justified by using L1 compactness tools, see e.g. [43, 47, 65]. However the energy method we exploited is capable of generalizations to system case, see e.g. [61].

5 Convergence of Relaxation Schemes Relaxation schemes to be considered here are a class of non-oscillatory numerical schemes for systems of conservation laws proposed by Jin and Xin [24]. These schemes provide a new way of perturbing, even regularizing, systems of conservation laws and approximating their solutions. The computational results that are available, see e.g. [24] as well as Aregba-Driollet and Natalini [2], indicate that the relaxed schemes obtained in the limit ǫ → 0 provide

470

H. Liu

a quite promising class of new schemes. We point out that the main assets of these schemes are that they neither require the computation of the Jacobians of fluxes for the conservation laws nor the use of Riemann-solvers. This important property is shared by other schemes such as for instance the high resolution central schemes introduced by Nessyahu and Tadmor [71], see also Kurganov and Tadmor [30] for references therein on recent developments. To make things more precise we want to approximate the scalar conservation law (11). We choose a time step ∆t, a spatial mesh size ∆x, a parameter a which will be related to the characteristic speed of the conservation law and a small relaxation parameter ǫ > 0. From these we define the mesh ratio ∆t λ = ∆x , the CFL parameter µ = aλ ∈]0, 1[, and the scale parameter k = ∆t ǫ . The mesh is given by the points (j∆x, n∆t) for j ∈ ZZ and n ∈ IN. The approximate solution takes the discrete values unj at the mesh points. Further, relaxation schemes involve the discrete relaxation fluxes vjn . We want to focus on the following semi-implicit relaxation scheme < µ n n n n n λ − un un+1 j ∈ ZZ, n ∈ IN j + 2 (vj+1 − vj−1 ) − 2 (uj+1 − 2uj + uj−1 ) = 0, j 2 µ n+1 n n n n )]. − f (un+1 vjn+1 − vjn + a 2λ (un j+1 − uj−1 ) − 2 (vj+1 − 2vj + vj−1 ) = −k[vj j (38)

The discrete ! initial data are given by averaging the initial data u0 over mesh cells Ij = (j − 12 )∆x, (j + 21 )∆x , i.e. taking 1 u0j = u0 (x)dx, and setting vj0 ∼ f (u0j ). (39) ∆x Ij 5.1 Optimal convergence rates With Gerald Warnecke in [49], we obtained a global error estimate for the relaxation scheme (38) approximating the scalar conservation law (11). The key idea is to decompose the error into a relaxation error and a discretization error. √ Including an initial error ω(ǫ) we thus obtained the rate of convergence as stated in Sect. 4.1. In the discretization of ǫ in L1 for the relaxation step, √ step a sharp convergence rate of ∆x in L1 is obtained. These rates are independent of the choice of initial error ω(ǫ). Thereby, we obtain the order 1/2 for the total error. Also our estimate is independent of the type of nonlinearity. Theorem 5.1 [51] Let u be the entropy solution of scalar conservation law (11) satisfying the initial data u0 , and let (uN , v N ) be a piecewise constant N representation of the data (uN i , vi )i∈IN generated by relaxation scheme (38). Then, for any fixed T = N ∆t > 0, there exists a constant CT , independent of ∆x, ∆t and ǫ such that 2√ √ 3 uN − u(·, T )1 ≤ CT ǫ + ∆x .

1. The total error bound is proved to be independent of the initial error ω(ǫ), which provides a theoretical support for the relaxation scheme that one could

Relaxation Dynamics and Scaling Limits

471

choose any initial v0 (x), not necessarily on the equilibrium manifold Γ (u). 2. Taking ǫ = 0 in discretization step, we immediately recover the optimal convergence rate of order 1/2 for monotone schemes of scalar conservation laws , see Sabac [78], Tang and Teng [84]. 3. Consult [87] for convergence analysis of second order relaxation schemes. For convergence results on some other relaxation schemes, see e.g. [1, 28, 80]. 5.2 Lip+ stability and error estimates As is well known if the flux function f is strictly convex, say f ′′ (u) ≥ α > 0, the entropy solution to the scalar conservation laws enjoys Oleinik’s one-sided regularity estimate ux ≤

1 , αt + u0 −1 Lip+

wLip+ := esssupx=y

w(x) − w(y) x−y

+

,

(·)+ = max(·, 0).

The natural question is whether one can obtain a better estimate for the relaxation scheme (38) with convex flux. With Wang and Warnecke, we provided a definite answer in [52] for (38). We first established the discrete lip+ -stability for the relaxation scheme (38), which is the heart of our effort and the proof is quite elegant. Such lip+ stability says that if u0 Lip+ = L < ∞, then unj − unj−1 ≤ 2L∆x

for j ∈ ZZ,

n ∈ IN.

(40)

Equipped with (40) we obtained global error estimates in the spaces W s,p for −1 ≤ s ≤ 1/p, 1 ≤ p ≤ ∞ and point-wise error estimates for the approximate solution obtained by the relaxation scheme (38). The proof uses the framework introduced by Nessyahu and Tadmor [72, Corollary 2.2, 2.4]. The resulting error estimates are summarized as follows. Theorem 5.2 [52] Consider the convex scalar conservation law (11) with Lip+ -bounded initial data u0 . Then the relaxation scheme (38) with discrete initial data (u0j , f (u0j ))j∈ZZ converges. The piecewise linear interpolants u∆,ǫ satisfy the convergence rate estimates u∆,ǫ (·, T ) − u(·, T )W s,p ≤ CT (∆x + ǫ) as well as

1−sp 2p

,

for

−1 ≤ s ≤

1 , 1 ≤ p ≤ ∞, p (41)

1

|u∆,ǫ (x, T ) − u(x, T )| ≤ Constx,T (∆x + ǫ) 3 , with

# # Constx,T ∼ 1 + #ux (·, T )#L∞ (x−(∆x+ǫ)1/3 , x+(∆x+ǫ)1/3 ) .

(42)

Remarks: 1. When (s, p) = (−1, 1) the error estimate (41) turns into the Lip′ error estimate

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u∆,ǫ (·, t) − u(·, t)Lip′ (IR) ≤ O(ǫ + ∆x). 2. When (s,√p) = (0, 1) the error estimate (41) yields an L1 -convergence rate of order O( ∆x + ǫ) which is consistent with the result in Theorem 5.1 for conservation laws with possibly non-convex flux functions. 3. In this context we also established a remarkable discrete maximum principle for (38) in the sense that if b1 ≤ u0 (x) ≤ b2 and v0 = f (u0 ) then b1 ≤ unj ≤ b2

for j ∈ ZZ,

n ∈ IN.

5.3 Convergence to discrete relaxation shocks Discrete shock profiles epitomize the propagation of solutions and structure properties of shocks in numerical solutions. The study of discrete shocks for conservative schemes have attracted much attention. Among researchers who joined this endeavor, let us list Jennings, Majda and Ralston, Michelson, Smyrlis, J.-G. Liu and Xin, T.P. Liu and Yu, Fan, Ying · · · . For relaxation schemes of the type (38) together with J. Wang and T. Yang in [34, 54, 57], we have obtained a series of positive results on discrete relaxation shocks. More precisely we considered the existence, the asymptotic stability and the decay rate of discrete relaxation shocks for relaxation schemes including (38), these results develop the stability theory of discrete relaxation shock profiles. To see the flavor a result in [34] is recorded here. Theorem 5.3 Assume that the CFL condition 0 < µ < 1, and the subcharacteristic condition (3) hold. Let (Uj , Vj )j∈ZZ be a stationary discrete relaxation shock wave connecting Γ (u± ). Assume that (u0j − Uj ) = 0 j∈Z Z

and for some α > 0 and some δ > 0 3 2 (1 + j 2 )α/2+1 |u0j − Uj |2 + (1 + j 2 )α/2 |vj0 − Vj |2 ≤ δ. j∈Z Z

Then the unique global solution (unj , vjn )j∈ZZ to (38) with the initial data (u0j , vj0 )j∈ZZ satisfies √ sup |(unj , vjn ) − (Uj , Vj )| ≤ C(1 + nh)−α/2 δ, j

n ≥ 0,

provided λ is suitably small, and the scale parameter k = ∆/ǫ ∈ IR+ . This result shows that there is a relation between the spatial decay assumed of the initial perturbation and the rate of decay in time. In this sense the theorem exhibits the transformation of spatial decay into temporal decay.

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5.4 The R-W model of Boltzmann equation Under the sub-characteristic condition (3), the relaxation model (7) as a balance law enjoys the quasi-monotone property, which plays essential role in the analysis of this model. The R-W model for the Boltzmann equation is, however, a typical model which does not share such a quasi-monotone property. With Wang and Warnecke in [53], we proposed a splitting scheme for such RW-model and proved the uniform convergence as both relaxation parameter and mesh size tend to zero; the convergence rate (local and global) is recovered following Tadmor’s Lip’ theory. One of the new elements in this study is that there is no monotonicity for this model, therefore a more careful and intrinsic analysis of its structure becomes essential. See [53] for details. Acknowledgement. I would like to express my deep thanks to the Institut f¨ ur Analysis und Numerik of Magdeburg University. I was proud and happy to honor my Humboldt research fellowship and to undertake the DFG research project there in the period of July 1997-December 1999. I particularly thank Professor Gerald Warnecke for his extremely valuable collaboration on much of the research described here, and for his constant encouragement during my exciting period in Germany. Many ﬁne results described in this article also belong to my other collaborators, to whom I am most grateful. Finally, I would like to acknowledge the support from the DFG-ANumE program and the Alexander von Humboldt Foundation.

References 1. D. Aregba-Driollet and R. Natalini, Convergence of relaxation schemes for conservation laws, Appl. Anal. 61 (1996), 163–190. 2. D. Aregba-Driollet and R. Natalini, Discrete kinetic schemes for multidimensional conservation laws, SIAM J. Numer. Anal. 37 (2000), 1973–2004. 3. C. Bardos, F. Golse and C.D. Levermore, Fluid dynamic limit of kinetic equations II: Convergence proofs for the Boltzmann equations, Comm. Pure Appl. Math. 46 (1993), 667-753. 4. S. Bianchini, A Glimm type functional for a special Jin-Xin relaxation model, Ann. Inst. H. Poincare’ Anal. Nonlinear 18-1 (2001), 19–42. 5. A. Bensoussan, J.L. Lions and G.C. Papanicolaou Boundary layers and homogenization of transport process, J. Publ. RIMS Kyoto Univ. 15 (1979), 437–476. 6. F. Bouchut, Construction of BGK models with a family of kinetic entropies for a given system of conservation laws, J. Statist. Phys. 95, (1999), 113–170. 7. F. Bouchut, F.R. Guarguaglini and R. Natalini, Diffusive BGK Approximations for nonlinear multi-dimensional parabolic equations, Indiana Univ. Math. J. 49 (2000), 723–749. 8. F. Bouchut and B. Perthame, Kruˇzkov’s estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc. 350 (1998), 2847–2870.

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29. S. Kawashima and A. Matsumura, Asymptotic stability of travelling wave solutions of systems for one dimensional gas motion, Comm. Pure Appl. Math. 101 (1985), 97–127. 30. A. Kurganov and E. Tadmor, New High-Resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys. 160 (2000), 241–282. 31. A. Kurganov and E. Tadmor, Stiff systems of hyperbolic conservation laws. Convergence and error estimates, SIAM J. Math. Anal., 28 (1997), pp. 14461456. 32. M. Katsoulakis and A. Tzavaras, Contractive relaxation systems and scalar multidimentional conservation laws, Commun. P.D.E., 22 (1997), pp. 195-233. 33. N.N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first order quasilinear equation, U.S.S.R. Comput. Math. and Math. Phys. 16 (1976), 105-119 34. H.L. Liu, Convergence rate to the discrete traveling wave for relaxation schemes, Math. Comp. 69 (2000), 583–608. 35. H.L. Liu, The Lp stability of relaxation rarefaction profiles, J. Diﬀer Equations, 171, (2001), 397-411. 36. H.L. Liu Asymptotic decay to the relaxation shock fronts in two dimensions, Proceedings of the Royal Society of Edinburgh: Section A 131A (2001), 1385– 1410. 37. H.L. Liu, Asymptotic Stability of Relaxation Shock Profiles for Hyperbolic Conservation Laws, J. Diﬀ. Equ. 192 (2003), 285–307. 38. T. P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108 (1987), 153-175. 39. T. Luo, Asymptotic stability of planar rarefaction waves for the relaxation approximation of conservation laws in several dimensions, J. Diﬀerential Equations, 133 (1997), 255-279. 40. E. Harabetian, Rarefactions and large time behavior for parabolic equations and monotone schemes, Commun. Math. Phys. 114 (1988), 527-536. 41. E.W. Larsen and J.B. Keller, Asymptotic solutions of the neutron transport problem, J. Math. Phys. 15 (1974), 75–81. 42. H. L. Liu and R. Natalini, Long-time diffusive behavior of solutions to a hyperbolic relaxation system, Asymptotic Analysis, 25 (2001) 21–38. 43. C. Lattanzio, R. Natalini, Convergence of diffusive BGK approximations to nonlinear parabolic systems, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), no. 2, 341–358. 44. T.P. Liu and K. Nishihara, Asymptotic behavior for scalar viscous conservation laws with boundary effect, J. Diﬀerential Equations, 133 (1997), 296–320. 45. P.L. lions, B. Perthame and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc., 7 (1994), 169–191. 46. C. Lattanzio and D. Serre, Convergence of a relaxation scheme for hyperbolic systems of conservation laws, Numer. Math. 88 (2001), 121–134. 47. P.L. Lions and G. Toscani, Diffusive limit for finite velocity Boltzmann kinetic models, Rev. Mat. Iberoamericana, 13 (1997), 473–513. 48. R. J. LeVeque, J.-H, Wang, A linear hyperbolic system with stiff source terms. Nonlinear hyperbolic problems: theoretical, applied, and computational aspects (Taormina, 1992), 401–408, Notes Numer. Fluid Mech., 43, Vieweg, Braunschweig, 1993.

