Lecture Notes on the Discretization of the Boltzmann Equation

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Series on Advances in Mathematics for Applied Sciences - Vol. 63

LECTURE NOTES ON THE DISCRETIZATION OF THE BOLTZMANN EQUATION

Editors

•

Nicola Bellomo and Renee Gatignol World Scientific

LECTURE NOTES ON THE DISCRETIZATION OF THE ROLTZMANN EQUATION

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Series on Advances in Mathematics for Applied Sciences - Vol. 63

LECTURE NOTES ON THE DISCRETIZATION OF THE ROLTZMANN EQUATION Editors

Nicola Bellomo Politecnico di Torino, Italy

Renee Gatignol Universite Pierre et Marie Curie, France

0 World Scientific •

New Jersey • London • Singapore • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Lecture notes on the discretization of the Boltzmann equation / editors Nicola Bellomo, Renee Gatignol. p. cm. - (Series on advances in mathematics for applied sciences ; v. 63) Includes bibliographical references. ISBN 9812382259 (alk. paper) 1. Transport theory. 2. Finite element method. 3. Differential equations—Asymptotic theory. I. Bellomo, N. II. Gatignol, Renee. III. Title. IV. Series. QC718.5.T7L43 2003 530.13'8-dc21

2002038059

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DEDICATION It is a pleasure for us to dedicate this volume on discrete and semicontinuous models in the kinetic theory Held to Professor Henri Cabannes, emeritus Professor in Universite Pierre et Marie Curie and member of Academie des Sciences de Paris, for his eighty birthday. In 1964, J.E. Broadwell has introduced the hrst discrete models with six and eight velocities. Immediately, these models set up a very large interest. The discrete kinetic theory deals with the description of a gas of a large number of particles moving in the space by a finite, may be however large, number of velocities (R. Gatignol 1970). Professor H. Cabannes imagines the coplanar semi-continuous models. They correspond to a gas having velocities with a finite number of magnitude but with a continuum of directions. During the past thirty years, a scientific community interested in these two types of kinetic models appeared and increased with a very large number of papers, some books, many meetings or specialized sessions in well known International Congresses (ICIAM, RGD Symposium...). Professor H. Cabannes has taken a prominent part in the development of the discrete kinetic theories and specially in the mathematical questions related to the kinetic equations. Discrete models offer relatively lower difficulties, analytic and computational, with respect to those involved by the Boltzmann equation. So nice results have been obtained on the existence or uniqueness of the solutions of the kinetic equations with initial or boundary conditions, among others, N. Bellomo, H. Cabannes, C. Cercignani, A. Kawashima, T. Nishida, R. Ulner. In 1990, C. Bardos, F. Golse, D. Levermore have described the connection between discrete velocity kinetic theory and fluid dynamics. They have given the conditions that formally lead to generalized compressible Euler equations or to generalized incompressible Navier-Stokes equations. In another connection, in lattice gases, the velocities are discretized like in the discrete models, but the space and time variables are also discretized. The nice simulations of flows by the lattice gas methods stimulated new works on the hydrodynamics of the discrete model gases and the modeling of some particular flows. The solutions so yielded are in good agreement with those given from different treatments of the Boltzmann equation. v

VI

Dedication

A progress concerning the convergence, in some adequate sense, of the discrete models to the Boltzmann equation has been recently obtained. The Erst is in 1994 by S. Mischler in his thesis. In his pioneer work, he has introduced one discretization schema for the velocities in the Boltzmann equation, and then he has proved, under some conditions, the convergence of the so-discretized solutions to the solutions of the Boltzmann equation. Recent studies, among others Y.H. Babovsky, V. Bobylev, D. Gorsch, A. Palcewski, L. Preziosi, J. Schneider, have developed the theory towards models with an arbitrary large number of velocities, the final target being an analysis of the convergence towards the full Boltzmann equation. It remains a conjecture according to it the positive "eternal" solutions of the full Boltzmann equation are only the Maxwellian solutions (eternal means valid for each time from minus to plus infinity). For a semi-continuous model of the Boltzmann equation, Professor H. Cabannes has proved in a very nice paper (1998) this conjecture.

Nicola Bellomo and Renee Gatignol

CONTENTS

Preface

ix

C h a p t e r 1. F r o m t h e B o l t z m a n n E q u a t i o n t o Discretized Kinetic Models 1 N. Bellomo and R. Gatignol C h a p t e r 2. Discrete Velocity Models for G a s M i x t u r e s C. Cercignani

17

C h a p t e r 3 . Discrete Velocity M o d e l s w i t h M u l t i p l e Collisions 29 jR. Gatignol C h a p t e r 4. Discretization of t h e B o l t z m a n n E q u a t i o n t h e Semicontinuous Model L. Preziosi and L. Rondoni

and 59

C h a p t e r 5. Semi-continuous E x t e n d e d K i n e t i c T h e o r y W. Roller

97

C h a p t e r 6. S t e a d y Kinetic B o u n d a r y Value P r o b l e m s H. Babovsky, D. Gorsch and F. Schilder

133

C h a p t e r 7. C o m p u t a t i o n a l M e t h o d s a n d Fast A l g o r i t h m s for B o l t z m a n n E q u a t i o n s 159 L. Pareschi vii

VU1

Contents

Chapter 8. Discrete Velocity Models and Dynamical Systems 203 A. Bobylev and N. Bernhoff Chapter 9. Numerical Method for the Compton Scattering Operator 223 C. Buet and S. Cordier Chapter 10. Discrete Models of the Boltzmann Equation in Quantum Optics and Arbitrary Partition of the Velocity Space 259 F. Schiirrer List of Contributors

299

PREFACE Classical kinetic theory of gases is based on the Boltzmann equation which describes the evolution of a system of equal particles undergoing collisions preserving mass, momentum, and energy. Mathematical problems, e.g. initial and boundary value problems asymptotic theories, and computational schemes, are objects of permanent interest of applied mathematicians. Solution of problems in Suid dynamics requires the computational treatment of the Boltzmann equation linked to suitable initial and/or boundary conditions. Discretization methods have been developed on the idea of replacing the original continuous Boltzmann equation by a finite set of differential equations corresponding to the densities linked to suitable sets of velocities. The general framework is the one of the so-called discrete Boltzmann equation. Different discretization methods have been developed and, consequently, different discrete models have been obtained. This method is generally developed to obtain a relatively simpler model with respect to the continuous Boltzmann equation which may be useful not only for computational treatment, but also to describe interesting phenomena such as gas mixtures or equation with multiple collisions. Recently various discretization methods of the full Boltzmann equation have been developed for arbitrary large, however finite, number of velocities. The discretization methods involve several problems, some of them still open, which are certainly a challenging, however difficult, research field of applied mathematicians. Some of the above relevant mathematical problems may be summarized as follows: i) Discretization methods in the velocity space and discretization of the Boltzmann equation or derivation of a suitable discrete velocity models; ii) Analysis of the conservation properties (mass, momentum and energy) of the collision scheme and the trend to equilibrium; Hi) Asymptotic convergence to the continuous equation when the number of velocities tends to infinity, and convergence rate for increasing, however finite, number of velocities; iv) Application of discrete models to physics, biology and applied sciences in general.

IX

X

Preface

This book aims to collect some relevant contributions on the topics which have been described above. It is organized into ten chapters, each authored by applied mathematicians who have been active in the field, and whose scientific contributions are well recognized by the scientific community. The first chapter provides a concise introduction to the continuous and discrete Boltzmann equation so that the reader can find suitable technical and bibliographic references. Then the chapters which follow deal with various research topics related to the above outlined field of applied mathematics. In particular, Chapter 2, by Carlo Cercignani, deals with modeling of gas mixtures according to some recent developments proposed by himself and co-workers. Chapter 3, by Renee Gatignol, deals with the derivation and qualitative analysis of discrete velocity models with multiple collisions. The contents of both above chapters pays special attention to the characterization of the Maxwellian state and its properties. Chapter 4, by Luigi Preziosi and Lamberto Rondoni, deals with the derivation of the equation and with the analysis of some quantitative results concerning the asymptotic trend to the hydrodynamic description. Chapter 5, by Wilfried Koller, shows how the semi-continuous equation can be derived in the framework of the extended kinetic theory. It also refers to some interesting applications and specifically the equations for a chemically reacting gas. Chapter 6, by Hans Babovsky, Daniel Gorsch, and Frank Schilder deals with the analysis of steady boundary value problems for discrete models. Chapter 7, by Lorenzo Pareschi, deals with the development of computational schemes and algorithms for numerical applications. Chapter 8, by Bobylev and Bernhoff, deals with the qualitative analysis of the relation between discrete models and the full Boltzmann equation. Chapter 9, by Claude Buet and Philippe Cordier, deals with the analysis of discretization and computational schemes for Compton Scattering Operator. Chapter 10, by Wilfried Schiirrer deals with discretization schemes for quantum optics models with special attention to arbitrary partition of velocities and scaling procedures. The Editors trust that this book is going to be a useful reference for applied mathematicians operating on discretization methods in kinetic theory and with applications of discrete velocity models.

Nicola Bellomo and Renee Gatignol

Chapter 1

From the Boltzmann Equation to Discretized Kinetic Models N. BellomoW and R. Gatignol^ ^ Dipartimento di Matematica, Politecnico di Torino, Italy (2) Laboratoire de Modelisation en Mecanique et UPCM-CNRS Universite Pierre et Marie Curie, Paris, France

1.1

Introduction

Classical kinetic theory of gases is based on the Boltzmann equation which describes the evolution of a system of equal particles undergoing collisions preserving mass, momentum, and energy. Mathematical problems, e.g. initial and boundary value problems, asymptotic theories, and computational schemes, are object of a permanent interest of applied mathematicians as documented in specialized books, e.g. [5], [9]. Solution of problems in fluid dynamics require the computational treatment of the Boltzmann equation linked to suitable initial and/or boundary conditions. Statement of mathematical problems and a review of analytic results and computational results can be recovered in [4]. Discretization methods have been developed on the idea of replacing the original continuous Boltzmann equation by a finite set of differential equations corresponding to the densities linked to a suitable finite set of velocities. Then, the discrete Boltzmann equation is obtained. The general framework of the above model can be recovered in the Lecture Notes [10], as well as in the review paper [17]. Various discretization methods have been developed and, consequently, different discrete models have been obtained. This method is generally developed to obtain a relatively simpler model with respect to the continuous Boltzmann equation which may be useful not only for computational treatment, but also to describe interesting 1

2

Lecture Notes on the Discretization

of the Boltzmann

Equation

phenomena such as gas mixtures or equation with multiple collisions, so far some dense gas effects may be described. Discretization methods involve several problems, some of them still open, which are certainly a challenging, however difficult, research field of applied mathematicians. Some of the above relevant mathematical problems may be summarized as follows: i) Discretization methods of the velocity space and derivation of a suitable discrete velocity models, ii) Analysis of the conservation properties (mass, momentum and energy) of the collision scheme and of the trend to equilibrium. iii) Asymptotic convergence to the continuous equation when the number of velocities tends to infinity, and convergence rate for increasing, however finite, number of velocities. iv) Asymptotic analysis to the continuum (hydrodynamic) limit for discrete velocity models. These Lecture Notes are proposed with the aim of reporting the state-ofthe-art on the discretization of the Boltzmann equation with special attention to the topics which have been listed in items i)-iv). This introductory chapter provides a concise description of the models of the kinetic theory related to the topics dealt with in the chapters which follow: the continuous Boltzmann equation, the discrete and semidiscrete equation. Then, after the above brief description, a presentation of the various contributions to the Lecture Notes will be given. It is worth mentioning that the description given in Sections 1.2 and 1.3 is very concise with the aim of introducing the reader to some basic notations and concepts. For a deeper understanding of the above topics, Section 1.2 refers to the book by Cercignani, Illner and Pulvirenti [9], while Section 1.3 refers to the Lecture Notes by Gatignol [10].

1.2

The Nonlinear Boltzmann Equation

A fluid is a disordered system of interacting particles moving in all directions within a space domain Q C H . When the position of each particle is correctly identified by the coordinates of its center of mass, the system may be reduced to a set of point masses and conveniently referred to a fixed frame of orthogonal axes. This is, for instance, the case of spherically symmetric particles. When the domain Q is bounded, the particles interact with its walls d£l.

From the Boltzmann

Equation to Discretized Kinetic

Models

3

It is generally believed in physics, that a complete understanding of the macroscopic properties of a fluid certainly follows from the detailed knowledge of the state of each of its atoms or molecules. In most fluids of practical interest, the evolution of the microscopic states is governed by equations of motion according to the laws of classical mechanics for a system of N particles. Then, in principle, the macroscopic state of the fluid can be known exactly, once the solution to the above equation of motions is known. In most cases it is sufficient to compute a rather limited number of macroscopic variables (density, global momentum, energy, pressure tensor, etc) in order to obtain a suitable picture of fluid evolution. However, it is very hard to implement this program unless a suitable simplification is introduced. Indeed, the large values of the total number N of molecules (about 10 20 for a cubic centimeter of gas under normal conditions), and unavoidable inaccuracies in the knowledge of initial conditions result in the impossibility of retrieving and manipulating the microscopic information contained in the total set of state vectors. An alternative to the above description is provided by the Boltzmann equation, the fundamental mathematical model of non-equilibrium statistical mechanics, which describes the evolution of a dilute monoatomic gas of a large number of identical particles undergoing elastic binary collisions. We take for granted that the reader already possesses the basic knowledge of the phenomenological kinetic equation of Boltzmann. Therefore, in the following, we limit ourselves to recalling the main features of the model, referring to the classical literature, e.g., [9] for the derivation and fundamental properties, and [1], [4] for analytic treatment, and [4], [5] for the development of computational schemes. The Boltzmann equation describes a dilute monoatomic gas of particles modeled as mass points identified by mass m, position x and velocity v. The equation refers the time evolution of the one-particle distribution function / = f(t, x, v),

/ : IR+ x M 3 x H 3 -» H + ,

with the meaning that f(t,x,v)dxdv gives the expected number of particles in the elementary volume centered at the phase point (x, v), at time t.