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Multidimensional Adaptive Staggered Grids S. Noelle1 , W. Rosenbaum1 , and M. Rumpf2 1

2

RWTH Aachen Templergraben 55, D-52056 Aachen, Germany, noelle|[email protected] Universitat Bonn, Institut f¨ ur Numerische Simulation, Nussallee 15, 53115 Bonn, Germany, [email protected]

Summary. A variety of numerical schemes operates on staggered grids, that is a pair of meshes of the same computational domain whose interior nodes (in 1D), edges (in 2D) and faces (in 3D) do not coincide. While the shape of a staggered grid is canonical on an uniformly reﬁned Cartesian mesh it becomes more complicated for an underlying adaptively reﬁned grid, in particular in higher spatial dimensions. Here we present both a construction technique for staggered dual grids exploiting the structure of the adaptively reﬁned Cartesian primal grids in 2D and 3D, and discuss the necessary modiﬁcations of a standard Finite Volume scheme which is originally formulated on uniform meshes.

1 Introduction On the field of computational fluid dynamics central schemes became quite popular during the last decade. Operating on staggered grids offers algorithmical simplicity. The ease of evaluating fluxes not on the border but inside a cell, hence no requirement for (approximate) Riemann solvers and the componentwise application of the scalar framework to solve systems of conservation laws make these schemes very convenient to work with. The prototype of all central schemes is the first-order Lax-Friedrichs scheme [7] in one spatial dimension. To increase its resolution and to widen the field of applications many authors ([11], [9], [2], [5], [1]) contributed to its extension to a higher order scheme in two and even three spatial dimensions on unstructured resp. uniformly refined Cartesian grids. [4] proposes a non-staggered version of central schemes by averaging the staggered solution over the non-staggered grid. This approach takes advantage of the uniform structure of the underlying grids and, since the handling of the (very simple) staggered grid is avoided, simplifies the treatment of boundary conditions. However it poses difficulties for nonuniformly refined meshes. For the use of adaptively refined Cartesian grids it becomes far more complicated to describe (and in a next step to circumvent) the corresponding staggered grids.

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Here we present a construction of structured adaptive staggered grids. The local refinement constellation on the adaptively refined Cartesian primal grid determines the local shape of the according dual grid. Since the primal grid is structured and the adaption in space is subject to grading restrictions the number of different local refinement constellations is bounded. This number can still be diminished by identifying local shapes that differ only by rotation or reflection. The dual grid construction is hence understood as an assembling of scaled and rotated copies of some predefined local pattern. Such staggered grids serve as a base for a modified version of the Finite-Volume scheme proposed by Jiang and Tadmor in [5]. The paper is organized as follows. After stating a general formulation of finite volume schemes on staggered grids in section 2, section 3 presents the staggered grid construction, necessary modifications of a standard FV scheme and some applications in 2D. The basic ideas are used to epitomize the extensions to three spatial dimensions, also mentioning new difficulties and still open questions in section 4. Throughout this paper we use the capital letters G, C, F , E and N for a grid, 7 cell, face, edge resp. node, capital calligraphic letters to denote sets, like N = N , subscripts p and d to distinguish between primal and dual objects, subscripts i, j and k for node indices, and the superscript + to label local objects on single cells.

2 Finite Volume scheme on staggered grids Two grids G and G ∗ are called staggered if

• both G and G ∗ are meshes for the same computational domain Ω • interior nodes, edges and faces of G and G ∗ do not coincide (however, they may and will intersect)

The general system of conservation laws reads ut + div F (u) = 0.

(1)

Here u ∈ Rm is the vector of conservative variables, and F = (f1 , f2 , . . . , fd ) with fi : Rm → Rm ∀i the d-component flux function. We denote the mean value of the approximate solution v at time tn on the cell C ∈ G by 1 ¯ nC = vn dC (2) v |C| C Using (1) and (2) an evolution step in time is described by

¯ n+1 v = C

1 |C|

1 = |C|

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vn+1 dC

0C

C

n

v dC −

tn+1

tn

div F (v) dC dt

C

<

⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ tn+1 ⎬ ⎨ 1 n = F (v) · n ds dt . v dC + ⎪ |C| ⎪ tn ∂C C ⎪ ⎪ ⎪ ?@ A⎪ ⎭ ⎩> ?@ A > =:I1

(3)

=:I2

Here n denotes the outer normal on C’s boundary ∂C. The evolution (3) applies componentwise on vn+1 . Depending on the cellwise presentation of v (constant, linear, . . . ), the value of the integral I1 can be determined exactly. In contrast, I2 includes the integration of the non-linear flux function F and should be approximated by an appropriate quadrature rule in time and space. Moreover, v is in general not continuous at the cell boundary ∂C, hence the evaluation of F is not declared there. In order to circumvent the latter problem, numerical solutions to successive timesteps are now defined on corresponding staggered grids. The temporal evolution for C ∗ ∈ G ∗ now reads ⎫ ⎧ tn+1 ⎬ ⎨ 1 n ¯ n+1 v v dC + . (4) F (v) · n ds dt = ∗ C ⎭ |C ∗ | ⎩ C∩C ∗ ∂C ∗ tn ∗ C∩C =∅

The application of (4) for a first-order scheme with cellwise constant data on a uniform grid in one spatial dimension leads to the well known Lax-Friedrichs scheme from [7]: = vn+1 j+ 1 2

1 n (v + vnj+1 ) − λ[f (vnj+1 ) − f (vnj )], 2 j

(5)

with λ = ∆t/∆x. A second-order scheme with a cellwise linear presentation of the numerical solution on uniform grids in 1D was given by Nessyahu and Tadmor in [11]. In I2 the integration in time is now achieved by using the midpoint rule. vn+1 j+ 1

=

2

1 n 1 [vj + vnj+1 ] + [(vnj )′ − (vnj+1 )′ ]− 2 8 ∆t ∆t n )) − f (v(xj , tn + ))]. λ[f (v(xj+1 , t + 2 2 1

In order to evaluate the flux argument at time tn+ 2 = tn + and the Taylor expansion of v to get v(xj , tn +

∆t 2

(6)

one uses (1)

∆t ∂ ∆t ) = v(xj , tn ) + v(xj , tn ) + O(∆t2 ) 2 2 ∂t 1 ≈ vnj − λ(fjn )′ . 2

(7)

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The piecewise linear reconstructions in space of v and f on cell j, v′j and fj′ resp., have to meet the approximation requirement 1 ∂ (vn )′ = v(xj , tn )+O(∆x), ∆x j ∂x

1 ∂ (f n )′ = f (v(xj , tn ))+O(∆x). (8) ∆x j ∂x

An extension of this scheme to uniform Cartesian grids in 2D was given by Jiang and Tadmor in [5]. Now, I2 contains additionally an integration in space along the edges of every cell which is accomplished by the rectangular rule. For details see [5].

3 Adaptive staggered grids in 2D Numerical simulations produce a large amount of data. In order to minimize memory requirements and computing time without sacrificing high spatial resolution in regions of interest numerical schemes are often based on the use of adaptive grids. Staggered grid schemes on unstructured grids were described by Arminjon, Viallon and Madrane in [2]. On the other hand, structured grids are particularly popular for reasons of simplicity. Our investigation restricts to the extension of the Jiang-Tadmor scheme to structured adaptive grids. 3.1 Dual grid construction in 2D We claim the Cartesian primal grid Gp to be - like widely used - an adaptively refined Cartesian grid in 2D fulfilling a 1-level transition constraint between cells which share a common edge, see figure 1. Cells which share a common node only are allowed to differ by more than one refinement level (however, due to the 1-level transition constraint referred above those cells might differ by at most two refinement levels).

Fig. 1. 1-level transition condition for an adaptive Cartesian grid.

Given a primal grid Gp we are now looking for a corresponding staggered dual grid Gd . The numerical algorithm claims at least: • Gp and Gd are meshes for the same computational domain Ω

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• interior nodes and edges of Gp and Gd do not coincide Moreover, the dual grid Gd should locally reflect the resolution of the primal grid Gp and, with regard to the timestep limiting CFL-number, try to maximize the distance between faces of primal and dual cells. Our approach consists of the following principle items: • construct dual cells always surrounding exactly one node Np ∈ N (Gp ) • the shape of the dual cells is determined by a Voronoi decomposition of the domain Ω respecting all nodes Np ∈ N (Gp ) It turns out to be sufficient to perform this Voronoi decomposition locally on every cell Cp of the primal grid. Depending on N + (Cp ) := N (Gp ) ∩ Cp , the nodes on the boundary of the primal cell Cp , we deduce its decomposition into local Voronoi regions. All these local regions together form a local pattern on Cp . Local pattern on adjacent cells of the primal grid match naturally, all local Voronoi regions corresponding to a fixed node Np of the primal grid finally form the dual cell Cd (Np ). Voronoi regions Given a set Ω ⊂ Rn and a finite set of nodes N = {Nk , k = 1, . . . , m | Nk ∈ Ω}, we 7 decompose Ω into m Voronoi regions VNk by the following conditions: m ˙ N = ∅, i = j, (iii) x − Nk q ≤ x − Nj q , x ∈ ˙N ∩V (i) k=1 VNk = Ω, (ii) V j i ˙ VNk , ∀j = k. Here VNi denotes the interior of VNi , and · q an arbitrary norm on Rn . Voronoi regions VNi (Nj ) and VNj (Ni ) originating by a decomposition of Ω ⊂ R2 referring to only two nodes Ni resp. Nj ∈ Ω share a common separating polygon Sij . This separating polygon is the locus of the intersection points of circles with same diameter centered at Ni resp. Nj . The shape of the separating polygon depends as well on the locus of Ni and Nj as on the used norm on R2 . BIn general, a Voronoi region VNi around the node Ni takes the form VNi = Nj ∈N VNi (Nj ) . Thanks to the structure of the primal grid and on that score rigidly prescribed location of its nodes the choice of the ·∞ -norm (over the ·2 -norm) simplifies the construction of local Voronoi regions on cells of the primal grid. Boundary faces of Voronoi regions are now aligned only with the axes or the plane diagonals of the primal grid (cf. figure 2(b) in contrast to the more complex figure 2(a)). Creating local pattern To create a local pattern on a cell Cp of the primal grid we have to assign Cp ’s ”volume“ the nodes on the boundary of Cp , i.e. all Nk ∈ N + (Cp ). These are the four corner nodes of Cp (they always exist) and possibly up to four nodes on the midpoint of Cp ’s edges (these hanging nodes only exist if adjacent cells

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(a) resp. the · 2 -norm

(b) resp. the · ∞ -norm

Fig. 2. Comparison of local Voronoi regions on a primal cell in 2D.

in the primal grid were refined). The local intersection of separating polygons boils down to only six different types of local Voronoi regions, cf. figure 3.

Fig. 3. Resulting local Voronoi regions on a primal cell in 2D.

Obviously, all cuts of the cell Cp run parallel to the axes or the plane diagonals. We could therefore describe the steps for the construction of local Voronoi regions on Cp alternatively as: 1. subdivide the cell Cp into 16 congruent squares 2. subdivide all squares containing the midpoint of Cp into two triangles. The cuts follow the face diagonals through Cp ’s midpoint. On Cp we get 8 triangles.

Fig. 4. Subdivision of a primal cell in 2D.

These squares and triangles, depicted in figure 4, will be denoted as elements of Cp . Finally we assign these 20 elements of Cp the nodes of N + (Cp ) simply by calculating the · ∞ -distance between the nodes and the element’s center of gravity. All elements being assigned the same node Nk ∈ N + (Cp )

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form the local Voronoi region VNk (Cp ) on Cp . The set of all local Voronoi regions on Cp forms the local pattern. Figure 5 collects all six essentially different local pattern in 2D. These are pattern that do not arise from each other by rotation. Figure 6 shows an adaptive primal grid and the corresponding dual grid.

Fig. 5. All essentially diﬀerent local pattern in 2D.

Fig. 6. Cartesian primal 2D-grid (solid) and corresponding dual grid (dashed).

3.2 Modifications of the Jiang-Tadmor scheme The general grid-independent formulation of a cellwise update in the staggered finite volume scheme is given by (4). Using adaptive Cartesian grids, cells of the corresponding dual grid are no longer guaranteed to be squares. Hence the quadrature rules for the integration in (4) have to be modified. In the first integral I1 , the size of overlapping regions of a primal cell C and a dual cell C ∗ now depends on the current local pattern on C. For I2 , the length |e| of an edge e of C ∗ , and hence the integrating weight λ = ∆t/|e|, can only be deduced by knowing both the local pattern on C as well as on C’s neighbour overlapped by the edge e. Moreover, for edges not aligned to the axes both flux components contribute to the flux integral. Finally the reconstruction

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of functions to higher order polynoms might become an intricate task, in particular on the dual grid where cells can have even more neighbours than on the primal grid. 3.3 Applications in 2D In [14] and [12] the construction technique for the dual grid slightly deviates from that described in section 3. Nonetheless, a variety of standard problems (including scalar-, Euler- and MHD-equations in 1D and 2D) solved here produce pleasant results. Figure 7 shows adaptive calculations for the forward facing step incorporating techniques of artificial compression from [8].

(b) t = 1.0 Fig. 7. Forward facing step + artiﬁcial compression, max. resolution 210 × 28 .

4 Adaptive staggered grids in 3D Picking up the idea of constructing dual grids by assembling local pattern described in section 3 we extend this approach to adaptively refined Cartesian grids in 3D. Now, the 1-level transition condition on the primal grid, mentioned in section 3.1, should be respected over faces and edges of the hexahedral cells. Appropriate refinement techniques are discussed in the context of AMR in [3], by using saturated element-based error indicators in [13], or by applying tools of multiscale analysis in [10]. The local subdivision of primal cells into Voronoi regions and the forming of local pattern follows similar principles like those in 2D and is discussed more in detail in [15]. Whereas in 2D the number of essentially different local pattern is easily determined as six (cf. figure 5), its counting is a more challenging task in 3D. We present its essential parts in section 4.2 below. The finite volume scheme deriving from the general formulation (4) causes similar difficulties like those listed in section 3.2. Spatial integration of the flux function has now to be performed over polygonal bounded cell faces. Linear reconstruction of functions complicates further by the increased number of neighbouring cells.