(1.1) dxdv

4

Lecture Notes on the Discretization

of the Boltzmann

Equation

If / is known, the macroscopic observable quantities can be computed as expectation values of the corresponding microscopic functions. In particular, denoting by n(t, x) the gas number density at time t then p[t,x) — m n(t,x) = ml f(t,x,v)dv, JWi3

(1.2)

and

u( x)=

'' ^A. v/( «' x ' v)dv '

(L3)

are, respectively, the mass density and the mass velocity, while the mean translational energy is given by

*-'-3^ojt.l T -^ / <^ T '*'-

(1 4)

'

As known, in equilibrium conditions, the energy can be related (according to the so-called Boltzmann's principle) to the temperature by the relation s(t,x) = 3/2kT(t,x), where k is the Boltzmann constant and T(t,x) is the fluid kinetic temperature. This relation has to be handled carefully in nonequilibrium conditions, where the above principle cannot be straightforwardly applied. The time dependence in the Boltzmann equation allows that, as time goes, the number of particles inside dxdv may change if the system is away from equilibrium. Obviously, such a change is computed by balancing the incoming and outgoing particles in dxdv. The idea behind this balance of losses and gains in dxdv is that particles are lost or gained in dx by free streaming, while they leave or enter dv as the result of collisions with other particles. This leads to the basic assumptions behind the phenomenological derivation of the Boltzmann equation: Assumption 1.2.1. Gas particles move freely in space between instant and local collisions, with conservation of momentum and translational (kinetic) energy according to the laws of classical mechanics. Assumption 1.2.2. Collisions involving more than two particles are negligible. Assumption 1.2.3. Colhsions between pair of particles are uncorrelated (molecular chaos hypothesis).

From the Boltzmann

Equation to Discretized Kinetic Models

5

It should be remarked that, according to the first hypothesis, variations of the distribution function, inside dx, are neglected. Therefore, the distribution function of the field particles which enter into the action domain of the test particles is actually approximated by the value of / in the position of the field particle. Obviously, the momentum and energy conservation relations for an encounter of two particles of simple gas are mv + mw = mv' 4- m w ' , m|v| 2 + m|w| 2 = m|v'| 2 + m|w'| 2 ,

(1.5)

where v, w denote the pre-collision velocities, and v', w' are the postcollision velocities. It appears that a collision process (v, w) —> (v',w') is characterized by a prescribed density probability V(v,w | v ' , w ' ) for the transition of particles from states with velocities v, w to those with velocities v', w', respectively. Due to mechanical reversibility, V(v,w

| v',w') = P(v',w' | v , w ) .

Moreover, the symmetry of collisions with respect to the interchange of v and w, or of v ' with w', imply that V(v, w | v', w') = V(v, w | w', v ' ) . V is a distribution with support defined by (1.5). Taking into account conservation of momentum, in addition to conservation of mass and momentum, yields V(v, w | v', w') = B(v - w, v ' - w')J(v + w - v ' - w') x <5(|v|2 + | w | 2 - | v ' | 2 - | w ' | 2 ) ,

(1.6)

where B(v — w, v' — w') is a smooth function depending only on |v — w| and the angle between v — w and v' — w', and is determined uniquely from the knowledge of the pair interaction potential. Once such an approach has been accepted, the celebrated Boltzmann equation, which defines the evolution of the distribution function / =

6

Lecture Notes on the Discretization

f(t,x,v)

of the Boltzmann

Equation

reads:

(jt + (v, Vx) + (F, Vv>) / = J[f, /] = G[f, f] - L[f, f],

(1.7)

where (•, •) denotes the internal product; F is the external force field acting on the particles; and G and L are the gain and loss terms respectively given by G[f, / ] = j V(v, w | v', w ' ) / ( t , x, v ' ) / ( t , x, w') dV dW dw, and L\f, / ] = f(?, x, v) J V(y, w | v', w ' ) / ( t , x, w') dv' dw. Performing the above integration technically yields G[/,/]=

J

B(n,v-w)/(t,xlv,)/(t,x,w')dndw,

(1.8a)

R3x§^ and L[f,f]

= f(t,K,v)

J

B(n,v-w)/(t,x)w)dndw,

(1.86)

m,3x§2+

where the post-collision velocities are given by

{

v' = v + ( w - v , n ) n ,

(1.9)

w' = w — (w — v , n ) n ,

where n is the unit vector in the direction of the apse-line bisecting the angle between w — v and w' — v', and §\ = {n G M 3 :

|n| = 1,

(w - v, n) > 0}

is the domain of integration of the variable n. The collision kernel B can be modeled under suitable assumptions on the pair interaction potential. Technical expressions of this term for inverse

From the Boltzmann

Equation to Discretized Kinetic

Models

7

power potentials are reported in the classical literature, e.g., Chapter 1 of [5]. A rigorous justification of Assumptions 1.2.1-1.2.3 behind the derivation of the Boltzmann equation is difficult to obtain, impossible for general thermodynamic conditions as documented in [9]. Moreover, the above picture on the variation of / , as a result of competition between free streaming and balance of losses and gains in dxdv, requires the size of the volume element dx dv be large enough that the number of particles contained in it justifies the use of statistical methods. On the other hand, this number must be small enough that information contained in it should have local character. Clearly, these two features are not compatible in general, hence problems are expected in justifying the whole procedure. Hopefully, in the cases of practical interest, the molecule size does fall in a range of values which are small when compared to those of the volume elements dx. which, in turn, can be considered as microscopic with respect to the observation scale. Mathematically, this is achieved in the Boltzmann-Grad limit, which allows the number of particles N tends to infinity, and the radius of action a tend to zero, in such a way that a —> 0

and

aN2 -> c € (0, oo).

In this respect, some progress in deriving the Boltzmann equation from the first principles of mechanics in the Grad limit [14], [13], [9] should be mentioned. With reference to the specialized literature, we are now interested in reporting some fundamental properties of the Boltzmann model. Formally, [

J[f,f}dv = 0,

(1.10)

[

vJ[f,f]dv = 0,

(1.11)

and

L»M2.7[/,/]dv = 0,

(1.12)

which refer to local conservation of mass, momentum and translational energy.

8

Lecture Notes on the Discretization

of the Boltzmann

Equation

In particular, we will concentrate on the existence, uniqueness, and stability of equilibrium solutions. Recall that J[/,/]=0

(1.13)

is a functional equation which admits the so-called Maxwellian equilibrium solution UJ = u)(t, x, v) given by

m

\

2ir

-^(v-u(*,x))

2

(1.14)

Tn

where f3 = — , and p, T, and u are the macroscopic observable quantities. kl Trend towards equilibrium, for a gas in a box with periodic boundary conditions, is described by the Entropy functional H[f]{t) = J / ( t > x , v ) l o g / ( * , x l v ) d x d ,

(1.15)

under the assumption that the term / l o g / is integrable. Indeed: H[f](ti)

> H[f](t2),

Vti.ta,

Q
(1.16)

and that equality holds at equilibrium.

1.3

The Discrete and Semicontinuous Boltzmann Equation

This section provides a brief description of two classical discrete models of the kinetic theory of gases. The Chapters which follow report the developments, mathematical problems, and applications concerning the said model. Due to the contents which follow, the description of the section will be concise as the one of Section 1.2. The aim is simply to introduce the reader, who has not yet operated in this field of applied mathematics, to the discrete Boltzmann equation and to a background bibliography. The reader who already possesses a sufficient knowledge in the field can skip this section and move to the next one which deals with the presentation of the contents of the chapters which follows.

From the Boltzmann

Equation to Discretized Kinetic

Models

9

The discrete Boltzmann equation is a nonlinear mathematical model of the kinetic theory of gases, which describes the evolution of a gas of particles allowed to move with a finite number of velocities. The model is an evolution equation for the number densities JVj linked to the admissible velocities v*, for i £ L = { 1 , . . . , n}. The set N = {ATj}"=1 corresponds, for certain aspects, to the one-particle distribution function of the continuous Boltzmann equation. The mathematical theory of the discrete kinetic theory was systematically developed in the Lecture Notes by Gatignol [10], which provides a detailed analysis of the relevant aspects of the discrete kinetic theory: modeling, analysis of thermodynamic equilibrium and application to fluiddynamic problems. The contents of [10] mainly refer to a simple monoatomic gas and to the related thermodynamic aspects. After such a fundamental contribution, several developments have been proposed, as we shall see, in order to deal with more general physical systems: gas mixtures, chemically reacting gases, particles undergoing multiple collisions and so on. Various mathematical aspects, namely the qualitative analysis of the initial value and of the initial-boundary value problem, have been object of continuous interest of applied mathematicians. The existing literature is reported in the review papers [17] and [3]. Considering that models for gas mixtures and models with multiple collisions will be dealt with, respectively, in Chapters 2 and 3, this section simply reports the discrete Boltzmann equation for a simple gas with binary collisions. The formal expression of the evolution equation is as follows: ^+Vi-V*}Ni

= Ji[N\,

(1.17)

where Ni = Ni(t,x)

:

(t,x) e [0,T] x f i " - . ] ^ ,

i = l,...,n,

(1.18)

where t and x S IR", for u = 1,2,3, are the time and the space variables, whereas J, [N] denote the binary collision terms

MN] = \J2 jhk=l

A™{NhNk - NiNj).

(1.19)

10

Lecture Notes on the Discretization

of the Boltzmann

The terms Ay are the so-called transition collisions (v»,Vj)<—>(v h ,Vfe),

Equation

rates referred to the binary-

i,j,h,keL,

(1.20)

and where the collision scheme must be such that momentum and energy are preserved. The transition rates are positive constants which, according to the indistinguishability property of the gas particles and to the reversibility of the collisions, satisfy the following relations Ahk

ij

Ahk

~

ji

Akh .

Akh

~~

ij

~~

ji

'

Ahk

ij

Aij

~

hk •

i-\

91 \

\l-

z l

)

A detailed computation of the transition rates can be performed specializing the velocity discretization and analyzing the related collision mechanics. In general, as shown in various papers e.g. [2] and [11], and as we shall see in details in Chapter 3, it is possible to deal with kinetic models with multiple collisions. These models may be suitable to describe some dense gas phenomena. Further developments are dealt with in Chapter 2 which deals with gas mixtures. Indeed, models with multiple collisions and mixtures can be developed toward the Boltzmann like modeling of chemically reacting gases. As for the Boltzmann equation, see, among others, [11] or [2], the qualitative analysis of discrete models needs the identification of the space of collision invariants and of the Maxwellian state. Specifically, the following definitions can be used: Definition 1.1. A vector <j> = {(f>i}i£L £ H m is called collision if (4>,J[N])=0,

J[N] = {JieL e M m } ,

where the inner product is defined in M set L.

m

invariant (1.22)

and m is the cardinality of the

Definition 1.2. The set of the totality of collision invariants is called space of the collision invariants and is a linear subspace of IRm. Such a space will be denoted, in what follows, by M.. Definition 1.3. Let Ni > 0 for any i £ L , then the vector N is defined Maxwellian if J[N] = 0. Moreover, the following propositions characterize the space of the collision invariants and the Maxwellian state:

From the Boltzmann

Equation to Discretized Kinetic

Models

11

Proposition 1.1. Let

S M, i.e. is a collision invariant; ii) (,J[N}) = 0. Proposition 1.2. Let Ni > 0 for any i 6 L, then the following four conditions are equivalent i) N is a Maxwellian; ii) {logN^i^eM; iii) J[N] = 0. Consider now the classical H-Boltzmann function defined as

H = '$2ciNi]agNi.

(1.23)

Then, the evolution equation for the H-Boltzmann equation can be derived by multiplying the discrete Boltzmann equation by 1 + log Ni and taking the sum over i £ L. It can be technically verified that the time derivative of the above functional is nonpositive and that the equality holds if and only if iV is a Maxwellian. Further technical analysis is needed for equations with multiple collisions as we shall see in Chapter 3. The above discretization corresponds to discretize the velocity space into a suitable set of points by linking a number density to each velocity. Several applied mathematicians have attempted in the last decade, e.g. [6][8], [12], [15]-[16], [18]-[20], to design models with arbitrarily large number of velocities and hence to analyze convergence of discretized models toward the full Boltzmann equation. Several technical difficulties have to be tackled as well documented in the paper by Gorsch [12], among others: i) The discretization schemes for each couple of incoming velocities do not assure a pair of outgoing velocities such that conservation of mass and momentum is preserved; ii) The discretized equation may have a number of spurious collision invariants in addition to the classical ones corresponding to conservation of mass, linear momentum and energy; iii) The discretized equation may not correspond to the discretization of the original Boltzmann equation obtained by suitable interpolation polynomials; iv) The convergence of the solutions of discretized equation to those of the full Boltzmann equation, when the number of discretization points tends to infinity, under suitable hypotheses which have to be properly defined.

12

Lecture Notes on the Discretization

of the Boltzmann

Equation

A technical difficulty in dealing with the above convergence proof consists of obtaining an existence theorem for the Boltzmann equation in a function space which can be properly exploited for the convergence theorem. Moreover a stability analysis of the Maxwellian state needs to be developed according to the methods developed by Kawashima [21], [22]. The papers which have been cited in this chapter deal with the above problems and some interesting results have been obtained, as documented in [12], although the above listed problems still remain, at least partially, open. An alternative to the discretization scheme we have seen above has been developed mainly to deal with items i)-ii). It consists of discretizing the velocity space into a suitable set of velocity moduli letting the velocity free to assume all directions in the space. The formal expression of the evolution model is as follows: (jt

+ v ( • V x ) Ni(t,x,Q)

= Ji[f}(t,x,Q),

(1.24)

where Ni=Ni(t,x,il):

(*,x)G [ 0 , T ] x l R ' / x l R 3 - > ] R + , i = l , . . . , n ,

(1.25)

and where Q is the unit vector which identifies the direction of the velocity, Vj = Vi ft.

The above discretization was proposed in [15] for the plane case, and then in [18] for the description in the whole space. It was shown, already in [14] that if the velocity moduli are properly chosen, then number of spurious collision invariants in addition to the classical ones corresponding to conservation of mass, linear momentum and energy. The above model has been developed by various authors as it will be properly documented in Chapters 4 and 5 of this book. The interested reader is addressed to the above cited chapters to recover the detailed expression of the model.