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4.1 Grid construction in 3D The construction of local Voronoi regions on cells of the primal grid resembles that of grids in 2D and is briefly described as: 1. subdivide the hexahedral cell Cp into 64 congruent cubes 2. subdivide all cubes containing a midpoint of Cp ’s faces into two prisms. The cuts follow the face diagonals through the face midpoints. On Cp we get 8 × 6 = 48 prisms. 3. subdivide all cubes containing the central point of Cp into six tetrahedra. The three cuts equal the diagonal cuts on their adjacent cubes from step 2. This results in 6 × 8 = 48 tetrahedra. These cubes, prisms and tetrahedra will again be denoted as elements of Cp . Figure 8(a) shows a cell Cp and some of its elements. Next we assign these 64 + 48 − 24 + 48 − 8 = 128 elements of Cp the nodes of N + (Cp ) by calculating the · ∞ -distance between the nodes and the element’s center of gravity. All elements being assigned the same node Nk ∈ N + (Cp ) form the local Voronoi region VNk (Cp ) on Cp . Figure 8(b) shows a cell Cp , the set N + (Cp )\N (Cp ) (i.e. the corners of Cp left out) and a few corresponding local Voronoi regions. Due to the · ∞ -norm local Voronoi regions might now happen to be nonconvex (figure 8(c)). The set of all local Voronoi regions on Cp form the local pattern. The shape of these three-dimensional local pattern on the boundary faces of a primal cell obviously matches the two-dimensional pattern from figure 5.

(a) subdivision of a pri(b) local Voronoi regions (c) non-convex Voronoi mal cell in 3D regions Fig. 8. Local construction of Voronoi regions in 3D.

4.2 Counting local pattern The distribution of the nodes N + (Cp ) on the boundary of a primal cell Cp determines the shape of the local pattern. Instead of intricately speaking of

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occurring nodes on the midpoint of an edge or a face of Cp we call this edge resp. face ’colored’. If a face f of Cp is colored, i.e. f is refined, consequently all four edges of f are also refined, i.e. automatically colored. The converse is in general not true. All information to determine the number of essentially different edge- and face-colored cubes, i.e. all colored cubes that do not arise from each other by rotation or reflection, is supplied by the group of symmetries D of the hexahedron. First, we list all 48 elements of D, the self-mappings of the cube by specifying the 10 representatives gk including the cardinality of the classes of conjugated elements c(gk ): ⎛ ⎞ ⎛ ⎞ +1 0 0 +1 0 0 g1 = ⎝ 0 +1 0 ⎠, c(g1 ) = 1, g2 = ⎝ 0 −1 0 ⎠, c(g2 ) = 3, 0 0 +1 0 0 +1 ⎛ ⎞ ⎛ ⎞ 0 0 +1 −1 0 0 g3 = ⎝ 0 +1 0 ⎠, c(g3 ) = 6, g4 = ⎝ 0 +1 0 ⎠, c(g4 ) = 3, −1 0 0 0 0 −1 ⎛ ⎞ ⎛ ⎞ −1 0 0 +1 0 0 g5 = ⎝ 0 0 −1 ⎠, c(g5 ) = 6, g6 = ⎝ 0 −1 0 ⎠, c(g6 ) = 1, 0 0 −1 0 −1 0 ⎛ ⎞ ⎛ ⎞ 0 0 −1 −1 0 0 g7 = ⎝ 0 0 +1 ⎠, c(g7 ) = 6, g8 = ⎝ −1 0 0 ⎠, c(g8 ) = 8, 0 +1 0 0 +1 0 ⎛ ⎞ ⎛ ⎞ 0 0 +1 0 0 −1 g9 = ⎝ 0 −1 0 ⎠, c(g9 ) = 6, g10 = ⎝ +1 0 0 ⎠, c(g10 ) = 8. 0 −1 0 +1 0 0 7

z

8 y

5

6

x

3 1

4 2

Fig. 9. Nodes of a cube.

Respecting figure 9 we label the edges of the cube by indicating their nodes by

Multidimensional Adaptive Staggered Grids

e1 = (1, 2),

e5 = (1, 3),

e9 = (1, 5),

e2 = (3, 4), e3 = (5, 6),

e6 = (5, 7), e7 = (2, 4),

e10 = (2, 6), e11 = (3, 7),

e4 = (7, 8),

e8 = (6, 8),

e12 = (4, 8).

489

The cycles of permutations on the set E(W) = {ej , j = 1, . . . , 12} induced by the mappings gk read as (e)

g1 = (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12), (e)

g2 = (1 2)(3 4)(5)(6)(7)(8)(9 11)(10 12), (e)

g3 = (1 9 3 10)(2 11 4 12)(5 6 8 7), (e)

g4 = (1 3)(2 4)(5 8)(6 7)(9 10)(11 12), (e)

g5 = (1 4)(2)(3)(5 11)(6 9)(7 12)(8 10), (e)

g6 = (1 4)(2 3)(5 8)(6 7)(9 12)(10 11), (e)

g7 = (1)(2 3)(4)(5 10)(6 12)(7 9)(8 11), (e)

g8 = (4 6 11)(10 1 7)(5 12 3)(2 8 9), (e)

g9 = (1 12 3 11)(2 10 4 9)(5 7 8 6), (e)

g10 = (1 6 10 4 7 11)(2 5 9 3 8 12). Clearly, in order to get an edge-colored cube mapped onto itself by a mapping (e) gk , all edges inside a cycle of gk have to wear the same color. For every (e) (e) cycle of gk one can choose its color independently. With N (gk ) denoting (e)

(e)

the number of cycles in gk we get 2[N (gk )] different (but not necessarily essentially different!) edge-colorings Cj that are mapped by gk onto itself, i.e. gk ◦ Cj = Cj . Hence the sum S=

g∈D

2[N (g

(e)

)]

=

10

(e)

c(gk ) 2[N (gk

)]

= 6912

k=1

counts all# different edge-colorings # Cj of the cube, each of them with multiplicity mj = #{g | g◦Cj = Cj , g ∈ D}#. On the other hand, if W = {wl , l = 1, . . . , q} denotes the set of the q (still unknown) essentially different edge-colorings of the cube, the set P = {g ◦ w, g ∈ D, w ∈ W} consists of all edge-colorings Cj of the cube. Each Cj is the image of exactly one w ∈ W and mj self-mappings g ∈ D. Consequently we get # # # # # # # # #{g | g ∈ D}# · #{w | w ∈ W}# = #D# · #W # = 48 q = S. Thus there are q = 144 essentially different edge-colored cubes. To answer the question on the number of essentially different cubes with

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exactly 3 colored edges, for example, one has to count the number S3 of (e) possibilities of choosing cycles in the gk where their lengths sum up to 3. Let (e) (e) nr (gk ) denote the number of cycles of length r in gk . Then we have . 1 (e) 2 (e) 3 (e) / 10 (e) n1 (gk ) n (gk ) n (gk ) n (gk ) S3 = c(gk ) + + =9 3 1 1 1 k=1

These 9 colorings are shown in figure 10.

Fig. 10. Essentially diﬀerent cubes with 3 colored edges.

The results derived above in detail are all supplied directly using the tools of P´ olya’s theory, for details we refer to [6]. Basing on the cycle representation (e) of the gk we get the cycle index on the set E(W) as Z(D, E(W)) =

1 12 (x + 3x41 x42 + 12x21 x52 + 4x62 + 8x43 + 12x34 + 8x26 ). 48 1

Here d · xri denotes r cycles of length i in d different mappings g ∈ D, e.g. (e) (e) (e) and g9 both supply x34 with coefficients g1 leads to the term x12 1 , g3 c(g3 ) = c(g9 ) = 6. Substituting xi by 1 + xi leads to ZE(W) (D, 1 + x) =

1& (1 + x)12 + 3(1 + x)4 (1 + x2 )4 + 12(1 + x)2 (1 + x2 )5 48 ' + 4(1 + x2 )6 + 8(1 + x3 )4 + 12(1 + x4 )3 + 8(1 + x6 )2

=1 + x + 4x2 + 9x3 + 18x4 + 24x5 + 30x6 + 24x7 + 18x8 + 9x9 + 4x10 + x11 + x12 .

The number of essentially different edge-colored cubes with r colored edges equals the coefficient of the monomial xr . We find again the number of 9 cubes with 3 colored edges. The whole number of essential different edge-colored cubes sums up to ZE(W) (D, 2) = 144. Applying this counting on the permutations of faces instead of edges leads to 10 essentially different face-colored cubes Wf . For all these Wf we consider the appropriate self-mappings g f , i.e. only those elements g ∈ D which map the colored faces of Wf onto each other. Obviously these self-mappings {g f } ˜ f ) which form a subgroup Df of D. The influence of all g f on the edges E(W f are not automatically colored by the colored faces of W is summarized in the ˜ f )). It supplies again the number of essentially (reduced) cycle index Z(Df , E(W different edge-colored cubes basing on the face-colored cube Wf .

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As an example we treat a cube Wf with exactly one colored face. Every selfmapping of Wf has to map Wf ’s only colored face onto itself. The set of self-mappings, again characterized by representatives gkf and the cardinality of their classes of conjugated permutations c(gkf ), reads as ⎛

⎞ +1 0 0 g1f = ⎝ 0 +1 0 ⎠, c(g1f ) = 1, 0 0 +1 ⎛ ⎞ +1 0 0 g3f = ⎝ 0 0 +1 ⎠, c(g3f ) = 2, 0 +1 0 ⎞ ⎛ +1 0 0 f g5 = ⎝ 0 0 −1 ⎠, c(g5f ) = 1. 0 −1 0

⎛

⎞ +1 0 0 g2f = ⎝ 0 +1 0 ⎠, c(g2f ) = 2, 0 0 −1 ⎛ ⎞ +1 0 0 g4f = ⎝ 0 0 +1 ⎠, c(g4f ) = 2, 0 −1 0

˜ f ) are The (reduced) cycles of the induced permutations on E(W f(e)

g1

f(e) g2 f(e) g3

f(e)

= (1)(2)(3)(4)(5)(6)(9)(11),

g4

= (1 3 4 2)(5 11 6 9),

= (1 3)(2 4)(5 6)(9)(11),

f(e) g5

= (1 4)(2 3)(5 6)(9 11),

= (1)(2 3)(4)(5 9)(6 11),

and we get f 2 3 4 5 6 7 8 ZE(W f ) (D , 1 + x) =1 + 2x + 6x + 10x + 13x + 10x + 6x + 2x + x , ˜

e.g. 10 (the coefficient of x3 ) colorings with one colored face and 3 (additional) colored edges. Table 1 summarizes the number of colorings. Table 1. Essentially diﬀerent cube colorings. colored faces 0 1 2 ess. diﬀ. colorings 144 51 20

3 7

4 3

5 1

6 1

All together we get 227 essentially different edge- and face-colored cubes, and hence 227 essentially different local pattern. They can all be assembled, analyzed and stored in a lookup-table in advance. Later, the numerical scheme gets instant access to them and evaluates all necessary local information rapidly. 4.3 First application in 3D In order to test the presented dual grid construction and to roughly estimate the numerical costs of the (yet to be implemented) staggered grid scheme we

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performed a mere volume integration over a subset of the dual grid Gd . The primal grid has been constructed by adaptively resolving the isosurface of a tilted elliptical paraboloid inside the unit cube. We successfully verified our algorithm by means of the Gauss Integral Theorem 1. by summing up the volumes of the involved Voronoi regions 2. by integrating 13 x, nf over all common faces f of neighbouring Voronoi regions (in view of flux integration over cell boundaries). Here nf denotes the oriented normal of f . The expected numerical effort is comparable to that of a cheap non-staggered finite volume scheme based on a 2-flux Riemann solver. For more details we refer to [15]. 4.4 Further work The extension of the staggered grid approach to a higher order 3D finite volume scheme is still challenging. It concerns the integration of non-constant functions over the boundary of the dual cells, the reconstruction of cell- and flux-values to end up with a higher order scheme, and the data handling for large 3D simulations. Additionally desirable would be the parallelization of the staggered grid scheme as well as the local adaptivity in time.

References 1. P. Arminjon, A. St-Cyr and A. Madrane. New two- and three-dimensional nonoscillatory central ﬁnite volume methods on staggered cartesian grids. Appl. Numer. Math., Vol. 40, 367–390, 2002. 2. P. Arminjon, M.C. Viallon and A. Madrane. A ﬁnite volume extension of the Lax-Friedrichs and Nessyahu-Tadmor schemes for conservation laws on unstructured grids. Int. J. Comput. Fluid Dyn., Vol. 9, No. 1, 1–22, 1997. 3. J. Bell, M. Berger, J. Saltzman, and M. Welcome. Three-dimensional adaptive mesh reﬁnement for hyperbolic conservation laws. SIAM J. Sci. Comput., Vol. 15, No. 1, 127–138, 1994. 4. G.-S. Jiang, D. Levy, C.-T. Lin, S. Osher and E. Tadmor. High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws. SIAM J. Numer. Anal., Vol. 35, No. 6, 2147–2168, 1998. 5. G.-S. Jiang and E. Tadmor. Nonoscillatory central schemes for multidimensionalhyperbolic conservation laws. SIAM J. Sci. Comput., Vol. 19, No. 6, 1892– 1917, 1998. 6. M.Ch. Klin, R. P¨ oschel and K. Rosenbaum Angewandte Algebra, VEB Deutscher Verlag der Wissenschaften, Berlin, 1988. 7. P.D. Lax. Weak solutions of non-linear hyperbolic equations and their numerical computations. CPAM, 7:159–193, 1954. 8. K.A. Lie, and S. Noelle. High resolution nonoscillatory central diﬀerence schemes for the 2D Euler equations via artiﬁcial compression. Progress in industrial mathematics at ECMI 2000 (Palermo), 318–324, Springer, 2002.