Prom the Boltzmann

1.4

Equation to Discretized Kinetic Models

13

Plan of the Lecture Notes

The preceding sections have summarized the reference equations of the mathematical kinetic theory of gases which are object of the contents proposed in these Lecture Notes, namely the continuous, the discrete and the semicontinuous Boltzmann equation. The chapters which follow are proposed with the aim of updating the state-of-the-art in the field with special attention to the developments which have been proposed in the last decade. Indeed, all chapters which follow can be regarded as surveys of applied mathematicians who are active in the field and who have given interesting results which have effectively developed the traditional approach to the discretization methods to the Boltzmann equation with special attention to the development of computational schemes and applications. Bearing all the above in mind, a brief summary of the contents of the chapter (Lectures) which follow can be given. While one examines the contents it is important to analyze items i)-iv) which have been listed at the end of Section 1.3. The reader will recognize that part of the problems which have been posed above have already found a satisfactory answer. This means that applied mathematicians can find several interesting and challenging problems to deal with. Moreover, the following additional problems can be brought to the attention of the reader: v) Are discretized models useful for the application? vi) Is it possible to establish some a priori advantages of the discretized equation with respect to the semicontinuous one, or vice versa? vii) Rather than studying convergence of the solutions to discretized equation to the ones of the full Boltzmann equation, when the number of discretization points tends to infinity, is it possible to provide estimates of the distance of the two solutions at given number of discretization points? The contents are organized into four parts and ten chapters: Specifically the first part, namely Chapters 1,2, and 3, deals with fundamental aspects on modeling and analysis of thermodynamical properties of discretized models. In particular, after this preliminary introduction, Chapter 2, by Carlo Cercignani, deals with modeling of gas mixtures according to some recent developments proposed by himself and co-workers. Chapter 3, by Renee Gatignol, deals with the derivation and qualitative analysis of discrete velocity models with multiple collisions. The contents

14

Lecture Notes on the Discretization

of the Boltzmann

Equation

of both the above chapters pays special attention to the characterization of the Maxwellian state and its properties. The second part, namely Chapters 4, and 5, is devoted to the semicontinuous Boltzmann equation, which is such that, as already mentioned in Section 1.3, the velocity moduli are discretized, while particles can move in all directions in the space. In detail, Chapter 4, by Luigi Preziosi and Lamberto Rondoni, deals with the derivation of the equation and with the analysis of some quantitative results concerning the asymptotic trend to the hydrodynamic description. Chapter 5, by Wilfried Roller, shows how the semi-continuous equation can be derived in the framework of the extended kinetic theory. This chapter after having examined some properties of the collision term, the characterization of the Maxwellian state and the relaxation trend to equilibrium, deals with some interesting applications and specifically the equations for a chemically reacting gas. The third part, that is Chapters 6, 7, and 8, deals with the development of computational schemes by discrete velocity models. Specifically, Chapter 6, by Hans Babowsky, Daniel Gorsch, and Frank Schilder deals with the derivation of discrete velocity models with arbitrary large number of velocities and with application to interesting fluid dynamic problems. Chapter 7, by Lorenzo Pareschi, with the development of computational schemes and algorithms for numerical applications. Chapter 8, by Alexander Bobylev, with the analysis of convergence problems of discrete models to the full Boltzmann equation. The final part, namely the last two chapters, deals with the development of discrete velocity models in quantum kinetic theory. Chapter 9, by Wilfried Schiirrer deals with discretization schemes for quantum optics models with special attention to arbitrary partition of velocities and scaling procedures. Chapter 10, by Claude Buet and Philippe Cordier deals with the analysis of discretization schemes for Fokker-Planck type equations. The same chapter deals with the development and convergence analysis of computational schemes. 1.5

References

[1] Arlotti L. and Bellomo N., On the Cauchy problem for the nonlinear Boltzmann equation, in Lecture N o t e s on the Mathematical

From the Boltzmann

Equation to Discretized Kinetic

Models

15

Theory of the Boltzmann Equation, Bellomo N. Ed., World Scientific, London, Singapore, (1995). [2] Bellomo N. and Kawashima S., The discrete Boltzmann equation with multiple collisions: Global existence and Stability for the initial value problem, J. Math. Phys., 31, (1990), 245-253. [3] Bellomo N. and Gustafsson T., The discrete Boltzmann equation: A review of the mathematical aspects of the initial and initial-boundary value problem, Review Math. Phys., 3, (1992), 137-162. [4] Bellomo N., Le Tallec P., and Perthame B., Nonlinear Boltzmann equation solutions and applications to fluid dynamics, Mech. Review, 48, (1995), 777-794. [5] Bellomo N. Ed., Lecture Notes on the Mathematical Theory of the Boltzmann Equation, World Scientific, London, Singapore, (1995). [6] Buet C., A discrete-velocity scheme for the Boltzmann operator of rarefied gas-dynamics, Transp. Theory Statist. Phys., 25, (1996), 33-45. [7] Buet C., Conservative and entropy schemes for the Boltzmann collision operator of polyatomic gases, Math. Models Methods Appl. Sci., 7, (1997), 165-184. [8] Bobylev A., Palczewski A., and Schneider J., On approximation of the Boltzmann equation by discrete velocity models, Comp. Rend. Acad. Sci. Paris, 320, (1995), 639-644. [9] Cercignani C., Illner R., and Pulvirenti M., Theory and Application of the Boltzmann Equation, Springer, Heidelberg, (1993). [10] Gatignol R., Theorie Cinetique d'un Gaz a Repartition Discrete des Vitesses, Springer Lecture Notes in Physics n.36, (1975). [11] Gatignol R. and Coulouvrat F., Description hydrodynamique d'un gaz en theorie cine tique discrete: le modele reguliers, C. R. Acad. Sci. Paris, II, 306, (1988), 168-174. [12] Gorsch D., Generalized discrete models, Math. Models Meth. Appl. Sci., 12, (2002), 49-76. [13] Illner R. and Pulvirenti M., Global validity of the Boltzmann equation for two- and three-dimensional rare gas in vacuum, Comm. Math. Phys., 105 (1986), 189-203, and Global validity of the Boltzmann equation for a two- and three-dimensional rare gas in vacuum:

16

Lecture Notes on the Discretization

of the Boltzmann

Equation

Erratum and improved results, Comm. Math. Phys. 121 (1989), no. 1, 143-146. [14] Lanford O. Ill, Time evolution of large classical systems, in Springer Lecture Notes in Physics, 38, Moser E.J. Ed. (1976). [15] Longo E., Preziosi L. and Bellomo N., The semicontinuous Boltzmann equation: Towards a model for fluid dynamic applications, Math. Models Methods Appl. Sci., 3, (1993), 65-84. [16] Palcewski A., Schneider J., and Bobylev A., A consistency result for discrete-velocity schemes of the Boltzmann equation, SIAM J. Num. Anal, 34, (1987), 1865-1878. [17] Platkowski T. and Illner R., Discrete velocity models of the Boltzmann equation: A survey on the mathematical aspects of the theory, SIAM Review, 30, (1988), 213-255. [18] Preziosi L. and Longo E., On a conservative polar discretization of the Boltzmann equation, Japan J. Ind. Appl. Math., 14, (1997), 399-435. [19] Preziosi L., and Rondoni L., Conservative energy discretization of Boltzmann operator, Quarterly Appl. Math., 57(4), (1999), 699-721. [20] Rogier F. and Schneider J., A direct method for solving the Boltzmann equation, Transp. Theory Statist. Phys., 23, (1994), 313-338. [21] Kawashima S., Global existence and stability of solutions for discrete velocity models of the Boltzmann equation, L e c t u r e N o t e s in N u m e r i c a l Applied Analysis, 6. R e c e n t Topics in P D E , H i r o s h i m a , 1983, (1983), 59-85. [22] Kawashima S., Large time behaviour of solution to the discrete Boltzmann equation, Comm. Math. Phys., 109, (1987), 563-589.

Chapter 2

Discrete Velocity Models for Gas Mixtures C. Cercignani Dipartimento di Matematica, Politecnico di Milano,Italy

2.1

Introduction

In the last 25 years a considerable amount of research was devoted to discrete velocity models, after the pioneering work of Carleman [1], Broadwell [2], Gatignol [3] and Cabannes [4]. In the last 12 years it started to become clear that the discrete velocity models (DVM) were also becoming a tool to approximate the solutions of the Boltzmann equation, at a theoretical if not at a practical level [5; 6; 7; 8]. This work culminated in the paper by Bobylev, Palczewski and Schneider authors, who proved a consistency result for the DVM as an approximation of the Boltzmann equation [9]. With the exception of the case when all the momenta have the same magnitude and hence conservation of energy follows from the conservation of the number of particles, or when the lack of conservation of energy is interpreted as the occurrence of a chemical reaction [10], discrete velocity models of the traditional kind for mixtures have been introduced much more recently [11; 12; 13; 14; 15; 16]. The extension of DVM to mixtures seems impossible when the ratio of masses is irrational, but poses no special problems for the case of a rational ratio (this limitation is, of course, irrelevant in practice). In this paper we follow two lines; (Sect. 2) we first discuss the extension of the result of [9] to mixtures, thanks to a transformation of variables, then (Sect. 3) we discuss the case of models with a small number of velocities 17

18

Lecture Notes on the Discretization

of the Boltzmann

Equation

and the problem of spurious collision invariants; then a technique to enlarge the number of velocities is discussed, following [12] and an example with infinitely many velocities is given, following [15].

2.2

D V M for mixtures

According to standard definitions, a discrete velocity model of a gas is a system of partial differential equations of hyperbolic type (discrete Boltzmann equation), having the following form: dfn/dt + vn-dfn/dx = Qn{f,f),

QM,

f) = J2 Ik

C

^ / l / k - 2 knm/n/m, m

(2.1)

(2.2)

where v n are the discrete velocities (vectors of 5Rd) belonging to a prearranged discrete set, cnik and fcni are positive constants and the vector indices run from —p to p (here p is a vector with integer components, possibly infinity), whereas / „ are the probabilities (per unit volume) of finding a molecule at time t at position x with velocity v n = /in. We occasionally write / , as done in (1.2), for the collection {/ n }The coefficients cnik, knm must vanish if the following conservation equations are not satisfied: 1 + k = m + n,

(2.3)

|1| 2 + |k| 2 = |m| 2 + |n| 2 .

(2.4)

We remark that there are three different kinds of such models for the general Boltzmann equation, plus a simple model for a very special case in two dimensions with a "perpendicular law of scattering" (see [17] for a review). The most natural and popular model was first proposed by Goldstein, Sturtevant and Broadwell in 1989 [5]. The proof of consistency for this model was provided in [19] (see also [18; 19]). Let us consider a mixture with integer masses m i , m 2 , . . . ,ms where s is the number of species. Let us consider any pair of molecules with masses m,, rrij and let us put rrii = m, rrij = M (m < M). The usual Boltzmann equation for mixtures

Discrete Velocity Models for Gas

Mixtures

19

(with continuous velocities) has the following form dfi/dt

+ v-dfi/dx

(2-5)

= Y/QiJ>

where Qy = | d v d o ; | u | a ( M | u | 2 / 2 , u . a ; / | u | ) [ / ( v ' ) F ( w ' ) - / ( v ) F ( w ) ] ,

(2.6)

where / ( v ) = /j(v), F(v) = fj{w), and u = v_w

v' =

w

'

_ +

=

(2 7)

-

" = ^TM' l l w

77

u

i i

W

2.8)

(2-9)

T7-

K

m+ M ' ' M ' The first step toward obtaining a form of the collision term suitable for arriving at a discrete velocity model is to adopt u = v — w as an integration variable. We obtain: Qa = Jdudu\u\a(»\u\2/2,u-W/|u|)[/(v')F(w')-/(v)F(v

- u)], (2.10)

where the primed variables must be expressed according to M V = V+

'

^TM(|U|U,-U)'

(2 n)

-

Ttl

m+M

(|u|w-u).

(2.12)

We change again the variables by letting u = (m + M ) u and then omitting the tilda. We have: Q y = (m + M)d+1

f dudu\u\a(mM(m

+ M ) | u | 2 / 2 , u • w/|u|)

x [/(v - M u + M|u|w)F(v - M u - m | u | w ) - / ( v ) F ( v - ( J l f + m)u)].

(2.13)

20

Lecture Notes on the Discretization

of the Boltzmann

Equation

It will be useful, for any vector u', to write: + M)|u|2/2, u • u'/|u|2)

$ ( u , u') = \u\a(mM(m x [/(v - M u + Mu')F(v

- M u - mu') - / ( v ) F ( v - (M + m)u)]. (2.14)

A statement first made in [19] for the case of a simple gas, related to classical problems of number theory, was slightly generalized in [11]. The problem is full of subtleties and one can also provide an estimate of the error in the quadrature formula given below. Here is the generalized statement: L e m m a . Let \T/(u, u') : $td x 5Jd => 5ft be continuous and have a compact support. Let, for any d > 3:

Sfc(¥) = £ hd-^Em n e z

,

r

E

*(ph,*°h),

(2.15)

where |fi
as

h -> 0.

(2.16)

This lemma provides the desired approximation and hence a rule to construct discrete velocity models for mixtures with arbitrarily many velocities. The proof is essentially the same as in [19]. We remark that a byproduct of the study of DVM for mixtures is the possibility of showing that the coefficient of |v| 2 in the exponent of the discrete Maxwellian must be a function of the absolute temperature multiplied by the molecular mass. In fact, we can introduce the "discrete Maxwellians", which are of the form exp(ip) where if> is any collision invariant. If the only collision invariants for a single gas are the classical ones (i.e. if the model is normal [20], then the discrete Maxwellian are exponential of a second degree polynomial in v and can be written as the usual Maxwellians). When we consider a nontrivial model for a mixture of normal discrete gases, then equilibrium can be attained if and only if the coefficients of the second and first degree terms in the Maxwellians of each gas are in the same ratio as their masses (because of conservation of mass and momentum). In particular if the mixture is at rest just the coefficients of the second degree terms count and the only macroscopic parameter which

Discrete Velocity Models for Gas

Mixtures

21

must be equal for the two gases is temperature and the statement above follows. Thus the proof of this basic fact can be carried out within discrete kinetic theory without the necessity of resorting to a general argument of equilibrium statistical mechanics as done in [21].

2.3

Models with a finite number of velocities and the problem of spurious invariants

In this section we abandon the general approach of the previous section and provide examples of models with a small number of velocities. In order to obtain nontrivial models and respect all the symmetry properties suggested by physics, Bobylev and Cercignani [11] suggested that one must have at least 5+8 velocities. Their first model is defined as follows: a) The heavy particles (mass M) can possess one out of five velocities: w 0 = 0,

wi, 3 = {±2m,0},

w2,4 = {0, ±2m} .

(2.17)

b) The light particles (mass m) can possess one out of eight velocities: vi,3 = {±(M - m), 0},

v2,4 = {0, ±(M - m)} ,

v5,7 = { ± ( M + m),0},

v 6 , 8 = { 0 , ± ( M + m)}.

(2.18)

All the Broadwell-type collisions between identical particles with the same speed are permitted. Moreover, the following nontrivial collisions (with the scattering angle © = n) are possible: (w 0 ,v 5 ) -B- ( w i , v 3 ) ,

(w 0 ,v 6 ) -H- (w 2 ,v 4 ),

(w 0 ,v 7 ) <-> (w 3 ,vi),

(w 0 ,v 8 ) <-»• ( w 4 , v 2 ) .

(2.19)

A drawback of the first model is that it becomes "unreasonable" in the limiting case of two non-interacting species. The resulting DVM are unsatisfactory because we obtain two independent Broadwell models for light particles, and a Broadwell model plus the non-interacting particles with zero velocity for heavy particles. The fact that the model is unsatisfactory is confirmed by the fact that, already in 1998, H. Cornille [22] found that the first one has spurious conservation equations, in addition to those of mass, momentum, energy. Cornille has also investigated the possibility of finding exact solutions of this model in closed form [22].

22

Lecture Notes on the Discretization

of the Boltzmann

Equation

Because of the above unsatisfactory feature, Bobylev and Cercignani introduced a second non-trivial model, which is free from the first drawback. The main idea behind it is to allow some new collisions between the identical particles of the previous model, with the consequence of constructing a "reasonable" model for each component of the mixture even in the limiting case discussed above. A price to be paid for this improvement is that the new model has 9+16 velocities. There remains another difficulty, as pointed out by Cornille [22]: the system of heavy particles is regular but the system of light particles has two spurious collision invariants. The new model is defined as follows: a) The heavy particles, having mass M, can possess one out of nine velocities, 5 of which are the five velocities of the previous model. The new velocities are: w 5>7 = ±{m,m},

w6,8 = ± { - m , m } .

(2.20)

The new velocities are the result of the following collisions (with scattering angle 0 = TT/2): (W0,WI)

-B- ( w 5 , w 8 ) , . . .

(2.21)

b) The light particles, having mass m, can possess one out of sixteen velocities, 5 of which are the five velocities of the previous model. The new velocities are: V9,n = ±{M,m}, vi3,i5 = ± { M , - m } ,

v 10 ,i2 = ± { - m , M } , vi4,ia = ±{m,M}.

(2.22)

The new velocities are the result of the following collisions (with scattering angle 0 = TT/2): (VI,V5)O(V9,V13),...

(2.23)

This model is more complicated, but rather rich (many collisions are possible) and much more realistic. Yet there are many more possible collisions. Ref. [11] had to give them zero probability in order to avoid having two spurious collision invariants (with the exception of the case M = 2m). One can also describe the equations for the second model (which includes the first one as a limiting case) and examine the form taken by the collision terms [11].

Discrete Velocity Models for Gas

Mixtures

23

It is clear that the second model can be extended by adding new velocities appearing as a result of new collisions between particles of this model (with scattering angle 0 = n and 0 = 7r/2). Following this procedure, we can construct a sequence of DVM which consistently approach (the proof is very easy in this case) a continuous model, i.e. a system of Boltzmann equations. The differential cross section will have the very particular form: a(u, 0 ) = ff||(u)<J(e - TT) + a±(u)S(Q - vr/2).

(2.24)

At first this statement seems to contradict another statement in Ref. [ll], according to which in two dimensions matter of convergence are more complicated than for three-dimensional models. The contradiction is only apparent because we are considering a very special class of models with a limiting singular cross section. The first models without any problem with spurious invariants were proposed in [14], where it was also shown that the minimal number for a planar DVM can be lowered to 5+6 instead of 5+8, if one of the symmetries is slightly relaxed, in a way acceptable for problems depending on just one space coordinate (semi-symmetric models). It is remarkable that this model has the same structure for any mass ratio. Moreover it holds for irrational values of the mass ratio as well. In the same paper, the problem of the structure of a shock wave in a mixture was given a closed form solution for this model.

2.4

Constructing D V M with arbitrarily many velocities

In a second paper on DVM for mixtures, Bobylev and Cercignani [12] introduced a recursion procedure to construct models with arbitrarily many velocities. Their method will be now described. It is a method that can be applied to more general situations, e.g. polyatomic molecules with discrete internal degrees of freedom, because it considers an abstract Discrete Model, with a set of discrete states VN = {v1,v2,...vN}.

(2.25)

In applications, each v is a vector in some 3?m and includes all the discrete variables (velocity, internal energy, etc.). If the set X is given, we can simply denote the states by the integrs { 1 , 2 , . . . } . Particles may change

24

Lecture Notes on the Discretization

of the Boltzmann

Equation

their states by pair reactions

(*) + 0') -> (*) + (0,

(2-26)

with probabilities r*- > 0. These coefficients satisfy the following restriction (A) Tfj = 0 unless p > 1 conservation laws i>a(Vi) + fax{Vj) = 1pa(vk) +

faM

(2.27)

are fulfilled for a given set of linearly independent functions fa,..., fa. The functions fa,..., fa are called the physical collision invariants. If K denotes the number of possible reactions (2.26) of the model, the K is the number of different combinations {i,j\k,l}, 1 < i,j,k,l < N, such that Tfj ^ 0. It is useful to introduce the following Definition. The finite DM described above is called a normal DM if any nontrivial solution of the K equations

4>{vi) + 4>(VJ) = {vk) + 4>(vi),

(2.28)

for 1 < i,j,k,l 0 is a linear combination of the p physical collision invariants

fa,...,

fa.

This definition was earlier introduced for usual discrete velocity models for a simple gas by the author [20]. Sometimes one can guess the simplest normal DM such as, e.g., the modified Broadwell model for a simple gas. Then, according to Bobylev and Cercignani [12], there is a procedure to "breed" new models, starting from a given normal model VN = {v1,v2,...vN}S±{l,2,...,N},

(2.29)

and constructing its extended version. The latter is defined by a new set of states VN = {VI,V2,...VN,VN+1,...,VN+S},

s>2,

(2.30)

where all the new states {VN+I, • • • ,VN+S} satisfy the following condition: for any 1 < r < s there exist three numbers 1 < i,j, k < N such that r * J V = r > 0. In other words, each new state (N + r) is the product of a certain reaction

(i) + (j)^(k)

+ (N + r),

(2.31)

Discrete Velocity Models for Gas Mixtures

25

which includes three states {i, j , k} already present in the initial DM (1.29). Then the following basic statement is rather obvious Lemma. If the initial DM is normal, the same is true for the extended DM. For the proof see [12]. The same reference should be consulted for examples of the use of the construction indicated above. In particular, one can retrieve the semi-symmetric model with 11 velocities introduced in [13]. Independently of this paper, Cornille and Cercignani [14; 16] developed a class of planar semisymmetric models with 9, 11, 13, 15 velocities, and, by a suitable superposition, symmetric models with 17 and 25 velocities. In a subsequent paper, Cornille and Cercignani [15] constructed models with arbitrarily many velocities, for the values 2, 3, 4, 5 of the mass ratio. 2.5

Concluding remarks

In this survey we have first reviewed the result of [10] showing that the procedure used to approximate the Boltzmann equation for a simple gas by DVM can be extended to mixtures. Then, always following [10], we have reviewed a general procedure and indicated how to construct two models with a small number of velocities. By following this general procedure, however, one may easily construct models having more collision invariants than the physical ones. When Bobylev and Cercignani [11] started their research, there were no models (except trivial ones) for the conservative case. They thought that they had found, by going as far as 25 (=9+16) velocities, a satisfactory, simple model. This model had clearly several limitations and was not taken as a typical instance of the large class of models discussed in Sect. 2. The renewed interest for DVM and their potential use as tools to produce approximate solutions to the Boltzmann equation made it necessary to clarify the area of mixtures as well. Subsequent work showed that there are simpler and more satisfactory models [13]-[16], which were even used to solve in a closed form the problem of the structure of a shock wave in a mixture [14]. DVM for mixtures appear to have reached now a satisfactory level of understanding, comparable to that of DVM for a single gas. The research described in the paper was performed in the frame of European TMR (contract n. ERBFMRXCT970157) and was also partially

26

Lecture Notes on the Discretization

of the Boltzmann

Equation

supported by MURST of Italy. 2.6

References

[1] T. Carleman, Problemes Mathematiques dans la Theorie Cinetique des Gaz, Almqvist & Wiksell, Uppsala, (1957). [2] J. Broadwell, Study of rarefied shear flow by the discrete velocity method, J. Fluid Mech., 19 (1964), 401-414. [3] R. Gatignol, Theorie Cinetique des Gaz a Repartition Discrete de Vitesses, Lectures Notes in Physics, 36, Springer, Berlin, (1975). [4] H. Cabannes, The discrete Boltzmann equation (Theory and application), Lecture notes, University of California, Berkeley, (1980). [5] D. Goldstein, B. Sturtevant, and J.E. Broadwell, Investigations of the motion of discrete-velocity gases, in "Rarefied Gas Dynamics: Theoretical and Computational Techniques", Edited by E.R Muntz, D.P. Weaver and D.H. Campbell, (1989), 100-117, AIAA Washington. [6] T. Inamuro and B. Sturtevant, Numerical study of discrete velocity gases, Phys. Fluids A., 2 (1990), 2196-2203. [7] F. Rogier and J. Schneider, A direct method for solving the Boltzmann equation, Transp. Theory Statist. Phys., 23 (1994), 313-338. [8] C. Buet, A discrete-velocity scheme for the Boltzmann operator of rarefied gas-dynamics, Transp. Theory Statist. Phys., 25(1996), 33-60. [9] A.V. Bobylev, A. Palczewski, and J. Schneider, Discretization of the Boltzmann equation and discrete velocity models, in "Rarefied Gas Dynamics 19", J. Harvey and G. Lord, eds., Vol. II, (1995), 857-863, Oxford University Press, Oxford. [10] G.L. Caraffini, and G. Spiga, Some remarks on the extended discrete kinetic equations, Transp. Theory Statist. Phys., 23 (1994), 9-25. [11] A.V. Bobylev, and C. Cercignani, Discrete velocity models for mixtures, J. Statist. Phys., 91 (1998), 327-341. [12] A. V. Bobylev and C. Cercignani, Discrete velocity models without nonphysical invariants, J. Statist. Phys., 97 (1999), 677-686. [13] C. Cercignani and H. Cornille, Shock waves for a discrete velocity mixture, J. Statist. Phys., 99 (2000), 115-140. [14] H. Cornille and C. Cercignani, A class of planar discrete velocity models for gas mixtures, J. Statist. Phys., 99 (2000), 967-991.

Discrete

Velocity Models for Gas

Mixtures

27

[15] C. Cercignani and H. Cornille, Large size planar discrete velocity models for gas mixtures, J. Phys. A (Math. Gen)., 34 (2001), 2985-2998. [16] H. Cornille and C. Cercignani, "On a class of planar discrete velocity models for gas mixtures", in Proceedings "WASCOM 99" 10th Conference on Waves and Stability in Continuous Media, V. Ciancio, A. Donato, F. Oliveri, S. Rionero, eds., World Scientific, Singapore, (2001). [17] V. Panferov, Convergence of discrete velocity models to the Boltzmann equation, Research report No. 1997-22/ISSN 0347-2809, Dept. of Mathematics, Goteborg University, (1997). [18] A.V. Bobylev, A. Palczewski and J. Schneider, On approximation of the Boltzmann equation by discrete velocity models, C.R. Acad. Sci. Paris, 320 (1995), 639-644. [19] A. Palczewski, J. Schneider and A.V. Bobylev, A consistency result for discrete velocity schemes for the Boltzmann equation, SIAM J. Num. Anal., 34 (1997), 1865-1883. [20] C. Cercignani, Sur des criteres d'existence globale en theorie cinetique discrete, C.R. Acad. Sc. Paris, 301 (1985), 89-92. [21] C. Cercignani, Temperature, entropy and kinetic theory, J. Statist. Phys., 87 (1997), 1097-1109. [22] H. Cornille, Private communication, February (1998).

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Chapter 3

Discrete Velocity Models with Multiple Collisions R. Gatignol Universite Pierre et Marie et Laboratoire

de Modelisation

Curie

en Mecanique

UPCM-CNRS

Paris - France

3.1

Introduction

In discrete kinetic theory, the main idea is that the velocities of the molecules belong to a given finite set of vectors. The Boltzmann equation is replaced by a system of equations with an interesting mathematical structure. The two first models with six or eight velocities have been introduced by Broadwell (1964) with the purpose to describe the shock wave structure [6] and the Rayleigh problem [5]. He was successful and since many other works are realized. In 1970, general models with an arbitrary number of velocities are settled up [23]. They allow the replacement of the Boltzmann equation by a system of partial differential equations having its main properties but more easy to analyze for a mathematical or numerical point of view. These first works deal with binary collisions only [23,24]. Without going into details, the very large number of papers on the discrete theory can be arranged in four groups. First, many papers concern the mathematical proofs of the existence and uniqueness of the solutions for the kinetic equations with initial data [2,7,28,33,36]; there are also some results on the boundary condition problems [9,10]. A second set of works discuss the trends of the kinetic equations to the hydrodynamic description [1,12,13,15,30], specially for small Knudsen numbers. A third group of papers are devoted to the mobilization of the physical phenomena like the 29

30

Lecture Notes on the Discretization

of the Boltzmann

Equation

structure of the shock waves [6,24,30], the flow or the heat transfer between two plates [17,20,26,27], the problem of evaporation and condensation on a condensed phase [8,18,19,31], and so on. According to the selected models, it is possible to obtain exact solutions [5,6,8,20,25,30] or numerical solutions [26,27,31]. In a fourth set, we arrange the papers interested in the gas mixtures with or without chemical reactions [4,16,29,30]. At last before going on, we pay particular attention to the recent mathematical works on the connection between the discrete models and the true Boltzmann equation. The paper of Bobylev, Palczewski and Schneider [3] gives a consistency result for a discrete velocity model of the Boltzmann equation. The aim of the present paper is to describe the general model of a discrete gas with multiple collisions. In Section 3.2, the multiple collisions are introduced and the kinetic equations are given. Due to the velocity discretization, it is well known that it is possible to have "spurious" summational invariants. Consequently the physical understanding of the physical results is confused. The multiple collisions contribute to the decrease of the dimension of the summational invariant space. So, our purpose is to reduce the number of summational invariants and, if possible, to obtain the physical invariants, mass, momentum and energy, as the only ones. In Section 3.3, we pay attention to the hydrodynamic description of the so-obtained gas. Specially, we shall see that the Euler equations associated with a class of quasi isotropic models are very closed to those of the continuum case. Then, in Section 3.4, the boundary conditions on a wall or on a condensed phase are investigated. For the Couette flow between two parallel walls, the evaporation / condensation problem between two parallel interfaces and the evaporation / condensation in an half space, the mathematical question of the existence of solutions is a little approached. On the contrary, numerical solutions are emphasized. We shall see that they are in a very good agreement with the solutions given by direct treatments of the continuum Boltzmann equation. We shall conclude that the discrete models can be used to understand the fundamental physical problems related to the rarefied gas dynamics.