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9. X.D. Liu and E. Tadmor. Third order nonoscillatory central scheme for hyperbolic conservation laws. Numer. Math., 79:397–425, 1998. 10. S. M¨ uller. Adaptive multiresolution schemes. Proceedings of Finite Volumes for Complex Applications III, June, 24 to 28, 2002, Porquerolles (France), Hermes Penton Science, p. 119 – 136. 11. H. Nessyahu and E. Tadmor. Non-oscillatory central diﬀerencing scheme for hyperbolic conservation laws. Journal of Computational Physics, 87:408–463, 1990. 12. S. Noelle, W. Rosenbaum and M. Rumpf. An adaptive staggered grid scheme for conservation laws. Int. Ser. of Numer. Math., Vol. 141, Birkh¨ auser, 2001. 13. M. Ohlberger and M. Rumpf. Hierarchical and adaptive visualization on nested grids. Computing, Vol. 59, No. 4, 269–285, 1997. 14. W. Rosenbaum. Ein zweidimensionales adaptives staggered-grid Verfahren zur L¨ osung von Systemen hyperbolischer Diﬀerentialgleichungen. Diploma thesis, Bonn University, 1999. 15. W. Rosenbaum, M. Rumpf, and S. Noelle. 3D Adaptive Central Schemes: part I, Algorithms for Assembling the Dual Mesh.

On Hyperbolic Relaxation Problems Wen-An Yong and Willi J¨ ager IWR, University of Heidelberg Im Neuenheimer Feld 368, 69120 Heidelberg, Germany yong.wen-an|[email protected]

Summary. This report summarizes our works on hyperbolic systems of ﬁrst-order partial diﬀerential equations with source terms. We discuss the introduction of our structural stability and entropy dissipation conditions for initial or initial-boundary value problems. For initial value problems, several systematic results are reviewed. These include the non-existence of (linearly stable) relaxation approximations to non-strongly hyperbolic systems of equations, the justiﬁcation of the formal zero relaxation limit, the existence of relaxation shock proﬁles, and the existence of global smooth solutions for balance laws.

1 Introduction This report is concerned with systems of first-order PDEs (partial differential equations) with source terms: Ut +

d

Fj (U )xj = Q(U ),

(1)

Aj (U )Uxj = Q(U ).

(2)

j=1

or more generally, Ut +

d j=1

Here U is the unknown n-vector valued function of (x, t) ≡ (x1 , x2 , · · · , xd , t) ∈ Ω × [0, +∞) with Ω ⊂ Rd , taking values in an open subset G of Rn (called state space); Q(U ), Fj (U ) and Aj (U )(j = 1, 2, · · · , d) are given n-vector or n × n-matrix valued smooth functions of U ∈ G; and the subscripts t and xj refer to the partial derivatives with respect to t and xj , respectively. As fundamental PDEs and as intermediate models [Ga75, Le96, MR98] between the Boltzmann equation [Ce88] and hyperbolic conservation laws [Da00], systems of first-order PDEs with source terms describe non-equilibrium

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processes in physics, for media with hyperbolic response. Important examples occur in chemically reactive flows [EG94], inviscid gas dynamics with relaxation [Wh74], radiation hydrodynamics [MM84, Po73], traffic flows [Wh74], nonlinear optics [HH00], the numerical solution of conservation laws with relaxation schemes [AN00, Bo99, CP98, JX95], and so on. Moreover, these systems seem indispensable to describe dissipative relativistic fluids [GL90, GL91]. In fact, the most straightforward generalizations of the NavierStokes equations (for simple or non-relativistic dissipative fluids) are physically unacceptable, because they fail to provide causal evolution equations. In those applications, the source term Q(U ) has, or can be transformed by a linear transformation into, the form 0 Q(U ) = q(U )/ǫ with q(U ) ∈ Rr consisting of r linearly independent functions of U and ǫ a small positive parameter. Accordingly, we rewrite (1) as d u fj (U ) 0 . (3) + = v t j=1 gj (U ) x q(u, v)/ǫ j

For such small parameter problems, a main interest is to investigate the limit as ǫ goes to zero, so-called zero relaxation limit or relaxation limit. To bring out the issues, we assume that (∗) There exists T > 0, independent of ǫ, such that, for each ǫ > 0, (3) with appropriate initial and boundary conditions has a unique solution U ǫ = U ǫ (x, t) defined for (x, t) ∈ Ω × [0, T ); U ǫ lies in a bounded subset 0 0 ǫ of L∞ loc (Ω × [0, T )); and there is a U = U (x, t) such that U converges to U 0 almost everywhere as ǫ goes to zero. Under this assumption, it is easy to verify that U 0 = (u0 , v 0 ) satisfies q(u0 , v 0 ) = 0, u0t +

d

j=1

fj (u0 , v 0 )xj = 0

(4)

in D′ (Ω × (0, T )) (the sense of distribution). Here q and Fj are assumed to be continuous. At this point, two fundamental questions arise. The first one is Q1. What conditions ensure the above assumption (∗)? Clearly, a necessary condition for (∗) to hold is that the so-called equilibrium manifold is not empty:

On Hyperbolic Relaxation Problems

497

E := {U ∈ G : Q(U ) = 0} = ∅. Further conditions will be presented in this report. Note that a complete answer to Q1 seems not available due to our limited knowledge of (multidimensional hyperbolic) conservation laws [Da00, Se00]. For the second question, we notice that (4) is valid only in the interior of Ω × (0, T ). Thus, if Ω has a non-empty boundary, then (4) together with initial data alone cannot determine U 0 , in general. In order to have a unique solution, (4) should be supplemented with an appropriate boundary condition. Q2. What is such a boundary condition (for (4) to be well posed)? Because U 0 is uniquely determined by (3) together with its initial and boundary conditions, the desired boundary condition should be induced from (3) and its boundary condition. In this report, we review our contributions from [Yo99]–[YZ00] in answering the above two questions. For Q1, we present a relaxation criterion, two stability conditions and an entropy dissipation condition for IVPs (initial value problems), and a generalized Kreiss condition for IBVPs (initial-boundary value problems). For Q2, the reduced boundary condition is derived for those satisfying the generalized Kreiss condition as well as the relaxation criterion. We remark that our entropy dissipation condition is essentially different from that introduced in [CLL94]. For balance laws, such an entropy dissipation condition is the same in spirit as the H-theorem for the Boltzmann equation [Ce88] and as the Lax entropy condition for conservation laws [Go61, FL71]. Moreover, this report contains the following results, which are consequences of the above criterion and conditions: (a) non-existence of (linearly stable) relaxation approximations to non-strongly hyperbolic systems of equations, (b) existence of relaxation shock profiles (structures), (c) existence of global smooth solutions for hyperbolic balance laws. Note that (a) prevents one from trying in vain to solve non-strongly hyperbolic conservation laws (e.g. conservation laws of mixed type and multidimensional Hamilton-Jacobi equations) with relaxation schemes. The shock structure problem (b) is proposed in [MR98]. This report is organized as follows. In Section 2 we consider linearized problems with constant coefficients and present the relaxation criterion as a necessary condition for the existence of controllable relaxation limits. This criterion implies (a). Section 3 records a few basic results for IBVPs. Here a previously proposed condition in [Yo92] is proved rigorously, for the first time, to be necessary for the existence of relaxation limit. In Section 4 we present the stability conditions and a systematic result on the zero relaxation limit for IVPs of nonlinear systems with smooth initial

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data. This section also contains a convergence-stability lemma for general hyperbolic singular limit problems. This lemma leads to a simple and efficient approach in studying such limit problems. Shock structure problems are discussed in Section 5. Section 6 contains the entropy dissipation condition and a general global existence theorem for IVPs of a class of hyperbolic balance laws satisfying the Kawashima condition [SK85].

2 On relaxation approximations As is pointed out in Introduction, conservation laws in (4) can be formally derived from balance laws (3) in the zero relaxation limit. This fact hints a possible approach to solve the former. Namely, instead of solving a given system of conservation laws, one may construct and then solve a system of balance laws (3) which approximates the given conservation laws when ǫ is small. Of course, it is expected that the balance laws can be solved easily. This idea was first implemented by Jin and Xin in [JX95] for hyperbolic conservation laws. Subsequent developments can be found in [CP98, AN00, Bo99]. For one-dimensional system ut + f (u)x = 0,

(5)

Jin and Xin introduced a semilinear system of balance laws (relaxation system): ut + vx = 0, (6) vt + aux = f (u)−v ǫ with v a new variable. The constant a is chosen as follows: Applying the Chapman-Enskog expansion [Li87, CLL94] to (6), we derive the following approximate system ut + f (u)x = ǫ(aux − fu (u)f (u)x )x ,

(7)

where fu (u) is the Jacobian matrix of the flux function f . This approximate system governs the first-order behavior of the relaxation system (6). By requiring that (7) be (strongly) dissipative or parabolic, a is chosen so large that the matrix [fu2 (u) − aIn ] is stable for all u under consideration, that is, a > fu (u)2 .

(8)

Notice that fu (u) can be diagonalized over R, for (5) is hyperbolic. When (5) is a scalar equation, the above choice of a ensures exactly that the relaxation system (6) satisfies the well-known subcharacteristic condition [Li87]. Remark that the approximate system (7) can be (strongly) dissipative even if the system of conservation laws (5) is not hyperbolic. On the other hand, the

On Hyperbolic Relaxation Problems

499

relaxation system (7) is strongly hyperbolic so long as a > 0. Thus, it seems that we have found a promising approach to (numerically or analytically) attack non-strongly or weakly hyperbolic systems—a challenging problem. Can the long-standing challenging problem be solved so easily? Our answer is “No”. Precisely, we will show in this section that there is no linearly stable relaxation approximations for non-strongly hyperbolic systems. Note that our answer also indicates that the above method in choosing a is not reliable. To this end, we consider linear systems with constant coefficients: Ut +

d

Aj Uxj = BU/ǫ,

(9)

j=1

which can be viewed as a linearization of the quasilinear system (3) about a constant state Ue ∈ E. Here B is the Jacobian matrix QU (Ue ) of Q evaluated at the constant state. Concerning this linearized system, we assume that its IVP is well-posed for each fixed ǫ > 0. It is well known that this assumption is equivalent to that the linearized system is (strongly) hyperbolic (see, e.g. [Se00]): hyperbolicity: there is a positive constant C such that # , -# # exp Hr (0, ξ) # ≤ C for all ξ = (ξ1 , ξ2 , · · · , ξd ) ∈ Rd .

Here and below | · | denotes the L2 -norm for matrices or vectors and Hr (η, ξ) is defined as ξj Aj Hr (η, ξ) = ηB + i j

for η ≥ 0 and ξ ∈ Rd . Furthermore, considering x-independent solutions for (9): U ǫ (x, t) = exp(tB/ǫ)U ǫ (0, 0), we see two essentially different limiting behaviors depending on whether or not B has non-zero purely imaginary eigenvalues. Here we restrict ourselves to the simple case where non-oscillation: B has no non-zero purely imaginary eigenvalues. This and the hyperbolicity assumption are clearly satisfied by the relaxation system (6). In [Yo02] (see also [Yo92, LS97]), we proved the following fact, which is analogous to the necessity part of the Lax equivalence theorem [LR56].

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Theorem 2.1 Under the hyperbolicity assumption, if # , -# sup # exp Hr (η, ξ) # = +∞, η≥0,ξ∈Rd

then, for any t > 0, there exists U0 ∈ L2 ≡ L2 (Rd ) such that the unique global solution U ǫ (x, t) to (9) with initial data U0 satisfies lim sup U ǫ (·, t)L2 = +∞. ǫ→0

Notice that the existence of unique global solutions with initial data in L2 is well known (see [Se00]). This theorem indicates a necessary condition for the linearized system to have a correctly behaved zero relaxation limit and, hence, would seem to be necessary also for the nonlinear system (3) with initial data near the constant state. Let us identify the following for (2): stability criterion: there is C(U ) > 0 such that # & '# # # ξj Aj (U ) # ≤ C(U ) # exp ηQU (U ) + i j

for all η ≥ 0 with ηQ(U ) = 0 and for all ξ ∈ Rd .

Note that this criterion is formally slightly stronger than the hyperbolicity and reduces to that if η = 0. In view of Theorem 2.1, it is natural to restrict our discussion to the systems satisfying the stability criterion. Thus, it is easy to verify (see [Yo92]) that B can be block-diagonalized as B = P −1 diag(0, S)P with S invertible. Moreover, P and S can be real if so is B. Hereafter we will always assume that, for the linear system (9), B is already in the block-diagonal form B = diag(0, S) with S an invertible r × r-matrix. Corresponding to this partition, we often write an n × n-matrix A or/and n-vector V as I 11 12 V u A A , V = = A= v A21 A22 V II with the same partition as that of B. With these notation, the linear system (9) can be rewritten as ! 11 ut + Aj uxj + A12 j vxj = 0, j

vt +

j

! 22 A21 j uxj + Aj vxj = Sv/ǫ.

Note that S is stable if the non-oscillation assumption holds.

(10)

On Hyperbolic Relaxation Problems

501

Suppose the solution U ǫ (x, t) of this linear system is convergent in the sense of distribution as ǫ goes to zero. It is immediate to see that the limit V (x, t) satisfies the so-called reduced or equilibrium system I VtI + A11 j Vxj = 0, j (11) V II = 0. Concerning this reduced system, we proved in [Yo02] the following result (see [Yo92, LS97] for other proofs): Theorem 2.2 Assume the linearized system (10) admits the stability criterion and the non-oscillation assumption. Then the reduced system in (11) is (strongly) hyperbolic. From this theorem, we see the non-existence of linearly stable hyperbolic relaxation approximations for non-strongly hyperbolic systems. See [Yo01a] for further discussions about this non-existence result. We conclude this section with the following result, which was first proved in [LS97] for one-dimensional problems. Theorem 2.3 Assume the linear system (10) admits the stability criterion and the non-oscillation assumption. Then, as ǫ goes to zero, the solution U ǫ to (10) with initial data U0 ∈ L2 converges in L2 for each t > 0. This theorem demonstrates the sufficiency of the stability criterion and the non-oscillation assumption for linear problems, while the necessity of the stability criterion is displayed in Theorem 2.1. Because of the last two theorems, we identify our relaxation criterion as the combination of the stability criterion and the non-oscillation assumption. Note that the non-oscillation assumption is independent of and consistent with the stability criterion, for the latter implies that B has no eigenvalues with positive real parts and does not imply that the real parts are negative. In [Yo01, Yo02], it was shown that the relaxation criterion is satisfied by the relaxation system (6) with (8) and is equivalent to the subcharateristic condition [Wh74] for one-dimensional 2 × 2 with r = 1.

3 Initial-boundary value problems In this section, we review a few basic results from [Yo92, Yo99, Yo01] for IBVPs of the relaxation system (9) or (10) in the half-space x ≡ (x1 , x ˆ) ∈ [0, +∞) × Rd−1 , together with boundary conditions of the form:

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BU (ˆ x, t)U (0, x ˆ, t) = b(ˆ x, t).

(12)

Here BU (ˆ x, t) is a given p × n-matrix, b(ˆ x, t) is a given p-vector, and p is the number of positive eigenvalues of the coefficient matrix A1 in (9). Additionally, we present a rigorous proof of the necessity of a structural condition, previously proposed in [Yo92, Yo99], for the existence of controllable relaxation limits. For IVPs, the sufficiency of the relaxation criterion has been shown in Theorem 2.3. The question here is whether or not the same criterion is also sufficient for IBVPs which are well-posed (for each fixed ǫ) in the sense of Kreiss [Kr70]. If not, what are the additional requirements? Furthermore, we have the question Q2: what is the reduced boundary condition? For Q2, we notice that the equilibrium system (11) may have completely different characteristics from the relaxation system (10). To answer these questions, we assume that (10) satisfies the relaxation criterion and the following three conditions: (a) the boundary x1 = 0 is non-characteristic for both the relaxation system and the equilibrium system, (b) A1−1 B has no non-zero purely imaginary eigenvalues, (c) the IBVP (10) with (12) satisfies the uniform Kreiss condition (see (16) below). Note that (b) is just the non-oscillation assumption in the x1 -direction, instead of the t-direction. It was shown in [Yo01] that (b) follows from (a) for a class of physically relevant relaxation systems. Moreover, (b) is obviously true if r = 1, A1 is real and ReS = 0. To state our results, we introduce ˆ ξ0 ) = i H = H(ξ,

d j=2

ξj Aj − ξ0 In

for complex number ξ0 and ξˆ = (ξ2 , ξ3 , · · · , ξd ) ∈ Rd−1 , and denote by p1 the number of stable eigenvalues of A−1 1 B. Here In denotes the unit matrix of order n. Note that H = −ξ0 In when d = 1. The first result is concerning the number of positive eigenvalues of the equilibrium system (11). Theorem 3.1 Assume the relaxation criterion, (a) and (b) hold. Then p1 ≤ p −1 11 ˆ ξ0 ) with and (A11 H has precisely (p − p1 ) stable eigenvalues for each (ξ, 1 ) Reξ0 > 0. The proof of this theorem can be found in [Yo92, Yo99] and needs ˆ ξ0 ) = A−1 (ηS + H) has Lemma 3.2 For Reξ0 > 0, the matrix M ≡ M (η, ξ, 1 p stable eigenvalues and (n − p) unstable eigenvalues.

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Taking ξˆ = 0 and ξ0 = 1 in Theorem 3.1, we have H 11 = −In−r and −1 −1 11 . On the other hand, by Theorem 2.2 A11 H = −(A11 therefore (A11 1 1 ) 1 ) has only real eigenvalues. Thus, we obtain Corollary 3.3 The conditions of Theorem 3.1 imply that A11 1 has precisely (p − p1 ) positive eigenvalues. This corollary has the following interesting consequence, which shows an important relation of characteristics for relaxation systems and their equilibrium ones. Theorem 3.4 Assume the relaxation criterion. Let Λ1 ≤ Λ2 ≤ · · · ≤ Λn and λ1 ≤ λ2 ≤ · · · ≤ λn−r be the eigenvalues of j ξj Aj and j ξj A11 j , respectively. If r = 1 or if r > 1 and ( j ξj Aj − aIn )−1 B has no non-zero purely imaginary eigenvalues for any real a∈ / {Λ1 , Λ2 , · · · , Λn , λ1 , λ2 , · · · , λn−r }, then the interlaced relation λk ∈ [Λk , Λk+r ] holds for k = 1, 2, · · · , n − r.

Let us mention that a similar result was first derived in [CLL94] under a certain entropy dissipation condition, instead of the relaxation criterion. The technical assumption that ( j ξj Aj − aIn )−1 B has no non-zero purely imaginary eigenvalues was shown in [Yo01] to hold for a class of physically relevant relaxation systems. For further results, we introduce Definition 3.1 Let n × n-matrix A have precisely k(0 ≤ k ≤ n) stable eigenS is called a right S-matrix of A if values. A full-rank n × k-matrix RA S S ARA = RA S− , U where S− is a k × k stable matrix. Similarly, we define the right U-matrix RA S U (unstable), left S-matrix LA and left U-matrix LA . S U Note that the existence of these RA , RA , LSA and LU A can be deduced from ˜ S can be expressed the Jordan canonical form theorem. Any right S-matrix R A S S ˜ = R S0 with S0 an invertible k × k-matrix. Theorem 3.1 and Lemma as R A A −1 11 ˆ ξ0 ) are of order H and M (η, ξ, 3.2 indicate that right S-matrices of (A11 1 ) ˆ (n − r) × (p − p1 ) and n × p for each (ξ, ξ0 ) with Reξ0 > 0, respectively. Next, we consider the following homogeneous system with constant coefficients: d Aj Uxj = BU/ǫ, Ut + j=1 (13)

BU U (0, x ˆ, t) = 0.

The following was proved in [Yo92, Yo99].

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S ˆ ξ0 ) be a right S-matrix of M (1, ξ, ˆ ξ0 ). If there exists Lemma 3.5 Let RM (1, ξ, S ˆ ξ0 with Reξ0 > 0 such that the p×p-matrix BU RM (1, ξ, ξ0 ) is singular, then the problem (13) with a bounded initial value admits an exponentially increasing solution for t > 0 as ǫ goes to zero.

Moreover, it was shown in [Yo99] that the relaxation criterion and the uniform S ˆ ξ0 ). Kreiss condition [Hi86] do not ensure the invertibility of BU RM (1, ξ, S ˆ ξ0 ) From Lemma 3.5, one could infer that the invertibility of BU RM (1, ξ, for any ξ0 with Reξ0 > 0 is necessary for the existence of controllable relaxation limit. However, the author did not claim the necessity, because the proof in [Yo92, Yo99] does not support such an inference. In fact, the initial value of the solution found in the proof is not independent of ǫ (see below)! The same argument was adopted in subsequent literature [XX00]–[Xu04] but the gap has not been filled yet. Here we will rigorously prove the necessity for one-dimensional problems. For this purpose, we firstly repeat the proof of Lemma 3.5 with d = 1. Proof of Lemma 3.5. Consider the following problem associated with (13): dφ dσ

= M (1, ξ0 )φ,

(σ > 0)

(14)

BU φ(0) = 0. By Lemma 3.2, it is easy to see that, for ξ0 with Reξ0 > 0, any solution φ(σ) to the first equation in (14), which is not exponentially increasing as σ goes to infinity, can be expressed as S φ(σ) = RM (1, ξ0 )ψ(σ).

Here ψ(σ) is a bounded p-vector function of σ and is determined by its initial value ψ(0). S Since BU RM (1, ξ0 ) is singular, there is a non-zero p-vector ψ(0) such that S BU RM (1, ξ0 )ψ(0) = 0.

Thus, the problem (14) has a non-zero bounded solution φ(σ). Having φ(σ), we can directly verify that & ' ξ0 t x ǫ (15) U ≡ exp φ ǫ ǫ solves (13) with a bounded initial value. This completes the proof. Now we are in a position to strengthen Lemma 3.5 as follows. Theorem 3.6 Assume (13) satisfies the Kreiss condition. If there exists ξ0 S with Reξ0 > 0 such that the p × p-matrix BU RM (1, ξ0 ) is singular, then for

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each t > 0 there is a U = U (x) ∈ H 1 (0, ∞), satisfying BU U (0) = 0, such that the unique solution U ǫ (x, t) to (13) with U as its initial value satisfies lim U ǫ (·, t)H 1 = ∞.

ǫ→0

Proof. Note that 5 6 := U = U (x) ∈ H 1 (0, ∞) : BU U (0) = 0

is a closed subspace of H 1 (0, ∞). Because (13) satisfies the Kreiss condition, for each U ∈ it has a unique solution U ǫ = U ǫ (x, t) ∈ C(0, ∞; ) with U as its initial value (see [LY01]). Moreover, there is a positive constant C = C(ǫ, t) such that U ǫ (·, t)H 1 ≤ C(ǫ, t)U H 1 . Thus we have a family of linear continuous operators Tǫ,t : −→ , defined as Tǫ,t U = U ǫ (·, t). Below we fix t > 0 and consider Tk := T1/k,t for k = 1, 2, · · · . In view of the solution given in (15), for each k there is a φk = φ(kx1 ) ∈ such that Tk φk H 1 = ektReξ0 φk H 1 . Thus, we have Tk ≥ ektReξ0 −→ ∞ as k goes to infinite. Note that is a Banach space. By the uniform boundedness principle, there is a U ∈ such that lim U ǫ (·, t)H 1 ≥ lim U 1/k (·, t)H 1 = lim Tk U H 1 = ∞.

ǫ→0

k→∞

k→∞

This completes the proof. Because of Theorem 3.6, the structural condition S det{BU RM (1, ξ0 )} = 0,

∀ξ0 with Reξ0 > 0,

is necessary for the existence of zero relaxation limit. Because the classical Kreiss condition can be formulated as S (0, ξ0 )} = 0, det{BU RM

∀ξ0 with Reξ0 > 0,

(16)

S and RM (η, ξ0 ) is homogeneous of order 0 with respect to (η, ξ0 ), we conclude that a necessary condition for the existence of zero relaxation limit is S det{BU RM (η, ξ0 )} = 0

∀η ≥ 0 and ∀ξ0 with Reξ0 > 0.

(17)

It is remarkable that, for IBVPs, this structural condition is as fundamental as the relaxation criterion for IVPs. The above necessary condition motivated us to propose the following new requirement in [Yo99].

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Generalized Kreiss Condition: there exists cK > 0 such that = S ˆ ξ0 )]| ≥ cK det[RS∗ (η, ξ, ˆ ξ0 )RS (η, ξ, ˆ ξ0 )] (η, ξ, | det[BU RM M M

for all η ≥ 0, ξˆ ∈ Rd−1 and ξ0 with Reξ0 > 0. Here the superscript * denotes the transpose.

Note that this generalized Kreiss condition does not depend on the special S ˆ ξ0 ). With η = 0, we recover the uniform Kreiss condition (η, ξ, choice of RM [Hi86] for IBVPs of first-order hyperbolic systems. Remark 3.1 Unlike the uniform Kreiss condition, the generalized one involves parameters η, ξˆ and ξ0 even for one-dimensional relaxation problems. Despite this, we expect more explicit expressions of this generalized Kreiss condition for special systems. See [XX00]–[Xu04] for examples. Finally, we turn to question Q2. Because of Corollary 3.3 and the classical theory for IBVPs, (p − p1 ) independent relations of boundary data should be imposed at the boundary x1 = 0 to solve the reduced system in (11) in the half-space. Our next result, also proved in [Yo99], will answer how to derive the required (p − p1 ) boundary conditions from the given relaxation problem in (10) and (12). Theorem 3.7 Assume the relaxation criterion, (a), (b), and the generalized Kreiss condition hold. Then there exists a (p − p1 ) × p-matrix Bp , unique up to an invertible (p − p1 ) × (p − p1 )-matrix multiplying Bp from right, such that = | det[Bp Bu R1S ]| ≥ c˜K det[R1S∗ R1S ] −1 11 with c˜K a positive constant and for any right S-matrix R1S of (A11 H , 1 ) in which ξ0 has positive real part. Here the partition BU = [Bu , Bv ] has been used.

According to Kreiss’ theory [Kr70, Hi86] for IBVPs of first-order hyperbolic systems, the equilibrium system in (11) together with the following reduced boundary condition x, t)Bu (ˆ x, t)u(0, x ˆ, t) = Bp (ˆ x, t)b(ˆ x, t) Bp (ˆ can constitute a well-posed problem. In [Yo99], this reduced boundary condition was also derived through formal asymptotic expansions.

4 Stability conditions and relaxation limits For IVPs of linear systems with constant coefficients, the question Q1 has been answered quite completely in Section 2. From this section on we consider physically relevant nonlinear systems.