Discrete Velocity Models with Multiple

3.2

Collisions

31

Discrete Models with Multiple Collisions

The gas is composed of identical particles of mass m whose velocities are restricted to a given finite set of p vectors: ~ui,li2, • • •,^u v - The number of particles with the velocity ~Uk (called particles "fc") per unit volume is denoted by Nk- The macroscopic quantities are defined by: n = J2Nk>

n~Tt = J2Nk~Uk,

k

ne = - ^ i V f c ( l ? f c - l ? ) 2 , (3.1)

k

k

and E-e+-'u2,

p = nm, Zt

pe=—nkT,

(3.2)

Zi

where J2k *s ^ or * n e summation from k = 1 to k = p. As usual, n is the total density, it the macroscopic velocity, E and e the total energy and the internal energy per unit of mass respectively, D the dimension of the physical space and T the temperature. This T is called kinetic temperature and is different from the thermodynamic temperature. As it is now well known, the definition of the temperature given by the continuous kinetic theory is no longer valid in discrete kinetic theory [11]. 3.2.1

The

"r-collisions"

The first theories [23,24] with the binary collisions are only generalized to the multiple collisions [12,13,20]. By definition, an r-collision involves r particles (r > 2). Before describing such a collision, we introduce some notations: Ir = (ii, 12, • • •,i r ) is for ii,h,... ,ir taken in the set ( 1 , 2 , . . . ,p) and £r is for the set of all the r-sets Ir. Let there be an r-collision where the r particles have respectively the velocities u j , , u , 2 , . . . , Uir before the collision and the velocities u j 1 , ~Uj2,..., ~Ujr after the collision. This r-collision is denoted by Ir —> Jr with Ir = (i\,i2, •. • ,ir) and Jr = (ji,J2,---,jr)Of course it must bear out the conservation of momentum and energy (the mass conservation is automatically borne out):

E ^ = E ^*' ifc6.Tr

jk&Jr

E ™i = E ^l • (3-3) ik€lr

jkGJr

32

Lecture Notes on the Discretization

of the Boltzmann

Equation

A transition probability Af is associated with each r-collision Ir —• Jr, so that the number of such r-collisions per unit volume and unit time is AjrNir where Nir denotes the product N^N^ ... Nir. As in the binary collision theory, the transition probabilities are strictly positive (they are taken equal to zero for the unrealizable r-collisions). It is assumed that these probabilities satisfy the hypothesis of the microreversibility (I), or more generally, of the semi-detailed balance (II): I:

AJi:=A%,

II:

£

AJ; = ] T ATfr.

It must be understood that all the physical phenomena present in the collisions are taken into account inside of these probabilities. Now the main point is to write a balance equation for the number density of particles "fc". It is important to remark that through the r-collision Ir —* Jr between r particles we can obtain three different situations [12]. The number of particles with the velocity ~Uk can be conserved, decreased or increased. We denote by 5(k, Ir) the number of indices fc present in the r-set Ir and by <5(fc, Jr, Ir) the algebraic number of particles "fc" created through the collision Ir —> Jr. Of course we have: 5{k, Jr, Ir) = S(k, Jr) — 5(k, Ir). So the algebraic gain of particles "fc" created through the r-collision Ir —• Jr per unit volume and time is: 6(k,Jr,Ir)AJ;NIr.

3.2.2

The kinetic

equations

The kinetic equations are the balance equations for the densities Nk, k = l,...,p. It is interesting to consider the set of kinetic equations with only the r-collisions (with r fixed), and those with all the r-collisions with r going from 2 to a fixed number P (r — 2 , . . . , P). They are now written:

^

+ ^-^tffc = ££«(Mr,/rM/;W,r,

fc

= i,...,p,

(3.4)

where ^Z 7 is for J2ir^s • * n *^ e particular case of the non-trivial binary collisions (r = 2), 5(k,l2) is always equal to 1 in a direct collision, and

Discrete Velocity Models with Multiple Collisions

33

S(k, J2) equal to 1 in an inverse collision. In that case, Eqs. (3.4) are the usual kinetic equations with binary collisions only [23]. The p Eqs. (3.4) can be written in a condensed form: dN -> - + A - V N = F(N),

(3.5)

where N = (Ni,N2, • •., Np) £ R p and where A • V is a diagonal matrix operator with the diagonal elements equal to ~Uk • V. By using the relation 6(k, Jr, Ir) = S(k, J r ) — S(k, Ir) and the assumption (II), it is easy to prove that the fc-component of J-"r(N) is written:

^ r w = E E *(*-J- wtNir = E E *(*• u (AliN->r - M:N^) = E E*(*' WJI ( ^ ~Nir)-

(3-6)

The kinetic equations with all the r-collisions (r = 2 , . . . , P) are:

Yl J2J£S(k,Jr,Ir)AJi:NIr=Ck(N),k = l,...,p,

^+Hk.VNk=

r=2,...,P IT

Jr

(3.7) or in a condensed form:

^4-A-VN-

Y,

^ r (N)=C(N).

(3.8)

r=2,...,P

All the properties established for the discrete Boltzmann equations with only binary collisions become general for the kinetic equations (3.5) and (3.8) [12,13]. Let us notice particularly that:

J2 Vk ^T(N) = E E E ^ *(*> V {AllNJr - AJ;NJr) fe

k

Ir

Jr

= E E E Vfc *(*« ^) (A£NIr - A%NJr) = lT,Y,T,^S(k>Jr>Ir)(AiNJr-Ai;NlT) k

Ir

Jr

. (3.9)

34

Lecture Notes on the Discretization

3.2.3

Summational

of the Boltzmann

Equation

invariants

At once we give the definition of the summational invariants: they are attached to the conservation properties through the r-collisions (with r fixed) or through all the r-collisions (r = 2 , . . . , P). They are defined as the pcomponent vector 4> = {fii • • • >¥p) £ W satisfying in the first case, the conditions AJfr^5(k,Jr,Ir)Vk fe

= 0,

VJr,

VJ r ,

(3.10)

and in the second case, A/;^<J(fc,Jr,/r)^=0,

Vr = 2 , . . . , P ,

V/r,

VJr.

(3.11)

k

The summational invariants associated with the r-collisions only and defined by (3.10) generate the linear subspace F r (of R p ) . Those associated with all the r-collisions (r = 2 , . . . , P ) and defined by (3.11) generate the linear sub-space F. Let an r-collision be given. It is possible to construct an (r-(-l)-collision by taking these r particles and one extra particle which does not change its velocity during the collision. In other words, let Ir —» Jr, it is always possible to add one "i r +i" particle and to construct the (r+l)-collision: (ilt.. .,ir, i r + 1 ) —> ( j i , . . . ,jr, ir+i)- So F r + 1 C F r (we recall that the transition probabilities are strictly positive for the realizable collisions). Consequently: F C WP C . . . C F r + i C F r . . . C F 2 .

(3.12)

The physical invariants (<£>& = l,~Uk, \~u\) belong to F. In contrast to the continuum kinetic theory for monatomic gases, the geometric character of the set of the given velocities may allow other summational invariants (socalled spurious invariants). By taking into account the multiple collisions with P increasing, we reduce the dimension of F. In other words, for some models it is possible to eliminate some spurious invariants. 3.2.4

A remark

about the dimension

of the space F

It is important to emphasize that it is possible to find the dimension of F without an explicit determination of all the collisions between the molecules. However, it is necessary to take into account all the multiple collisions. We pay also attention to the fact that the number of independent multiple

Discrete Velocity Models with Multiple

Collisions

35

collisions is finite [13,14], (the definition of the independent collisions will be given further). The original ideas and proofs are in the paper of Chauvat [14]. To each vector u k of the discrete model, we associate the vector Uk belonging to R D + 2 , (k = 1 , . . . ,p), by setting: C/j5 = {~uk)j for j = 1 , . . . , D, Up+1 = ~u\ and UQ+2 = 1, where (~uk)j denotes the j-th component of the velocity ~uk. Then Eqs. (3.3), considered for all the r-collisions, r = 1,2,..., can be written: A/;^J(fc,Jr,/r)[/fe=0, fe

Vr = 2 , 3 , . . . ,

,Wr,

VJr.

(3.13)

Let ( m , . . . , rip) be a p-uplet of Z p . Equations (3.13) can also be written in the form: ^ n f c = 0,

Y^n^k=0,

]Trafe"u£=0,

(3.14)

by taking nk particles with velocity ~vtk before collision if nk > 0, whereas the coefficients nk < 0 correspond to the velocities after the collision. Let us note that a vanishing coefficient nk = 0 can be related to any number of velocities conserved through the collisions. Now, let ( n i , . . . , np) be a p-uplet of Z p such that: Y,nkUk

= Q.

(3.15)

The knowledge of the solutions ( n i , . . . , np) G Z p of Eq. (3.15) is equivalent to the knowledge of the rational number sets (qi,..., qp) of Q p such that: X>tf*=0.

(3.16)

k

We define G p as the set of the solutions (qi,. • • ,qp) G Q p of System (3.16). With the usual addition and multiplication operations, G p has the structure of a Q-linear space (it is a sub-space of Q p ). Of course dim G p < p. Let G = (qi,..., qp) be one element of G p . There exists a sequence of p integers ( n i , . . . , np) such that nk = Xqk, where A is an integer independent of k. Consequently, there exists at least one collision Ir —• Jr between r

36

Lecture Notes on the Discretization

of the Boltzmann

Equation

particles which is associated with the set {n\,..., np) and consequently to G. To this collision, it corresponds a sequence of p integers: 5(k,Jr,Ir), k = 1 , . . . ,p. This sequence satisfies relations (3.13). In other words, it is a particular element of G p . Now let a number a of elements of G p which are assumed Q-linear independent. Let a collisions ITa —> JTa which correspond to them. The a sequences S(k,Jra,Ira) of p integers associated with these a collisions are also Q-linear independent elements of G p . The associated collisions are defined as independent collisions. We return to the definition of F. All collisions are taken into account, therefore: cf> = (ipi,...,ipp)

6 F ^ A / ; ^ ( 5 ( f c , ; r , 7 r ) W = 0, Vr = 2 , . . .

,VIr,VJr,

k

or equivalently: 4> = ('52qk
V(9I,...,9P)GGP.

(3.17)

k

The range of the linear system (3.17), where the unknowns are (
SQ = {X>tffc,

ft£Q

,

SR = J5>fc[/fc, rfeGR

We have: dim S R < D + 2 and

dim S R < dim S Q .

Let us introduce the linear application £ from Q p on S Q defined as follows:

C{qi,...,qp)

= ^2qkUk. k

Discrete Velocity Models with Multiple

Collisions

37

The kernel of £ is the previously introduced space G p . So: dim S Q — p — dim G p , and consequently: d i m S R < d i m S Q = dimF

(3.18)

Thus, we have obtained the dimension of the summational invariant space F without explicitly determining the collisions between the particles. This dimension is equal to dimSQ. It is important to note that all the multiple collisions must be taken into account. If all the components of the vectors ~Uk are rational, then dim S R = dimSQ. Indeed it is easy to see that if the set (f/ 1 ,..., Us) is a basis in the space S R , it is also a basic in S Q . Then: dim S R = dim S Q = d i m F .

(3.19)

If we assume that dim S R = D + 2, then the space F is reduced to the physical invariants (i.e. mass, momentum and energy invariants). This is the case of the Broadwell models [5,6], or the Cabannes models [8], (the details are given in the paper of Chauvat [14]). More, let us consider the space models such that: ~Uk = &k i + &fc j + Ckk , k = 1 , . . . ,p, with (ak,bk,Ck) € Z and such that ~u\ ^ ~uf for at least two indices k and I. Then the main result is the following [13,14]: the dimension of F is exactly 5. Of course we have a similar result for the coplanar models with a dimension for F equal to 4. As examples, the dimension of F is 5 for the space models related to the cube, that is, when i , j , k are three unit orthonormal vectors. The dimension of F is 4 for the coplanar models constructed on a square lattice ( i , j are two unit normal vectors) or an hexagonal lattice (i , j are two unit vectors with an angle of 7r/3).

3.3

Macroscopic Description

Prom now on, the considered models are such that dimF = 5 (space models) or dimF = 4 (coplanar models). In other words, we make the assumption that there exists an integer P for which the dimension of F is 5 or 4. (Of course, depending on the model, it is possible or not.) The summational invariants are reduced to the physical invariants only: (fk = m, (pk = m~Uk,
38

Lecture Notes on the Discretization

3.3.1

Mean

balance

of the Boltzmann

Equation

laws

By multiplying the kinetic equation (3.7) for the density JVfc by the physical summational invariants, by making the summation on k = 1 , . . . ,p, and by using the formula (3.9), we obtain the macroscopic balance laws for the mass, momentum and energy in a classical form: d_p + V • (pu) = 0,

(p = nm),

(3.20)

+ ^ • P = 0,

(3.21)

di

jt{pu) + ^ • (p^) d_ P[e + dt

+ V- p\ e

u +F-u +~q

0,

(3.22)

with:

V

= Y1 Nk^k ~ ^ 0 " * - ^ ) '

(3.23a)

and ~q = ir^NkCuk

3.3.2

H-Theorem,

Maxwellian

~ ~u)2CiJtk - ~u) •

(3.236)

state

The Boltzmann H-function is denned by H = £)fc AffclniV/.. For a spatial uniform gas, the computation of dH/dt yields the H-Theorem: dH_ <0. dt Indeed due to the assumption (II), we have:

£5>$;(JVJr-JVJr)lntf/r If

Jr

T,T,AjMr-N^laN^+Njr-NIr]

= Ir

Jr

j2^2Aj:{NJ^Nir-^Njr)+NJr-NIr}. ir

j

r

Discrete Velocity Models with Multiple

Collisions

39

Moreover we have the equality: £*(Mr)lnJV f c =lnJV J r . k

So by using these last equalities, the kinetic equations (3.7) and the first equality in (3.9), we can write: dH

dNk

dN

k — = ^^ r( •l , .+^ l n AT\ A r f e ) _ = ^Y l n^iiV fATe — k

=2

k

Y,J2{AliNJr-A£Nu)lnNIr

E r=2,...,P

Ir

Jr

= 2 E £X>£(^-^)lniV/, r=2,...,P Ir

=2

E

Jr

J2HAi:N^^Njr-lnNIr)

r=2,...,P Ir

+

NIr.-Njr.

Jr

Each term of the sum of the last right hand side is negative or zero. So the H-Theorem is proved. By similar arguments to those used in the binary collision theory [23], it is easy to establish that the four following properties are equivalent: a) l n N e F ,

InN = (lniVi,lniV 2 ,... ,lniV p ),

b)J2lnNkCk(N)

=0

k

c)C(N)=0 d)AJ;(NJr-NIr)^0,

Vr = 2 , . . . , P ,

V/ r , V Jr.