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In [Yo92, Yo01], it was shown that many equations of classical physics of the form (2) admit the following structure alternatively called the second stability condition: (i.) there is an invertible n × n matrix P (U ) and a stable r × r(0 < r ≤ n) matrix S(U ), defined on the equilibrium manifold E, such that 0 0 P (U )QU (U ) = P (U ) for U ∈ E. 0 S(U ) (this is the non-oscillation assumption); (ii.) as a system of first-order PDEs, (2) is symmetrizable hyperbolic, that is, there is a positive definite Hermitian matrix A0 (U ) such that A0 (U )Aj (U ) = A∗j (U )A0 (U )

for all j

and U ∈ G;

(iii.) the hyperbolic part and the source term are coupled in the following fashion 0 0 P (U ) for U ∈ E. A0 (U )QU (U ) + Q∗U (U )A0 (U ) ≤ −P ∗ (U ) 0 Ir This stability condition was shown in [Yo01] to be stronger than the relaxation criterion. Moreover, it implies the condition (b) in the previous section and the technical assumption in Theorem 3.4. Indeed, in [Yo99] we proved Proposition 4.1 Assume the second stability condition. If Aj (U ) is invertible, then A−1 j (U )QU (U ) has no non-zero purely imaginary eigenvalues. Next we show the sufficiency of the above stability condition in justifying the zero relaxation limit for quasilinear problems. Such a task usually has two aspects: existence of solutions in an ǫ-independent time interval and convergence to solutions of reduced problems as ǫ goes to zero. To begin with, we recall the convergence-stability lemma [Yo01] for general limit problems of IVPs for quasilinear symmetrizable hyperbolic systems depending (singularly) on parameters: Ut +

d

j=1

Aj (U, ǫ)Uxj = Q(U, ǫ),

(18)

¯ (x, ǫ). U (x, 0) = U

Here ǫ represents a parameter in a topological space, Aj (U, ǫ)(j = 1, 2, · · · , d) ¯ (x, ǫ) is a and Q(U, ǫ) are sufficiently smooth functions of U ∈ G ⊂ Rn , and U ¯ (x, ǫ) is periodic given initial value function. For simplicity, we assume that U with respect to x ∈ Rd . Denote by H k the usual L2 -Sobolev space of order k (a non-negative integer) on the d-dimensional torus, its norm is · k , and C(J, H k ) denotes the space of continuous functions on the interval J with values in H k .

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¯ (x, ǫ) ∈ G0 for all (x, ǫ) Let G0 ⊂⊂ G be convex and open. Assume U ¯ (·, ǫ) ∈ H s with s > d/2 + 1 an integer. Fix ǫ. According to the local and U existence theory for IVPs of symmetrizable hyperbolic systems (see [Ma84]), there is a time interval [0, T ] so that (18) has a unique H s -solution U ǫ ∈ C([0, T ], H s ). Define

" $ Tǫ = sup T > 0 : U ǫ ∈ C([0, T ], H s ) .

(19)

Namely, [0, Tǫ ) is the maximal time interval of H s existence. Note that Tǫ depends on G and may tend to zero as ǫ goes to a certain singular point, say 0. In order to show that limǫ→0 Tǫ > 0, which means the stability (see [KM82, Ma84]), we make the following convergence assumption: there exists T∗ > 0 and Uǫ ∈ L∞ ([0, T∗ ], H s ) for each ǫ, satisfying 8" $ Uǫ (x, t) ⊂⊂ G, x,t,ǫ

such that for t ∈ [0, min{T∗ , Tǫ }),

sup |U ǫ (x, t) − Uǫ (x, t)| = o(1), x,t

sup U ǫ (·, t) − Uǫ (·, t)s = O(1) t

as ǫ tends to the singular point. With such a convergence assumption, we are in a position to state the following fact established in [Yo01]. ¯ (x, ǫ) ∈ G0 ⊂⊂ G for all (x, ǫ), U ¯ (·, ǫ) ∈ H s with Lemma 4.2 Suppose U an integer s > d/2 + 1, and the convergence assumption holds. Let [0, Tǫ ) be the maximal time interval such that (18) has a unique H s -solution U ǫ ∈ C([0, Tǫ ), H s ). Then Tǫ > T∗ for all ǫ in a neighborhood of the singular point. Thanks to Lemma 4.2, our task is reduced to find a Uǫ (x, t) such that the convergence assumption holds. Below, we will use this lemma with G replaced by its compact subsets. It is remarkable that similar lemmas can be formulated for other evolution equations. For IVPs of (2) satisfying the above stability condition, such a Uǫ (x, t) was constructed in [Yo92, Yo99a] with the classical matched expansion method. The construction additionally needs the following two technical assumptions.

On Hyperbolic Relaxation Problems

509

(a) the limiting inner problem (x is a parameter here) dI˜ ˜ = Q(I) dτ

˜ 0) = U ¯ (x, 0) with I(x,

has a unique solution I˜ ∈ C([0, +∞), H s+3 ), with s > d/2 + 1 an integer, which takes values in a convex compact subset of G and decays exponen˜ +∞) in H s+3 (Ω) as τ goes to infinity; tially to a function I(x, (b) the equilibrium manifold can be expressed as $ " E = U = E(u) : u ∈ U ⊂ Rn−r , where U is open and E is a smooth diffeomorphism from U to E.

Note that (a) relies on the non-oscillation assumption, while (b) is convenient and for many examples also appropriate. With the above technical assumptions, we seek formal asymptotic approximations, to the initial-layer solution U ǫ of (2), of the form Uǫm (x, t) =

m

ǫk Uk (x, t) +

m

ǫk Ik (x, t/ǫ)

(20)

k=0

k=0

with 1 ≤ m ≤ s − 1 − [d/2]. By our construction, the leading term I0 (x, t/ǫ) of the initial-layer correction (the second sum) is ˜ t/ǫ) − I(x, ˜ +∞) I0 (x, t/ǫ) = I(x,

(21)

and the leading term U0 of the outer expansion (the first sum) satisfies the initial condition ˜ +∞). U0 (x, 0) = I(x, Moreover, U0 solves the reduced problem Q(U0 ) = 0 or U0 = E(u), ' & d Aj (U0 )U0xj = 0, P I (U0 ) U0t +

(22)

j=1

˜ +∞). U0 (x, 0) = I(x,

Here P I (U0 ) = P I (resp. P II ) denotes the (n − r) × n (resp. r × n) matrix consisting of the first (n − r) (resp. last r) rows of P (U0 ). The system of equations in (22) is the equilibrium system. Under the second stability condition, we use energy methods to prove Theorem 4.3 Suppose the system (2) satisfies the second stability condition, ¯ =U ¯ (x) satisfies U ¯ (x) ∈ G0 ⊂⊂ G for all x and U ¯ ∈ Hs the initial value U with an integer s > d/2 + 1, and the assumptions (a) and (b) concerning the limiting inner problem and the equilibrium manifold hold. Then there are two

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positive constants T∗ and K, independent , of ǫ, such- that all terms Uk , Ik (k = 0, 1, · · · , m) in (20) can be found in C [0, T∗ ], H s+1 , Uǫm (x, t) takes values in a convex compact subset of G, the H s -solution U ǫ exists in the uniform time interval [0, T∗ ], and ||U ǫ (t) − Uǫm (t)||s ≤ Kǫm+1/2 for ǫ sufficiently small and t ∈ [0, T∗ ]. This theorem is only for the simple case, where the coefficients in (2) are independent of (x, t, ǫ), the initial data are periodic and independent of ǫ, and the second stability condition holds. The interested reader may consult [Yo92] for the more general cases, where the second stability condition is replaced with the first one. The latter consists of (i), (ii) (of the second stability condition) and (iii)’. the hyperbolic part and the source term are coupled in the following fashion A0 (U )QU (U ) + Q∗U (U )A0 (U ) ≤ 0

for U ∈ E.

A direct consequence of Theorem 4.3 is U ǫ (x, t) =

m

ǫk Uk (x, t) +

k=0

m

ǫk Ik (x, t/ǫ) + O(ǫm+1/2 )

(23)

k=0

for x ∈ Rd and t ∈ [0, T∗ ]. In particular, it follows from (23) and (21) that, out of the initial-layer, the solution U ǫ of the original problem (2) converges to the unique smooth solution U0 of the reduced problem (22) as ǫ goes to zero. The proof of Theorem 4.3 can be found in [Yo92, Yo99a]. Let us remark that our approach is valid only for smooth solutions but works also for other singular limit problems (see [LaY01, JY03, Yo04]). One advantage of this approach is that it allows us to characterize the exact limiting behaviors and to obtain the convergence rate, as shown by the expansion (23).

5 Shock structure problems This section is devoted to the existence of traveling wave solutions to the relaxation system (3) satisfying the second stability condition. We assume that q(u, v) = 0 uniquely determines v in term of u, say v = h(u). Thus, the equilibrium manifold can be expressed with u E(u) = h(u)

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and the equilibrium system in (22) becomes a closed system of conservation laws d ut + (24) fj (u, h(u))xj = 0. j=1

Note that P (U ) in the stability condition can be simply taken to be 0 In−r P (U ) = . qv−1 qu (U ) Ir For conservation laws (24), the simplest discontinuous solutions are of the form 0 u− , if ξ · x < s∗ t; u(x, t) = (25) u+ , otherwise. Here the quantities u± , s∗ and ξ = (ξ1 , ξ2 , · · · , ξn ) are given and satisfy the Rankine-Hugoniot relation s∗ (u+ − u− ) =

d j=1

ξj [fj (u+ , h(u+ )) − fj (u− , h(u− ))].

Suppose the characteristic matrix j ξj fj (u, h(u))u of (24) has an isolated eigenvalue λk (u). The piecewise constant weak solution (25) is called a k-th shock-front if it satisfies Liu’s strict entropy condition [Li76]: s∗ = sk (ρ+ ) < sk (ρ)

(26)

for all ρ strictly between 0 and ρ+ . Here ρ parametrizes the k-th Hugoniot curve u(ρ) = u(ρ, u− ) passing through u− = u(0), u+ = u(ρ+ ), and sk (ρ) is uniquely determined by sk (0) = λk (u− ) and the Rankine-Hugoniot relation sk (ρ)(u(ρ) − u− ) =

d j=1

ξj [fj (u(ρ), h(u(ρ))) − fj (u− , h(u− ))].

The question here is when the relaxation system (3) has a smooth traveling wave solution (uǫ , v ǫ )(x, t) = Φ((ξ ·x−s∗ t)/ǫ) converging to (u, h(u)) as ǫ goes to zero, where u is the discontinuous solution (25). Set ω = (ξ · x − s∗ t)/ǫ. Such a Φ(ω) should satisfy −s∗ Φω +

2 d

j=1

3 ξj Fj (Φ) = Q(Φ), ω

(27)

Φ(±∞) = E(u± )

for ω ∈ (−∞, ∞). Such a traveling wave solution is called a relaxation shock profile and its existence is of independent interest physically (see

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[BR98, MR98]). We recall that the analogous question was asked by Gelfand for viscosity approximations of hyperbolic conservation laws and was satisfactorily answered by Majda and Pego in [MP85]. For the case where r = 1, (27) can be directly reduced to a one-dimensional connection problem, for which a general existence theorem is well known. With n = 2, this case was considered by Liu in [Li87] and it was shown there that the existence follows from the well-known subcharacteristic condition imposing on (3). As shown in [Yo01], this subcharacteristic condition has various generalizations for (3) with general n and r. Our interest is in the general case: r ≥ 1. The result is Theorem 5.1 Assume (3) satisfies the second stability condition, and λk (u− ) with u− fixed is an isolated eigenvalue for the equilibrium system (24) but not an eigenvalue for the relaxation system (3). Then the following three statements are true: (a) if u+ = u− and s∗ = sk (ρ) for some ρ strictly between 0 and ρ+ , then there is no smooth profile Φ(ω) of (27) close to E(u− ); (b) if s∗ < sk (ρ) for all ρ strictly between 0 and ρ+ , then there exist δ1 > 0 and δ2 > 0 so that, for any u+ satisfying |u+ − u− | < δ1 and lying in the k-th Hugoniot curve for u− with some speed s∗ , a unique smooth trajectory Φ(ω) of (27) satisfying |Φ(ω) − E(u− )| < δ2 exists connecting E(u− ) from left to E(u+ ) at the right; (c) if s∗ > sk (ρ), then there exist δ1 > 0 and δ2 > 0 so that, for any u+ satisfying |u+ − u− | < δ1 and lying in the k-th Hugoniot curve for u− with some speed s∗ , a unique smooth trajectory Φ(ω) of (27) satisfying |Φ(ω) − E(u− )| < δ2 exists connecting E(u− ) from right to E(u+ ) at the left. Moreover, if s∗ = λk (u± ) then Φ(ω) approaches E(u± ) exponentially as ω → ±∞.

The proof of Theorem 5.1 was motivated by the work of Majda and Pego [MP85]. Indeed, when r > 1, (27) is a genuinely multi-dimensional connection problem, for which we do not know a general existence theorem. As in [MP85] for viscosity problems, our strategy is to use the standard technique in dynamic bifurcation theory to reduce (27) to a one-dimensional problem. The possibility of this reduction is guaranteed by the preconditions in Theorem 5.1. Note that (3) is structurally different from the viscosity approximations and thus needs a delicate analysis of its algebraic structure. Moreover, our proof involves a new parametrization of Hugoniot curves. The details can be found in [YZ00].