(3.24)

If we take into account the r-collisions only (r fixed), we obtain similar results. In a) F is replaced by F r . In b) and c), C(N) is replaced by Jrr(N). And in d) the equality to 0 is written only for any Ir and Jr . As usual, the Maxwellian state is defined by In N £ F, with InN = (In JVi,...,InN p ).

40

Lecture Notes on the Discretization

of the Boltzmann

Equation

Consequently we can write: Nk = - exp (a + •/? • Hk + 7 "t?2) .

(3.25)

The correspondence between the variables (a, /3,7) and the macroscopic quantities (n,lt,e) defined in (3.1) is a bijection [23]. There exists one and only one Maxwellian state associated with the given quantities

(n,li,e). 3.3.3

Euler equations

Let us consider a discrete gas in a local Maxwellian state: The densities Nk are Maxwellian and depend on time and space variables through the macroscopic variables (n, ~u, e) only. In that case the balance laws (3.20) to (3.22) are the Euler equations associated with the model. To compare the Euler equations derived from the discrete kinetic description with the equations of the continuum fluid dynamics, there are some difficulties due to the lack of isotropy of the models. So, we have proposed in [12], a particular class of models called good models which generalize the models used in the lattice gas theory [21,22]. Now, we are going to give a description of these models. First we arrange the set U = (l?i, 1?2, • • • > ~u*p) of the velocities in L subsets Ul, (1 = 1,...,L), and we denote by ~u\, (i = 1 , . . . ,pi) the velocities of Ul (p± + p2+ •• • +PL = p)- For the velocities ~u\ 6 Ul we assume the following properties: a) \\~uli\\ = ci b) ~u\ and c

)p\ ^

depending on / only, — ~u\

£

are both in Ul,

uW = ±c*I,

(3.26)

i=i,...,P,

where I is the unit tensor such that Iap = Sap (where 8ap is the component of the Kronecker tensor). These hypotheses are useful to have some quasiisotropic properties for the macroscopic gas. In addition we put: ,

l=l,...,L

y

k

vl/2t

««= j £ l W \

y

k

J

Discrete Velocity Models with Multiple

Collisions

41

and consequently a\ = c. Moreover, we assume:

p

k

\

p

k

/

In general, for an arbitrary model, it is not possible to give the Euler equations in an explicit form. However, for some particular models, we can write these equations in a nice form as now we show it here-after. Let be consider a model of gas with the properties (3.26). If all the densities Nk are equal to n/p, the gas is in a Maxwellian state with the mean density n, the mean velocity ~u = 0 and the total energy equal to EQ. Here: 2

2p^

Now, let there be a flow having a small mean velocity ~u (\\li\\ < c ) and a total energy E closed to EQ (E — EQ
Y,Nk = n,

^JV f c l? f c =nl?,

,

(3.28)

^Y,N^l=nE-

(3-29)

The parameters /3 and 7 do not depend on n and are given by the two relations: ~u = - Y^~™kexp p k

f 0 • ~uk + -ylxl)

I ^ e x p (/3 • "ufc + l~u2k) ) V k J

(3.30) 2p

E=

•„

fyH ^

exp

(J* " ^k + ^2k) I 5 Z e x p (J* """ k + T""(3.31) *) ) •

The correspondence between (/3,7) and (~u, E) is a bijection [24]. The values u = 0 and E = EQ correspond to /3 = 0 and 7 = 0. As in lattice gas theory [21,22], it is possible to obtain explicit expressions for f3 and 7 by

42

Lecture Notes on the Discretization

of the Boltzmann

Equation

performing expansion about the values u = 0 and E = EQ. It is natural to take EQ as a reference energy and c = y/2Eo as a reference velocity. Then, we denote by 6U and 8e the orders of magnitude of ~u/c and (E — EO)/EQ. By analogy with the lattice gas theory, we look for the densities Nk some expressions in ~u and e = E — Eo- So, we look for /3 and 7 some expansion in ~u and e. In order to give a very simple presentation of the calculus, we try for /? and 7 the following expansions: + 0{53u) + 0(8e52u)),

? = ft «T + fcell + c-HOtf) 7 =

7lg

+

72 1?2 + c - 2 ( 0 ( j 2 )

+ 0(J3)

+

(3.32)

0(^2)) ;

(3 33)

where the scalars fix, /?2, 7i and 72 are to be determined. To obtain these scalars, we put (3.32) and (3.33) into (3.30) and (3.31). Then, we perform the expansion of the exponential terms. At last, we identify the terms in 1?, e, e~u and I ? 2 . The calculus are in detailed. At the orders of magnitude previously given and after some algebraic calculus, we obtain the expressions: - l + /3iw>fc-!? + 7i«>fce + 72« >2 : l? 2

expf/3 •itk+'T^lj + 02{^k-^)e

- J 3 exp (j3 -llk

+ Pil\^l(^k-^)e

+ ylt£)

~ 1+

+ -0(likUk

7lc

2

e +

72c

2

•• ~u~u ,

l?2 + ^ l ^ ^

(3.34)

, (3-35)

fe ~ 5 Z ^k

exp

( ^ ' ^k

+

1~Uk) -

"Q/^IC 2

w* + -p/?2C2el? +

—fluxajelt, (3.36)

ex

- J3 "1 P P ' ^ + 7^l) ~°

2+

2

wfc + 72fl2^ + 2Q ft4^2 • (3.37)

We put these expressions in (3.30) and (3.31) and we arrive at: (1 + 7ic 2 e)l? ~ -^(ft + ^)(?lt

+

—pincfiet?,

Discrete Velocity Models with Multiple

43

Collisions

and 1 + 7ic 2 e +

72c

2

l ? 2 + ^ / ? 2 c 2 " " 2 ) ( e + •£?

+ 7ia^e + 72a^ u 2 + j ^ / 3 ! ^

U

J.

and finally: #1

£> ~

(£> 7l - /32)c2 = /? l 7 l a*,

~2 Cz >

and, recalling that c = a\, one has: A ^ Pl = ~2 ,

2 71 = - 4

D_

2£> 4 .

P2 =

4- ,

72

(3.38)

It remains to calculate expa. By using the definition of n, we have: - ( e x p a ) ^ ] e x p ( / 3 • l u f c + 7 ' u 2 . J = 1. With (3.34) and (3.38), this relation yields: (3.39)

expa ~ 1 — 71c e.

We return to the expressions (3.28) for the densities Nk- After some algebraic calculus and by using the approximations (3.34) and (3.39) with the values (3.38) for the parameters /?i, /3 2 , 7i and 7 2 , we arrive at: 2[1?2 - a 2 ]

n f D ^ _^ JVi• = - s i + a T ^ p I i n ("2L

+

u t — at a 2 — a-f

4„4

2"1 aza

D2 2a\

«fc

H • l?fe] e + 0(<5e2) + O ( ^ ) + 0 ( M af (3.40)

With these expressions for the densities, we can give explicit expressions for the pressure tensor P and the heat flux vector ~q . They are obtained

44

Lecture Notes on the Discretization

of the Boltzmann

Equation

by replacing in the definitions (3.23) of P and ~q, the densities Nk by their expressions (3.40). So: P = m^2Nk

(~Uk - ~u) C^k -li)

= m^2Nk~Uk~Uk — pu~u

1 )I, +, 2 ^ - a j ] .

"'?<

74

- r t

4

-^2 1 D2 ""Wfc - ^ 1 : u u > ukuk + 1a\ L

—puu

and finally: P=

/ 9

/la2l+|eI+^

K--^n

: -tinX-pHH,

(3.41)

with:
•

In the same way: q = -^l^Nk{uk-

u) (uk-

= — 2_^Nk(Uk - u) 1

> pDflr.^^.,

1 +

P

t

\

2

.4 "2

uk-peu

k

= y ^2Nk(uk

~^2^

u)

_

_»

- 1~uk • ~u)Ttk ~ peu

2^/^-af

„ 4 e ) "fe^fel /

a

U

-P^U,

1 \__

kUkUk-U

(3.42)

Discrete Velocity Models with Multiple Collisions

45

where only the terms in 0(6%) and 0(5e5u) are conserved. With (3.27) we arrive at: 1«2

q=

-* , * (

pu+

a

Q a

\

a

l 2

2

^[A^-4^r^)reu

2^

o? _> 2ai __> 2a\ ~n^u-7v^4 i\Peu+Jv~^—-zrpeu

_

L>

__> _> -peu ,

,„ , „ , (3.43)

J-'(a2 — a i )

iJ(o,2 — a[)

and finally:

'

2

- M pit + -o L>y

n

-F7 + 4/ 1

af V£>

J N P"? - pel?. (3.44)

af(a|-a?)

In the general case, the tensor P is not spherical. In the same way ~~q is not equal to zero. The Euler equations are the balance laws (3.20), (3.21) and (3.22) with these expressions (3.41) and (3.44) for the pressure tensor P and the heat flux vector if. We are continually constructing explicit expressions for the Euler equations. To this end, we give an explicit expression to P • ~u + ~q . At the chosen order of magnitude one has: _> _> /a? 2 \ _> 1 (a\ P-+Y = p(^+^elJ.1? ^a\\f -_>^ + m

/ +

2

a\a\ - a\ \ __> _> H f\ p e u - p e u

"TI + T

= 7T^P U + - 4 7 - 4 2a\ af(a2

_ a

4TPeu -peu . i)

Then, always at the considered order of approximation, one has: B i t = [ e + ~ui \u=eu, id

I

e = E- -a{ = e + -uz - -a{ , (3.45a) Jit

Zi

= ^EI + p(J^K--^m) :ltlt-pltlt,

£t

(3.456)

46

Lecture Notes on the Discretization

of the Boltzmann

Equation

and u + q

a a

a

i( 2 ~ \)

a\a\ + a%d\ 2a% pit. 2a$a\ - a$

\)pEli

(3.45c)

The momentum equation, using the above expression for P, can be written as follows: 4

|G>T?) + ?

2of

2pE D

u u +V

2af

0.

Finally, the two Euler equations (3.21) and (3.22) are written: D2

d_

di (pit) + V

a%

2

-Vw,

u u

(3.46)

a\a\ + a%a\ - 2a\ pu 2a\{a*-a\)

d_ dt

**>+HM3>>**)-H

(3.47) with a pressure defined by w = 2p e/D. Equation (3.46) is the balance law for the momentum with a modified material derivative in the left member. This modification is similar to those present in the lattice gas theory [21,22]. The evolution of the total energy is given by Eq. (3.47) in which the right member contains the divergence term V • (pit) and not, as usual, the divergence term V • (wit). This is correct at the first order: Indeed in the first approximation: 1 1 v * —>2

•*• 2

= - - z > u k% = - c 2p -f 2

j

and

c

w = —p, DH'

and then, the sound velocity is c. If the discrete models are such that the tensor K is isotropic,

that is:

4

Ka/3yS

= (Sa/3<)-t6 + ficrySpg +

SaS^f}",)

D(D + 2)

then Eq. (3.46) becomes: — (/>!?) + r$ • (p~ult) = -V

(w +

— -pu

(3.48)

Discrete Velocity Models with Multiple

Collisions

47

with Ba\ (D + 2)a\ ' On the left of (3.48), there is a modified material derivative and on the right, the pressure is modified. If r] = 1, Eq. (3.48) is exactly the continuum momentum equation. The parameter rj seems important to measure the ability of regular models to take advection phenomena into account. Now, it is interesting to give one example [15]. So, we take 18 velocities defined in the following way: ~u\ = ci (cos(i7r/3)T + sin(i7r/3)y) ,

i = 1,2,...,6,

Z = 1,2,3

c 3 = 2c2 = 4ci where ( i , j J are two normal unit vectors. This coplanar model has the physical summational invariants only (dim F = 4) if we take into account all the multiple collisions. As a matter of fact, it is sufficient to consider the five-collision only (F5 = F). For this model:

77

D4 _ 1 iiEJI^fcll 4 - {D + 2)a\-2U u

13 n o o Y~U~

Applying the Chapman Enskog method, it is also possible to obtain the Navier-Stokes equations associated with the models described in the beginning of Section 3.3.3. Always with the hypothesis 6U ), where T is the kinetic temperature, E> the deformation rate tensor and K, A and fi the usual transport coefficients. Shortly, the regular models with their symmetries and their basic assumptions about the space of summational give conservation equations which are simple and very closed to the classical ones, especially if the tensor K is isotropic.

48

3.4

Lecture Notes on the Discretization

of the Boltzmann

Equation

Boundary Conditions for Discrete Models

We investigate in this section, the conditions which must satisfy the densities Nk of the gas on an impermeable wall or on its condensed phase. The wall and the interface in the second situation are denoted by S. At each point M of S we introduce the unit vector ~n normal to S pointing into the gas, the velocity Ttw and the temperature Tw of the wall or of the condensed phase. The total energy Ew in M of S is then defined as the sum of the kinetic energy and the internal energy of a gas having the kinetic temperature Tw, that is:

We define a discrete gas in Maxwellian equilibrium with the boundary S as the discrete gas whose microscopic densities are the densities Nkw of the Maxwellian state associated with 1, uw and Ew [25]. Near the point M of S, we arrange the p velocities of the model into two groups, corresponding to impinging molecules and emerging molecules. This leads to the partition of the set of velocity numbers into two sets: / = {i; ("ui - "u) • "n < 0},

R = {r; (~ur - ~u) • it > 0} .

(3.50)

Doing so, we have assumed that the models in consideration do not possess any velocity parallel to the boundary. In this paper, we limit our investigation to very simple boundary conditions but taking into account exchanges of momentum and energy between the condensed or solid medium bounded by S and the gas. The adopted presentation for discrete models of gas is similar to that made in the theory of continuum rarefied gases. 3.4.1

Diffuse

reflection

on a wall

We can easily write the boundary conditions for an impermeable wall by calling the following arguments: First, the properties of the reflected molecules are independent of their properties before the impact. They depend on the typical properties of the wall only. So, we write on the wall: Nr = XNrw ,

Vr € R,

(3.51)

Discrete Velocity Models with Multiple

49

Collisions

where the densities Nrw correspond to the discrete gas in Maxwellian equilibrium with the wall. Second, the re-emitted stream has not completely lost its memory of the incoming stream: per unit surface element and per unit time, the number of emerging molecules is equal to the number of impinging molecules, or equivalently, the normal component of the macroscopic velocity at the wall is zero. Then letting: ni = ] P Ni,

nIw = ^2 Niw ,

i€l

nRw = ^

i£l

niM = ]>Z-Wt|0?i - ~uw) • ~n\, nRw^Rw = 2 J Nrw\(iir

Nrw ,

(3.52)

r€R

niw)jLIw = ^-/Vj W |(l?j - ~uw) • ~n\ - ~uw) • T?|,

HRw

Od =

r6fi

^

The mean density of the gas near the point M is:

n = ]T Ni + J2 Nr = m + XnRw , iel

(3.53)

reR

and the relation (li — ~uw) • it — 0 is written: -nifii

+ A URWfiRw = 0.

(3.54)

A straightforward computation yields: A

=

TT—TT^ •

(3-55)

In expression (3.55), A is expressed by means of the macroscopic density n of the gas near the wall, the density of the reflected molecules and the ratio 6d of mean values of the reflected and the incident molecules fluxes. The quantities n and 6d are unknown and should normally be determined after the resolution of the kinetic equations. However, the geometry of the models can simplify the determination of Oj,- For instance 9d is equal to 1 for the models whose the numbers of velocities belonging to / and belonging to R are equal, and whose velocities have the same absolute value for their

50

Lecture Notes on the Discretization

of the Boltzmann

Equation

normal components with respect to the wall (i.e. \~Uk • ~n\ independent of k). Furthermore if the velocities of the model are symmetrical with respect to the wall and u w = 0, then: nRw = -

and

A = n.

(3.56)

We also notice that a change of n in f3n (/? > 0) leads to a change of A in p\. This allows to adimensionalise A with respect to n. Such boundary conditions have been used to analyze various problems. An example is the momentum and energy transfer between two parallel plates having different temperatures and different velocities. By using models with very few velocities, some parametric studies are allowed [17,19], and by using models with a more important number of velocities, the numerical simulations are easy to used [26,27]. The results are always in very good agreement with the results obtained from the treatment of the continuous Boltzmann equation.

3.4.2

Boundary

conditions

on a condensed

phase

The boundary conditions we adopt for evaporation and condensation give the physical property that the molecules are emitted from the condensed phase in Maxwellian equilibrium with it. So, as previously, we write the equalities (3.51) [18,20]. Conditions (3.51) are not sufficient because the parameter A is unknown. So, we have to introduce an additional hypothesis. For an impermeable wall, we have written the nullity of the normal component of the relative macroscopic velocity at the wall. On an interface between a liquid and its vapor, this argument is no longer valid. Let be consider the flux j of molecules passing through the interface: nj = y^^NjCuj - ~ u w ) • it + \y^iNrw(~ur iel

-Tty,) • it = —nifii + \URWHRW

r€R

(3.57) Relations (3.53) and (3.57) give x=

n(j + m) nRw(m + fJ.Rw)

(3.58)

.

51

Discrete Velocity Models with Multiple Collisions

We notice that except URW and /J,RW which depend on the discrete model and the macroscopic variables of the condensed phase, all the remaining parameters depend on the impinging molecules whose properties are unknown in general when the kinetic equations are unsolved. However, we can completely determine A in the two cases of the evaporation in the vacuum and of the evaporation / condensation at equilibrium. First, let us consider the evaporation flow in vacuum. It is unsteady: At the initial time when particles start to evaporate, there are no impinging particles. The total density of the vapor at the interface and the flux passing through it are respectively equal to the total density and the flux of emerging molecules. So, at the initial time and at any point M of the interface, we have: raj = HI = 0 ,

ra

= \URW ,

nj = XnRW(iRW .

However, the total density of the vapor at the interface in the absence of impinging molecules is equal to the sum of the Maxwellian densities of emerging molecules associated with the saturated density of vapor corresponding to the temperature of the condensed phase. Denoting by nsat the saturated density, we get: n = n3at 2_^ ^ n o = n3atnRw • r£R

Then by identification of the two expressions of ra, we have: A = naat.

(3.59)

For the evaporation/condensation at equilibrium, the fluxes of evaporating and condensing molecules counterbalance each other. The vapor gas is in Maxwellian equilibrium with the condensed phase and its total density is nsat. The exponent e denoting quantities at equilibrium, we have: f = 0,

N? = nsatNiw

iVre = n3atNrw

Viel,

Vr

Therefore at equilibrium, relation (3.58) becomes: n

nsatWw

"i nRw(fJ,iw+

s" jJLRw)

sat

r=

7i nRw(l

... TIPT

+ Oed)

wren

oe

^Rw

aJ =

. MW

eR.

52

Lecture Notes on the Discretization

of the Boltzmann

Equation

In the general case, we make the additional assumption that for given temperature and velocity, the number of emitted particles is the same in equilibrium as well as in nonequilibrium. Consequently, it is sufficient to identify the expressions of the densities of the emitted particles in equilibrium and in nonequilibrium to find: A=

n at

°

nRw(l

nps

+ 0ed)

with

0£ = — • d fiIw

(3-61) '

V

So A is known as soon as the geometry of the model and the macroscopic variables of the condensed phase are known. For models having velocities symmetrical with respect to the interface S, the sums of the densities of impinging and emerging particles of the gas in Maxwellian equilibrium with the interface are equal to 1/2 [20] (when the interface is at rest). With these models we can prove that 0% = 1. Then, relation (3.61) gives A = nsat and we obtain the conventional boundary condition of the continuous kinetic theory for evaporation and condensation, that is, the total density of the emerging molecules is equal to nsat/2 [32,34,35]. In that case, the molecules leaving the condensed phase constitute the corresponding part of the Maxwellian density distribution describing the saturated gas at the surface temperature of the condensed phase. Such boundary conditions have been used to analyze the evaporation / condensation problems between two parallel plane surfaces [18,20], and in a half space limited by a plane condensed phase. Here also, the results are in very good agreement with those obtained from other treatments of the continuum Boltzmann equation. Let us give some details on the half space evaporation and condensation investigated on the basis of the discrete kinetic theory [19,31]. We consider the formation and propagation of disturbances in an initially uniform gas bounded by its plane condensed phase in non-equilibrium with the gas, when the evaporation or condensation is taking place from the condensed phase [31]. For the condensation case, as in continuum theory [34], we yield four different types of unsteady solutions. First, the gas is compressed on the condensed phase, and this compressed region diffuses toward upstream infinity. A steady state with the prescribed condition at infinity is established. Second, the gas is compressed on the condensed phase and a compression wave propagates to upstream infinity. Third, a rarefaction region develops on the condensed phase and diffuses as time goes on. Finally a steady state

Discrete Velocity Models with Multiple

Collisions

53

with the prescribed condition at infinity is established. Fourth, a rarefaction region develops on the condensed phase and an expansion wave propagates to infinity. For the evaporation case, we also have different types of solutions [31]. They are described in the thesis of Nicodin [31]. More, it is possible to determine the condition relating the macroscopic variables at infinity and these of the condensed phase that allows a steady solution. All the results have been obtained from numerical computations by a splitting method. The discrete models used are very simple with 16, 36, 64 or 100 coplanar velocities. With 16 velocities only, the results are not all bad. The results with 64 or 100 velocities are very closed [31]. We emphasize that all these results are very closed to those obtained by other treatments of the Boltzmann equation from qualitative and also quantitative points of view.

3.4.3

Some remarks

on the boundary

condition

problem

Let us consider the mathematical problem of the existence of solutions for the kinetic equations with boundary conditions. First we pay attention to the steady problem between two plates. We look for the densities Nk, the solution of the boundary problem: dNk

C fc (N),

k =

l,...,p

Nk(0)

a*, Vfc such that

Vk > 0,

Nk(d)

bk

vk < 0,

Vfc such that

(3.62)

where y € [0, d] and where ak and bk are positive. All the quantities Vk are nonzero constants. We have to study the existence of solutions for the p differential equations (3.62) for the p unknowns Nk(y) and with p boundary conditions. When the collision term is reduced to binary collisions, from the works of Cercignani, Illner and Shinbrot [9], there exists one solution. The solution is bounded and uniformly continuous independent of the value of d. This result has been extended by similar arguments when the collision term takes into account the multiple collisions. Furthermore this result has also been proved when there exist some indices fc such that Vk = 0. In this last case, the models are such that their summational invariant linear

54

Lecture Notes on the Discretization

of the Boltzmann

Equation

spaces and those of the corresponding models obtained by canceling the velocities parallel to the boundaries have the same dimension [20,25]. Now we pay attention to the steady problem in a half space. We look for the densities Nk the solutions of the following boundary problem:

vk^=Ck(N), JVfe(O) = ak Nk{oo)=bk

k=

l,...,p

Vfc such that

vk > 0,

Vk

vk<0,

such that

(3.63)

where y € [0, oo) and where ak and bk are strictly positive. With binary collision term only, it is established that there exist positive solutions of the p differential equations (3.63) with the boundary conditions on the interface y = 0. At infinity (y —• oo), these solutions belong to the set of the Maxwellian states of the model. Let us consider the problem (3.63) with ak and bk defined as Maxwellian densities associated with the macroscopic variables of the condensed phase and of the gas at infinity. This new half space problem (3.63) does not necessarily have a solution for an arbitrary set of the two sets of the macroscopic variables. For a particular model with ten velocities [19], analytical results have been obtained. But for an arbitrary model, the problem is more difficult. For some particular models, the results have been established by computations [31] as it has been previously seen with the evaporation / condensation in a half space.

3.5

Conclusion

The aim of this paper has been to describe a discrete gas where the multiple collisions are taken into account. It is important to see that the multiple collisions contribute to eliminate the spurious invariants, and in some favorable cases to obtain only the physical invariants. We emphasize that with very simple models, it is possible to obtain a very good description of the physical phenomena. The example of the evaporation / condensation on a plane condensed phase and in a half space is convincing: All the physical aspects are present, and it is sufficient to consider a model with very few velocities. Due to the fact that the mathematical structure of the kinetic equations is very nice, and that the boundary conditions on a surface S are very

Discrete Velocity Models with Multiple

Collisions

55

convenient, it seems that the mathematical question of the existence of solutions is well posed. Some progresses are still possible. And by rebound some new investigations on the true Boltzmann equation are expected. 3.6

References

[1] Bardos C , Golse F. and D. Levermore, Fluid dynamic limits of discrete velocity kinetic equations, in Advances in Kinetic Theory and Continuum Mechanics, R. Gatignol and Soubbaramayer Eds, Springer-Verlag, (1991), 57-72. [2] Bellomo N. and Kawashima S., The discrete Boltzmann equation with multiple collisions: Global existence and stability for the initial value problem, J. Math. Phys. 3 1 , (1990), 245-253. [3] Palcewski A., Schneider J., and Bobylev A.V., A consistency result for discrete-velocity schemes of the Boltzmann equation, SIAM J. Num. Anal. 34, (1987), 1865-1878. [4] Bobylev A.V. and Cercignani C., Discrete velocity model for mixtures, J. Stat. Phys. 9 1 , (1998), 327-341. [5] Broadwell J.E., Study of rarefied shear flow by the discrete velocity method, J. Fluid Mech. 24, (1964), 401-414. [6] Broadwell J.E., Shock structure in a simple discrete velocity gas, Phys. Fluids. 7, (1964), 1243-1247. [7] Cabannes H., On the initial value problem in discrete kinetic theory, European J. Mech. B/Fluid. 10, (1991), 207-224. [8] Cabannes H., Evaporation and condensation problems in discrete kinetic theory, European J. Mech. B/Fluid. 13, (1994), 685-699. [9] Cercignani C , Illner R. and Shinbrot M., A boundary value problem for discrete velocity models, Duke Math. J. 55, (1987), 889-900. [10] Cercignani C , Illner R., Pulvirenti M. and Shinbrot M., On nonlinear stationnary half space problems in discrete kinetic theory, Transp. Theory Statist. Phys. 54, (1988), 885-896. [11] Cercignani C , On the thermodynamics of a discrete velocity gas, Transp. Theory Statist. Phys. 23, (1993), 1-8.

56

Lecture Notes on the Discretization

of the Boltzmann

Equation

[12] Chauvat P., Coulouvrat F. and Gatignol R., The Euler description for a class of discrete models of gases with multiple collisions, in Advances in Kinetic Theory and Continuum Mechanics, Gatignol R. and Soubbaramater Eds, Springer-Verlag, (1990), 139-154. Chauvat P., Lois de comportement en theorie cinetique discrete et applications. These, Universite Pierre et Marie Curie, Paris, (1991). Chauvat P., Summational invariants in discrete kinetic theory with multiple collisions, Mech. Res. Communications, 18, (1991), 11-16. Coulouvrat F. and Gatignol R., Description hydrodynamique d'un gaz en theorie cinetique discrete : les modeles reguliers, C. R. Acad. Sci. Paris, 306, serie II, (1988), 393-398. Cornille H. and Cercignani C , A class of planar discrete models for gas mixtures, Journ. Stat. Phys. 99, (2000), 967-991. D'Almeida A. and Gatignol R., Boundary conditions for discrete models of gases and applications to Couette flows, in Computational Fluid Dynamics, Leutloff D. and Srivastava R.C. Eds, SpringerVerlag, (1995), 115-130. D'Almeida A. and Gatignol R., Boundary conditions in discrete kinetic theory and evaporation and condensation problems, European J. Mech. B/Fluids, 16, (1997), 401-428. D'Almeida A. and Gatignol R., The half space problem in discrete kinetic theory, Math. Models Meth. in Appl. Sci. 13, (2003), (to appear). D'Almeida A., Etude des solutions des equations de Boltzmann discretes et applications, These, Universite Pierre et Marie Curie, Paris, (1995). Frisch U., Hasslacher B. and Pomeau Y., Lattice gas automata for the Navier-Stokes equation, Phys. Rev. Lett. 56, (1986), 1505-1508. Frisch U., D'Humieres D., Hasslacher B., Lallemand P., Pomeau Y. and Rivet J.P., Lattice gas hydrodynamics in two and three dimensions, Complex Systems, 1, (1987), 649-707. Gatignol R., Theorie cinetique d'un gaz a repartition discrete de vitesses, Z. Flugwissenschaften, 18, (1970), 93-97. Gatignol R,. Theorie cinetique d'un gaz a re partition discrete de vitesses, Lectures Notes in Physics n.36, Springer-Verlag, (1975).