6 Entropy and global solutions This section is based on [Yo04a]. Notice that, without the source term Q(U ), (3) reduces to a system of conservation laws. In that case, it is well known

On Hyperbolic Relaxation Problems

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that solutions generally develop singularities (e.g. shock waves) in finite time, even when the initial data are smooth and small [Da00]. In [Yo04a], we presented a set of structural conditions under which source terms may prevent the breakdown of smooth solutions. Namely, we developed a existence theory of global smooth solutions for balance laws (3). Let Ue ∈ E. The structural conditions read as follows. (i.) The Jacobian qv (Ue ) is invertible. (ii.) There is a strictly convex smooth function η(U ), defined in a convex compact neighborhood G of Ue , such that ηU U (U )FjU (U ) is symmetric for all U ∈ G and all j. (iii.) There is a positive constant cG such that for all U ∈ G, [ηU (U ) − ηU (Ue )]Q(U ) ≤ −cG |Q(U )|2 . , (iv.) The kernel of the Jacobian QU (Ue ), Ker QU (Ue ) , contains no eigenvector of the matrix j ωj FjU (Ue ) for any ω = (ω1 , ω2 , · · · , ωd ) ∈ Sd−1 (the unit sphere in Rd ). Here and below, the subscripts v, U, · · · denote the corresponding partial derivatives with respect to these variables, so ηU U (U ) is the Hessian matrix of η(U ). About these conditions, we make several comments. Together, (i)–(iii) constitute an entropy dissipation condition for balance laws (3). Alternative conditions in the same spirit were previously introduced by Chen, Levermore and Liu [CLL94], M¨ uller, Ruggeri and Boillat [MR98, BR98], and Yong [Yo01], with the aim of studying the zero relaxation limit. However, these conditions do not seem to provide the proper setting for proving global existence. See [Ze99] for a similar comment on the Chen-Levermore-Liu condition. Further comments need some consequences of Condition (i). Introduce u ≡ V (U ) (28) V = q(U ) and set Ve = V (Ue ). Since qv (Ue ) is invertible, V = V (U ) has an inverse U = U (V ) for U close to Ue . In particular, v can be viewed as a function v(u, q) of u and q. Set Aj (V ) = UV−1 (V )FjU (U (V ))UV (V ). Then, for smooth solutions, (3) can be rewritten as Vt +

d j=1

, Aj (V )Vxj = diag 0, qv (U ) V,

(29)

which is called the normal form of the balance law (3). Note that the transformation (28) does not destroy the conservative form of the u-equation in (3):

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ut +

d

fj (u, v(u, q))xj = 0.

j=1

The normal form (29) of (3) enables us easily to use the Chapman-Enskog expansion (or the Maxwell iteration [MR98]) to derive a first-order approximation to our system. With the partition in Section 2, we rewrite the q-equation in (29) as 2 3 q = ǫqv−1 (U (V )) qt + A21 A22 j (V )uxj + j (V )qxj , j

j

with a positive parameter 0 =1, for the present problem ǫ ≪ 1, for the zero relaxation limit problem, and then iterate once to obtain q = ǫqv−1 (U (u, 0))

2 A21 j (u, 0)uxj + O(ǫ ).

j

Substituting the truncation into the u-equation in (29), we arrive at ! fjv (U (u, 0))qv−2 (U (u, 0))A21 ut + fj (U (u, 0))xj + k (u, 0)uxk x = 0. j

j

jk

(30) Thus, we have a short and logically simple version of the classical ChapmanEnskog theory for balance laws [CLL94]. Note that no entropy function is involved here. In [Yo04a] we proved Theorem 6.1 Assume the entropy dissipation condition (i)–(iii) holds. Then UV∗ ηU U UV evaluated at U = Ue is a block-diagonal matrix (under the aforementioned partition) and there is a positive definite r × r-matrix Λ(Ue ) such that 0 0 ∗ ∗ VU (Ue ). ηU U (Ue )QU (Ue ) + QU (Ue )ηU U (Ue ) = −VU (Ue ) 0 Λ(Ue ) By this theorem, the present entropy dissipation condition implies the second stability condition. Thus, it provides a proper setting for studying the zero relaxation limit as well as the existence and stability of relaxation shock profiles [Zu01, MZ02]. Moreover, as was shown in [Yo04a, Yo01], our entropy dissipation condition is satisfied by many equations of classical physics of the form (3). Condition (iv), which is generally referred to as the “Kawashima condition”, was first formulated by Shizuta and Kawashima [SK85] for symmetric

On Hyperbolic Relaxation Problems

515

hyperbolic-parabolic systems of the form (30) with source term, in order to derive decay estimates on solutions. Unlike these authors, we do not require that ηU U (Ue )QU (Ue ) is symmetric. In [Ka84, SK85, UKS85], Kawashima et al. have verified that this condition is satisfied by many classical systems of hyperbolic-parabolic type governing fluid flow, in particular by the Navier-Stokes equations. At the same time, we have seen that such hyperbolic-parabolic systems can be derived as firstorder approximations to hyperbolic systems of balance laws, of the form (3), through the Chapman-Enskog expansion. Responding to this connection, we proved Theorem 6.2 Under the entropy dissipation condition (i)–(iii), a system of balance laws (3) satisfies the Kawashima condition if and only if its first-order approximation (30) satisfies it. Now, we turn to state the global existence result. Theorem 6.3 Let s ≥ s0 +1 ≡ [d/2]+2 be an integer and Conditions (i)–(iv) hold at Ue ∈ E. Then there are two constants c1 , c2 such that if U0 = U0 (x) ∈ Ue + H s (Rd ) satisfies U0 − Ue s ≤ c1 , then the system of balance laws (3) with U0 as its -initial value has a unique , global solution U = U (x, t) ∈ Ue + C 0, ∞; H s (Rd ) satisfying U (·, T ) − Ue 2s +

T

0

Q(U )(·, t)2s dt +

T

0

∇U (·, t)2s−1 dt ≤ c2 U0 − Ue 2s

(31)

for any T > 0. 2 d In this theorem, H s,(Rd ) denotes the - usual L -Sobolev space of order s on R , s d its norm is · s , C 0, ∞; H (R ) denotes the space of continuous functions on [0, ∞) with values in H s (Rd ), A(·, t)s is the norm of A = A(x, t) taken with respect to x while t is viewed as a parameter, and ∇ is the gradient operator with respect to x. Note that the estimate (31) implies # , -# as t → ∞, sup #∂ α U (x, t) − Ue # → 0, x∈Rd

for all multi-indices α satisfying |α| ≤ s − s0 − 1. Here we should mention that a similar existence result has been obtained independently by Hanouzet and Natalini [HN03] for one-dimensional problems. Moreover, conditions of the form (i)–(iv) have been used by Ruggeri and Serre in [RS04] to show that constant equilibrium states are timeasymptotically L2 -stable in a certain class of weak entropy solutions, again for one-dimensional problems. Theorem 6.3 is proved by the standard continuation argument together with the local-in-time existence theory for symmetrizable hyperbolic systems

516

W.-A. Yong, W. J¨ ager

(see, e.g. [Ma84]) and is based on the a priori estimate (31). This estimate is derived by the energy method. For that purpose, we employ the normal form (29) of (3), together with Theorem 6.1 expressing a basic consequence of Conditions (i)–(iii). Moreover, we apply the following skew-symmetrizer theorem, in Fourier space, due to Shizuta and Kawashima [SK85]: Theorem 6.4 (Shizuta-Kawashima, 1985) Assume (i) and (ii) hold. Condition (iv) is equivalent to the existence of a constant cS > 0 and a skewsymmetric real matrix K = K(ω) ∈ C ∞ (Sd−1 ) satisfying K(−ω) = −K(ω) and K(ω)A(ω) − A∗ (ω)K(ω) ≥ 2cS In − 2 diag(0, Ir ), for any ω ∈ Sd−1 .

The above global existence theory generalizes that due to Kawashima [Ka83] for discrete velocity Boltzmann equations and can be used to many other equations of classical physics. As a new application, we consider discrete velocity BGK models constructed in [AN00, Bo99] to approximate hyperbolic systems of conservation laws: ut +

d

gj (u)xj = 0,

(32)

j=1

where u ∈ U ⊂ Rp (a convex open set) and gj maps U to Rp . The BGK models are of the form fkt + a(k) · ∇x fk = Mk (u) − fk for k ∈ Ξ := {1, 2, · · · , N }. Here fk = fk (x, t) ∈ Rp is unknown, a(k) ∈ Rd is a constant vector with components aj (k) (j = 1, 2, · · · , d), a(k) · ∇x = d N p k=1 fk , and Mk : U → R satisfies the consistency j=1 aj (k)∂xj , u = relations N

k=1

Mk (u) = u,

N

aj (k)Mk (u) = gj (u),

k=1

∀u ∈ U.

(33)

Clearly, a BGK model is determined by choosing the a(k)’s and Mk (u)’s so that the consistency relations (33) hold. In [Bo99], the Mk (u)’s are taken as linear combinations of u and the gj (u)’s: Mk (u) = αk0 u +

d

αkj gj (u).

j=1

The coefficients αkj (j = 0, 1, · · · , d) are chosen so that the consistency red lations (33) hold and the matrix αk0 Ip + j=1 αkj gju (u) has only positive eigenvalues.

On Hyperbolic Relaxation Problems

517

With the Mk (u)’s thus obtained, Conditions (i)–(iv) were verified in [Yo04a] with the entropy function η = η(U ) constructed in [Se00a, Bo99], provided that (a) the approximated conservation laws (32) are endowed with a strictly convex entropy function η˜ = η˜(u); (b) the a(k)’s are chosen so that the d × (N − 1)-matrix ! a(2) − a(1), a(3) − a(1), · · · , a(N ) − a(1) has rank d.

Finally, we remark that, unlike the entropy dissipation condition (i)–(iii), (iv) does not hold for all physical models. An example is the system of equations of gas dynamics in thermal non-equilibrium (see [Ze99, Yo01]). Even so, a global existence theorem was established in Zeng [Ze99] for this model. This indicates that Condition (iv) is not quite necessary for global existence. It would be interesting to weaken that condition while preserving the global existence.

References [AN00] [BR98]

[Bo99]

[Ce88] [CLL94]

[CP98]

[Da00] [Dr87] [EG94] [FL71] [Ga75]

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W.-A. Yong, W. J¨ ager Geroch, R., Lindblom, L.: Dissipative relativistic ﬂuid theories of divergence type. Phys. Rev. D, 41, 1855–1861 (1990) Geroch, R., Lindblom, L.: Causal theories of dissipative relativistic ﬂuids. Ann. Phys., 207, 394–416 (1991) Godunov, S.K.: An interesting class of quasilinear systems. Dokl. Akad. Nauk SSSR, 139, 521–523 (1961) Hanouzet, B., Huynh, P.: Approximation par relaxation d’un syst`eme de Maxwell non lin´eaire. C. R. Acad. Sci., Paris, Ser. I, Math., 330, 193–198 (2000) Hanouzet, B., Natalini, R.: Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Arch. Rational Mech. Anal., 169, 89–117 (2003) Higdon, R.L.: Initial-boundary value problems for linear hyperbolic systems. SIAM Rev., 28, 177–217 (1986) Jin, S., Xin, Z.: The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Comm. Pure Appl. Math., 48, 235–276 (1995) Junk, M., Yong, W.-A.: Rigorous Navier-Stokes limit of the Lattice Boltzmann equation. Asymptotic Anal., 35, 165–185 (2003) Kawashima, S.: Global existence and stability of solutions for discrete velocity models of the Boltzmann equation. In: Recent Topics in Nonlinear PDEs. Kinokuniya, Tokyo (1983) Kawashima, S.: Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics. Ph.D. Thesis, Kyoto University, Kyoto (1984) Kawashima, S., Shizuta, Y.: The Navier-Stokes equation in the discrete kinetic theory. J. M´ec. Th´eor. Appl., 7, 597–621 (1988) Kawashima, S., Yong, W.-A.: Dissipative structure and entropy for hyperbolic systems of balance laws. Arch. Rational Mech. Anal., to appear Klainerman, S., Majda, A.: Compressible and incompressible ﬂuids. Comm. Pure Appl. Math., 35, 629–651 (1982) Kreiss, K.O.: Initial boundary value problems for hyperbolic systems. Comm. Pure Appl. Math., 23, 277–298 (1970) Lattanzio, C., Yong, W.-A.: Hyperbolic-parabolic singular limits for ﬁrstorder nonlinear systems. Commun. Partial Diﬀer. Equations, 26, 939–964 (2001) Lax, P.D., Richtmyer, R.D.: Survey of the stability of linear ﬁnite diﬀerence equations. Comm. Pure Appl. Math., 9, 267–293 (1956) Levermore, C.D.: Moment closure hierarchies for kinetic theories. J. Statist. Phys., 83, 1021–1065 (1996) Liu, H., Yong, W.-A.: Time-asymptotic stability of boundary-Layers for a hyperbolic relaxation system. Commun. in Partial Diﬀer. Equations, 26, 1323–1343 (2001) Liu, T.-P.: The entropy condition and the admissibility of shocks. J. Math. Anal. Appl., 53, 78–88 (1976) Liu, T.-P.: Hyperbolic conservation laws with relaxation. Comm. Math. Phys., 108, 153–175 (1987) Lorenz, J., Schroll, H.J.: Stiﬀ well-posedness for hyperbolic systems with large relaxation terms. Adv. in Diﬀ. Eqns., 2, 643–666 (1997)

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Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer, New York (1984) [MP85] Majda, A., Pego, R.: Stable viscosity matrices for systems of conservation laws. J. Diﬀer. Equations, 56, 229–262 (1985) [MZ02] Mascia, C., Zumbrun, K.: Pointwise Green’s function bounds and stability of relaxation shocks. Indiana Univ. Math. J., 51, 773–904 (2002) [MM84] Mihalas, D., Mihalas, B.W.: Foundations of Radiation Hydrodynamics. Oxford University Press, Oxford (1984) [MR98] M¨ uller, I., Ruggeri, T.: Rational Extended Thermodynamics. Springer, New York (1998) [Na98] Natalini, R.: Recent results on hyperbolic relaxation problems. In: Freist¨ uhler, H. (ed) Analysis of Systems of Conservation Laws. Chapman & Hall/CRC, Boca Raton (1998) [PI88] Platkowski, T., Illner, R.: Discrete velocity models of the Boltzmann equation: a survey on the mathematical aspects of the theory. SIAM Rev., 30, 213–255 (1988) [Po73] Pomraning, G.C.: The Equations of Radiation Hydrodynamics. Pergamon Press, Oxford (1973) [RS04] Ruggeri, T., Serre, D.: Stability of constant equilibrium state for dissipative balance laws system with a convex entropy. Quarterly of Appl. Math., 62, 163–179 (2004) [Se00] Serre, D.: Systems of Conservation Laws. Cambridge University Press, Cambridge (2000) [Se00a] Serre, D.: Relaxation semi-lin´eaire et cin´etique des syst`emes de lois de conservation. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 17, 169–192 (2000) [SK85] Shizuta, Y., Kawashima, S.: Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation. Hokkaido Math. J., 14, 249–275 (1985) [UKS85] Umeda, T., Kawashima, S., Shizuta, Y.: On the decay of solutions to the linearized equations of electro-magneto ﬂuid dynamics. Japan J. Appl. Math., 14, 435–457 (1985) [Wh74] Whitham, J.: Linear and Nonlinear Waves. Wiley, New York (1974) [XX00] Xin, Z., Xu, W.-Q.: Stiﬀ well-posedness and asymptotic convergence for a class of linear relaxation systems in a quarter plane. J. Diﬀer. Equations, 167, 388–437 (2000) [XX02] Xin, Z., Xu, W.-Q.: Initial-boundary value problem to systems of conservation laws with relaxation. Quarterly of Appl. Math., 60, 251–281 (2002) [Xu02] Xu, W.-Q.: Initial boundary value problem for a class of linear relaxation systems in arbitrary space dimensions. J. Diﬀer. Equations, 183, 462–496 (2002) [Xu04] Xu, W.-Q.: Boundary conditions and boundary layers for a multidimensional relaxation model. J. Diﬀer. Equations, 197, 85–117 (2004) [Yo92] Yong, W.-A.: Singular perturbations of ﬁrst-order hyperbolic systems. Ph.D. thesis, Universiy of Heidelberg, Heidelberg (1992) [Yo99] Yong, W.-A.: Boundary conditions for hyperbolic systems with stiﬀ source terms. Indiana Univ. Math. J., 48, 115–137 (1999) [Yo99a] Yong, W.-A.: Singular perturbations of ﬁrst-order hyperbolic systems with stiﬀ source terms. J. Diﬀer. Equations, 155, 89–132 (1999)

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W.-A. Yong, W. J¨ ager Yong, W.-A.: Basic aspects of hyperbolic relaxation systems. In: Freist¨ uhler, H. (eds) Advances in the Theory of Shock Waves. Birkh¨auser, Boston (2001) Yong, W.-A.: Remarks on hyperbolic relaxation approximations. In: Freist¨ uhler, H. (eds) Hyperbolic Problems: Theory, Numerics, Applications. Birkh¨ auser, Basel (2001) Yong, W.-A.: Basic structures of hyperbolic relaxation systems. Proc. R. Soc. Edinb., Sect. A, Math., 132, 1259–1274 (2002) Yong, W.-A.: Diﬀusive relaxation limit of multidimensional isentropic hydrodynamical models for semiconductors. Siam J. Appl. Math., 64, 1737– 1748 (2004) Yong, W.-A.: Entropy and global existence for hyperbolic balance laws. Arch. Rational Mech. Anal., 172, 247–266, (2004) Yong, W.-A., Zumbrun, K.: Existence of relaxation shock proﬁles for hyperbolic conservation laws. Siam J. Appl. Math., 60, 1565–1575 (2000) Zeng, Y.: Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation. Arch. Rational Mech. Anal., 150, 225–279 (1999) Zumbrun, K.: Multidimensional stability of planar viscous shock waves. In: Freist¨ uhler, H. (eds) Advances in the Theory of Shock Waves. Birkh¨ auser, Boston (2001)

Appendix: Color Plates

522

Appendix: Color Plates

Plate 1: (Fig. 2 on page 38) HAT ﬁring in the laboratory, 2000 A, 0.8 g/s.

Plate 2: (Fig. 4 on page 39) HAT: Heavy particle temperature Th , 2000 A, 0.8 g/s.

Appendix: Color Plates

Plate 3: (Fig. 7 on page 41) HAT: Density log10 (ρ), 4000 A, 0.8 g/s.

Plate 4: (Fig. 8 on page 41) HAT: Density log10 (ρ), 5000 A, 0.8 g/s.

523

524

Appendix: Color Plates

Plate 5: (Fig. 11 on page 43) DT2: Density log10 (ρ), 4000 A, 0.8 g/s.

Plate 6: (Fig. 12 on page 43) DT2: Density log10 (ρ), 5000 A, 0.8 g/s.

Appendix: Color Plates

525

Plate 7: (See also Fig. 9 on page 66) Density plot of a supersonic ﬂow propagating into a channel.

Plate 8: (See also Fig. 10 on page 67) Density plot of a thermally driven ﬂow.

526

Appendix: Color Plates

Plate 9: (Fig. 10 on page 89) Color plots of the temperature and schlieren plots of the density on reﬁnement regions in the ﬁrst (left) and second half (right) of a detonation cell.

Plate 10: (Fig. 11 on page 89) Schlieren plots of ρ (upper row) and YOH (lower row) in the ﬁrst (left) and second (right) half of detonation cell, mirrored at x2 = 0 cm, 5.0 cm < x1 < 7.0 cm. The plots of YOH are overlaid by a blue isosurface of ρ that visualizes lig .

Appendix: Color Plates

Plate 11. to be continued

527

528

Appendix: Color Plates sequel to Plate 11.

Plate 11: (Fig. 7 on page 126) Time evolution of magnetic ﬁeld (ﬁeld lines are shown in black) and temperature (color coded) from a 2D simulation run of solar magneto-convection near the visible solar surface (located at z ≃ 100 km height). A homogeneous vertical ﬁeld of 100 Gauss has been introduced at t = 0 after a statistically stationary convection pattern has evolved. Within a few minutes, most of the magnetic ﬂux is transported by the converging horizontal ﬂows to the cool downﬂow region (ﬂux expulsion). Owing to the suppression of the convective energy transport, the gas in the ﬂux concentrations cools and sinks; lateral compression by the external gas pressure then leads to a strong intensiﬁcation of the ﬁeld strength, which reaches kilogauss values. About 20 minutes after the introduction of the magnetic ﬁeld, the ﬂux concentrations have merged into three large ﬂux sheets (labeled I,II,III), which start to determine the surrounding ﬂow pattern with strong downﬂows surrounding the ﬂux sheets. After about 30 minutes a quasi-stationary situation has developed. The velocity ﬁeld in this state is shown in the form of velocity vectors in the last panel.

Appendix: Color Plates

(a) Pressure contours

(b) Density contours

(c) Mach contours

(d) Streamlines and pressure ﬁeld

(e) Fluid phases: liquid (blue), wet steam (green), vapor (grey) Plate 12: (Fig. 12 on page 158) Expansion of a liquefaction shock.

529

530

Appendix: Color Plates

p [N/m2]

y

x

x

(a) t = 3.01 µs

(b) t = 3.01 µs

p [N/m2]

y

ρ [kg/m3]

y

x

x

(c) t = 6.66 µs

(d) t = 6.66 µs

p [N/m2]

y

ρ [kg/m3]

y

x

(e) t = 11.50 µs

ρ [kg/m3]

y

x

(f) t = 11.50 µs Plate 13. to be continued

Appendix: Color Plates

531

sequel to Plate 13.

p [N/m2]

y

x

x

(g) t = 13.93 µs

(h) t = 13.93 µs

p [N/m2]

y

ρ [kg/m3]

y

x

x

(i) t = 21.48 µs

(j) t = 21.48 µs

p [N/m2]

y

ρ [kg/m3]

y

x

x

(k) t = 28.30 µs

ρ [kg/m3]

y

(l) t = 28.30 µs Plate 13. to be continued

532

Appendix: Color Plates sequel to Plate 13.

p [N/m2]

y

ρ [kg/m3]

y

x

x

(m) t = 30.30 µs

(n) t = 30.30 µs

p [N/m2]

y

y

3

ρ [kg/m ]

x

x

(o) t = 48.10 µs

(p) t = 48.10 µs

3

ρ [kg/m ]

2

p [N/m ] y

y

x

(q) t = 56.30 µs

x

(r) t = 56.30 µs Plate 13: (Fig. 13 on page 161) Bubble collapse.

Appendix: Color Plates

533

Plate 14: (Fig. 9 on page 198) Results using the mixed GLM-MHD scheme for a 2d magnetic ﬂux tube simulation, on the left with the Bgﬁx modiﬁcation, on the right using only the base scheme. The Bz component is shown in a small region of the domain (top) and a color representation of the grid reﬁnement (bottom), red indicating a very high grid resolution and blue a very coarse grid. With the Bgﬁx correction the grid consists of about 12000, without the correction of about 32000 elements. Note also the loss of symmetry caused by the base scheme.

534

Appendix: Color Plates

(a) Entropy isosurface and levelsurface in the central part of the domain (left) and the grid partitioning and reﬁnement (right) at times t = 0.0, 6.0, 9.0, and t = 12.0.

simulation time 70

execution time

60 50 40 30 20 10 0 0

1

2

3

no. of elements, maximum: 6687440 minimum runtime for numerics average runtime for numerics

4

5

6

7

8

maximum runtime for numerics runtime, total: 233928 s

(b) Eﬀect of the load balancing on the runtime. Top: grid partitioning, reﬁnement, and relative load of partitions at t = 6.2, 6.6, 6.65. Bottom: graphs of runtime. Load balancing takes place about every 200 steps (peaks in runtime graph) and requires approximately the runtime of three normal time steps. For example, load balancing occurs shortly before t = 6.2. Due to grid reﬁnement we observe an increase of the load, e.g., for the light green and the brown partition between t = 6.2 and t = 6.6. Thus the maximal and minimal runtime diverge. Consequently, the total runtime increases more than it would be justiﬁed by the growth of the number of elements. Following our load balancing strategy the grid is therefore repartitioned between t = 6.6 and t = 6.65. On average the total runtime increases by the same amount as the number of elements in the grid. Note that this is the optimal behaviour for an explicit ﬁnite volume scheme. Plate 15: (Fig. 11 on page 199) Simulation of an exploding flux tube in 3d on 8 processors of an IBM-SP2.

Appendix: Color Plates

Plate 16: (Fig. 7 on page 251) Initialization of a 3D bow shock with shock, subset grid, inner sphere and constant pressure values.

Plate 18: (Fig. 9 on page 251) Snapshot of an inviscid bow shock with Mach=3 and three cylinders. Pressure values are plotted

535

Plate 17: (Fig. 8 on page 251) Final state of the 3D bow shock calculation with front, subset mesh and pressure values in the cut plane.

Plate 19: (Fig. 10 on page 251) Zoom of area between both lower cylinders; pressure values, front, subset mesh and velocity vectors.

536

Appendix: Color Plates

Plate 20: (Fig. 11 on page 252) Snapshot of an inviscid unstable 2D CJ-detonation with pressure values calculated on an unstructured grid.

Plate 21: (Fig. 12 on page 252) Masterslave combination for a restricted recompression shock wave on a transonic airfoil: Colors of constant Mach number and front position.

Appendix: Color Plates

537

Plate 22: (Fig. 3 on page 329) Spatial appearance of the number of dust particles (log nd [cm−3 ]; false color background) and the vorticity (∇ × v; black and grey contour lines) of the 2D velocity ﬁeld for t = 0.8s of a simulation with Tref = 2100 K, ρref = 3.16 10−4 g cm−3 , vref = cS .

538

Appendix: Color Plates log |ζ| 0.0

y [cm]

(a)

(b)

(c)

(d)

1.0

2.0

3.0

4.0

2.5 × 104

2.0 × 103

1.5 × 103

1.0 × 104

5.0 × 103

0 y [cm]

2.5 × 104

2.0 × 103

1.5 × 103

1.0 × 104

5.0 × 103

0 0

5.0 × 103

1.0 × 104

1.5 × 103

2.0 × 103

2.5 × 104

x [cm] 0

5.0 × 103

1.0 × 104

1.5 × 103

2.0 × 103

2.5 × 104

x [cm]

Plate 23: (Fig. 12 on page 381) Flame propagation into quiescent fuel at ρu = 5 × 107 g cm−3 , resolution: 200 × 200 cells, snapshots taken at (a) 0, (b) 7.5, (c) 15, and (d) 30 growth times τLD of a perturbation with λ = 3.2 × 104 cm. Color-coded is the vorticity of the ﬂow ﬁeld.

Appendix: Color Plates

539

log |ζ| 0.0

(a)

(b)

(c)

(d)

1.5

3.0

4.5

6.0

y [cm]

6000

4000

2000

0

y [cm]

6000

4000

2000

0 0

2.0 × 103 4.0 × 103 6.0 × 103 8.0 × 103 1.0 × 104 1.2 × 104

x [cm] 0

2.0 × 103 4.0 × 103 6.0 × 103 8.0 × 103 1.0 × 104 1.2 × 104

x [cm]

Plate 24: (Fig. 13 on page 382) Flame propagation into vortical fuel at ρu = 5×107 g cm−3 ; velocity ﬂuctuations at the right boundary: v ′ /ulam = 2.5; resolution: 300 × 200 cells; snapshots taken at (a) 0 s, (b) 8.0 × 10−3 s, (c) 1.6 × 10−2 s, and (d) 4.8 × 10−2 s. Color-coded is the vorticity of the ﬂow ﬁeld.

Plate 25: (Fig. 1 on page 395) Snapshots of the E1 field component obtained

with uncorrected (left) and corrected (right) computation.

540

Appendix: Color Plates

Plate 26. to be continued

Appendix: Color Plates sequel to Plate 26.

Plate 26. to be continued

541

542

Appendix: Color Plates sequel to Plate 26.

Plate 26: (Fig. 11 on page 426) Three-dimensional propagation of a premixed ﬂame over an obstacle: Representation of the isoline G=0 as well as the velocity ﬁeld (vector arrows); local burning speeds are modeled over G-LEM ﬂame modules.

Plate 27: (Fig. 12 on page 426) Two-dimensional cut plane (x-y-direction) through the ﬂow ﬁeld: The velocity ﬁeld (vector arrows) and the turbulent kinetic energy are shown.