Discrete Velocity Models with Multiple

Collisions

57

[25] Gatignol R., Kinetic theory boundary conditions for discrete models of gases, Physics of Fluids, 20, (1977), 2022-2030. [26] Goldstein D., Sturtevant B. and Broadwell J.E., Investigations of the motion of discrete velocity gases, in Rarefied Gas Dynamics, E.P. Muntz, Weaver D.P. and Campbell D.H.Eds, AIAA, Washington, (1989), 100-117. [27] Inamuro T. and Sturtevant J., Numerical study of discrete velocity gases, Physics of Fluids A, 2, (1990), 2196-2203. [28] Kawashima S., Global existence and stability of solutions for discrete velocity models of Boltzmann equations, Lectures Notes Num. Appl. Anal. 6, (1983), 59-96. [29] Longo E. and Monaco R., On the thermodynamics of the discrete models of the Boltzmann equation for gas mixtures, Transp. Theory Statist. Phys. 17, (1988), 423-442. [30] Monaco R. and Preziosi L., Fluid Dynamic Applications of the discrete Boltzmann equation, Advances in Mathematics for Applied Sciences n2, World Scientific, (1991). [31] Nicodin I., La modelisation par des gaz discrets des phenomenes d'evaporation-condensation, These, Universite Pierre et Marie Curie, Paris, (2001). [32] Pao Y.P., Application of kinetic theory to the problem of evaporation and condensation, Physics of Fluids, 14, (1971), 306-312. [33] Platkowski T. and Illner R., Discrete velocity models of the Boltzmann equation: A survey on mathematical aspects of the theory, SIAM Review, 30, (1988), 213-255. [34] Sone Y., Aoki K. and Yamashita I., A study of unsteady strong condensation on a plane condensed phase with special interest in formation of steady profile, in Rarefied Gas Dynamics, V.C. Bom and C. Cercignani Eds, Teubner, Stuttgart, Vol. 2, (1986), 323-333. [35] Sone Y., Ohwada T. and Aoki K., Evaporation and condensation on a plane condensed phase: Numerical analysis of the linearized Boltzmann equation for hard-sphere molecules, Phys. Fluids, 1 (1989), 1398-1405. [36] Toscani G., Global existence and asymptotic behaviour for the discrete velocity models of the Boltzmann equation, J. Math. Phys. 26, (1986), 2918-2921.

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Chapter 4

Discretization of the Boltzmann Equation and the Semicontinuous Model L. Preziosi and L. Rondoni Dip. Matematica - Politecnico di Torino - Italy

4.1

Introduction

The Boltzmann equation provides an accurate description of the behavior of a rarefied gas. Unfortunately, it is very hard to numerically exploit all its capabilities and all the physical information it contains. This is essentially related to the structure of the collision term, a five-fold integral which is very expensive to compute. Furthermore, this term needs to be evaluated with great precision, because all the mechanical properties characterizing particle collisions, e.g. conservation of mass, momentum, and energy, are included in it. A lack of conservation, even very small, may lead to serious consequences on the integration of the equation. For this reason the deduction of intrinsically conservative models results particularly important. On the other hand, as addressed in the chapter by Cercignani, in deducing discretized models one should avoid the appearance of spurious collision invariants, which is characteristic of many discrete velocity models and discretization procedures. For this reason, in some numerical schemes proposed in the literature to integrate the Boltzmann equation, the solution is suitably corrected at each time step in order to preserve mass, momentum and energy (see for instance [1; 2]). In the last decade some attention has been paid to the identification of discretization procedures that lead to models possessing most of the properties characteristic of the Boltzmann equation, namely, 59

60

Lecture Notes on the Discretization

of the Boltzmann

Equation

conservation of mass, momentum, and energy, validity of an H-theorem, and stationary states described by a Gaussian distribution. Most of the papers [3; 9] deal with a discretization of the velocity space in a uniform cubic lattice, as described more in detail in other chapters of this volume. Another approach consists of uniformly discretizing the energy space, corresponding to a discretization of the velocity space in an ensemble of spheres. The idea of a polar discretization, already present in [10; 12] has been developed in [13] where the definition of a suitable discretization of particle speeds and the introduction of a suitable interpolation procedure allows to deduce a model in which the collision operator is a sum of integrals over finite domains. These are the cartesian products of a unit circle and the portion of the unit sphere between two parallels symmetric with respect to the equatorial plane perpendicular to the velocity of the field particle. One can then exploit the fact that integration over spherical domains can be performed with good precision and small computational effort. The resulting kinetic model preserves all the mentioned physical properties. The same characteristics are enjoyed by the model deduced in [14] where particle velocities are defined through their energy and velocity direction, the range of allowed energies is subdivided into intervals of equal width, and a stepwise interpolation is introduced. The main point addressed in this paper is the identification of the parameters involved in the discretization. This is based on the criterion that the spurious terms, which are present in the Euler equations related to the discretized models, are as small as requested by the application. The model has then been generalized by Koller and Schiirrer [15; 17] to handle gas mixtures, inelastic collisions, chemical reactions and photon excitation (see also the chapters by the authors in this volume). In this chapter a particular attention is paid to the formulation in terms of energy and velocity direction, and to the consequences of each step of the discretization procedure. In particular the discretization procedure is divided into two main steps: 1) the definition of a limited subset of the velocity space containing all allowed velocities (Section 3); 2) its discretization (Section 4). In Section 5 it is shown that: • The model preserves mass, momentum and energy; • The modelling procedure is constructive, so that it is possible to obtain estimates on the "distance" between the discretized collision operator and the continuous Boltzmann operator, i.e. consistency

Discretization

of the Boltzmann

Equation and the Semicontinuous

Model

61

of the quadrature rule. • There exists an H-functional describing trend towards an equilibrium described by a Gaussian distribution. • The fluid dynamic limit related to the discretized kinetic model tends rapidly towards the usual Euler equations with an isotropic pressure tensor, and vanishing heat flux when the number of allowed energies grows to infinity and the discretization interval for the particle energies becomes J?+. For the sake of completeness, Section 6 gives the details of the collision dynamics in terms of energy and velocity direction. Finally, in Section 7 we draw our conclusions giving some hints for possible developments.

4.2

Splitting and Energy Formulation

In kinetic theory, the behaviour of a neutral, rarefied gas is usually described through the evolution of a distribution function, / : [0, T] x M3 x JR3 —+ R+, which satisfies the Boltzmann equation

% = A(f) + Q(f,f),

(4-1)

-4(/) = - v - V x / ,

(4.2)

where

is a linear differential operator and Q(/,/)= /

/

a(7,fl)3[/(t,x,v,)/(t,x,v;)-/(i,x,v)/(t,x,v»)]dg'dv*

(4-3) is a nonlinear integral operator. In (4.3) (7(7, g) is the differential cross section, which depends on the interaction potential, 7 = g • g' is related to the angle between the pre and the post collisional plane, and g is the modulus of the relative velocity g of colliding particles g = |v» - v| = |v„ - v

(4.4)

62

Lecture Notes on the Discretization

of the Boltzmann

Equation

The rule for collision between two particles is expressed by v' = v + ( g . n ) n = - + | g ' , (4.5) , , A. A R g „, v» = v , - (g • n ) n = — 2 - 2- g', where R HI v + v , = v ' + v i ,

(4.6)

and n and g' are unit vectors that span S2. In particular, g' represents the direction of the relative velocity after collision. In this chapter hatted quantities refer to unit vectors and tilded ones to dimensionless quantities. In the entire chapter, particles are considered as characterized by their energy per unit mass, e = v2/2 (simply named energy for sake of brevity), and by their velocity direction fi. It is then useful to introduce the following quantities E d=f e + e. = e' + < ,

(4.7)

5 d = / 2 V ^ O - f i . = 2 N / e 7 c i n / • fy ,

(4.8)

g = ^2(E

- S),

,def

R =J |R| = y/2(E + S),

(4.9)

which, together with R, are invariant during collision. The description of the collision dynamics using the energy formulation is given for the sake of completeness in Section 6. The two operators in Eq. (4.1) present problems of different type to be addressed properly for a correct numerical evaluation. For this reason the so-called "splitting algorithm", first applied by Temam [18] in the study of several evolution problems, like the Broadwell model, results are very useful. This method, when applied to the Boltzmann equation, consists of decoupling the collision contributions from the free streaming ones, and

Discretization

of the Boltzmann

Equation and the Semicontinuous

Model

63

looking for iterative solutions of the following problems (4.10)

/n°(* = 0) = / ° ( t = 0) = / O , dt

®LL

dt

k = Qf UJn

'

for k > 0.

(4.11)

for k > 0.

(4.12)

. #(«*) = tf(«fc+l) : The convergence of the algorithm has been proven by Desvillettes and Mischler [19], under the same assumptions used by R.J. DiPerna and P.L. Lions [20] for their existence results for the Boltzmann equation. Theorem 4.1 Under assumptions 1 and 2, given below, the sequences fn and fn converge up to extraction of subsequences to the same nonnegative limit f € L°°([0,T];L x (JR 3 x R3)) weak-*, and this limit satisfies Q±(f) eLJ ({0,T}xM3xR3)., oc i + / and

(h^h^-m in the sense of distributions. Assumption 1.1 The function B = o~(-,-)g is in Ljoc(M3 x S2), and depends only on g and 7. Moreover, the function

A(g)= f B{g,i)dt>, Js2 satisfies, for all R > 0, 7-—2 /

A(|g + v | ) d v - > 0 ,

where B^ = {v e R3 : |v| < R}.

as

g-^oo,

64

Lecture Notes on the Discretization

Assumption 1.2

of the Boltzmann

Equation

The function /o is such that

/o(x,v){l + |x| 2 + |v| 2 + |log/ 0 (x,v)|}dxdv < oo. /IRS. In the following we will focus on the collision step for the splitting algorithm, Eqs.(4.12), mainly for two reasons. On the one hand this step is rather heavy, from a computational viewpoint, because it involves the evaluation of a five-fold integral. On the other hand, the evaluation has to be done with great care and accuracy because the mechanical properties characterizing particle collisions, e.g. conservation of mass, momentum, and energy, are included in it. Our main concern will in fact be the link with the conservation equations, classically deduced from the moments of the Boltzmann equation

rg +v .(,u) = o, -(/>U) + V - ( n +

/9U®U)

Ue + \^]+v-

= 0,

(4.13)

q + IIU+(£ + ^[/2)U

where p= m

f(v)dv JR3

=m

del JR+

d&V2e~f(e,n), JS2

is the mass density, U = — / v / ( v ) d v = — I del P JR3 P JR+ JS2

dtl2eilf{e,tl),

is the drift velocity, U is its modulus, and U = JJ/U its direction 8 = ^ 1 |v-U|2/(v)dv = ^ / 2 JR3 2 JR

de/ dn|V2in-U|V2i/(e,n), Js2

Discretization

of the Boltzmann

Equation and the Semicontinuous

Model

65

is the internal energy, n = m [ (v - U) ® (v - U ) / ( v ) dv JR3

= m

de J

d£l(V2en-U\®

f v ^ e f l U ) y/2ef(e,

fi),

(4.14)

is the pressure tensor, and |v-U|2(v-U)/(v)dv

q = ?2 /

JR3

de

=?/ 2

JR+

f dn\Vten-u\

JS2

(•s/2en-V\V2ef(e,n),

(4.15)

is the heat flux. The Maxwellian equilibrium, considered as a function of energy and velocity direction, can be written as /(*, x, e, tl) = Aexp[\/BeU

-il-Ce],

(4.16)

where the Maxwellian parameters A, B and C are related to the macroscopic observables through 3/2

r

*-£(&) -

o TT9I

3/>C/2

•, / o . r r \ 2

45 (4.17)

An important dimensionless parameter is the following:

which measures the ratio between kinetic and internal energy. It is also important to notice explicitly that if the dimensionless parameters B

'-U 2

and C = CU2

(4.19)

are introduced, one has B = C = r,

(4.20)

66

Lecture Notes on the Discretization

of the Boltzmann

Equation

and the Maxwellian distribution can be written as = Aexp r

f(t,x,e,Sl)

e U • ft 3/2

L(*P-\ m \Air£J

exp

-r-(eti-ijy

(4.21)

where \/2e

(4.22)

e =

The above definition of e is preferred to others characterized by a proportionality between e and e (e.g. e = 2e/U2) simply because of the simplifications in the following formula (in particular the presence of square roots is minimized). Referring to [13] for further details, one can compute the macroscopic quantities (4.2-4.16) at Maxwellian equilibrium given as in (4.16) in terms of energy and velocity direction and, in particular, write the pressure tensor and the heat flux as

n = n(i + ^fi n),

(4.23)

q = n(e,-j^)u,

where

/ %

e -C(e+e')[(£ e

+ 2)ss' - yfifecs' - ByfeT'cc'] dede'

JR\

, f

(4.24)

e-c(e+e')(y/]tec-s)s'dede'

JR\

/

y/Be(e - e^e'^+^cs'

de de'

JR\

-5-1. /

e-Ce(JTec-s)de

JR+

and the following abbreviations have been used s = sinh vBe,

c = cosh VBe,

s' — sinh VBe',

c' = cosh VBe'

(4.25)

Discretization

of the Boltzmann

Equation and the Semicontinuous

Model

67

The Euler equations can then be written in the following form

' ! + v.(,u) = o, ^( P u) + v • [(i + e^pv ® u] + vn = o,

li{£+2I^) + V-{

(1 + eq)U + S + -pU7

(4.26)

m = o,

where /

e-c^e+e'\{Be

+ 2)ss' - VWecs' - BV^

JR\ £*•

cc'] dede' (4.27)

=

J •e-

Ce

(VBec-s)de

JR+

The integrals in (4.24), (4.25), (4.27) can be computed analytically, and give Tfr = en = eq = 0, corresponding to an isotropic pressure tensor, a vanishing heat flux, and no extra terms in the Euler equations (4.26), as it must be. Obviously, that is not the case in numerical calculations, as explained in the next sections. 4.3

Working in a Finite Energy Interval

In this section we discuss the effect of working in a finite energy interval. The discretization procedure will be addressed in the following section. The very first step of any discretization procedure consists of denning a limited subset of the velocity space containing the allowed velocities. As a consequence of this step, the two spurious terms in the momentum and energy Euler equations (4.26) related to the Boltzmann equation (one extra pressure tensor, in the form of a Reynolds stress term, and one extra heat flux term, in the form of a pressure) do not vanish any longer. The presence of these terms may be directly related to the introduction of the "boundary" in the velocity space which corresponds to an approximation of the original equations, which breaks the Galilean invariance characteristic of both the continuous Boltzmann equation, and of the related Euler equations. To clarify this statement, one can consider the viewpoint of the particle which sees (or feels) this artificial boundary as a reference

68

Lecture Notes on the Discretization

Fig. 4.1

of the Boltzmann

Equation

The domain S ^ .

point. Therefore, one cannot expect that the Euler equations related to any discretization scheme have an isotropic pressure tensor and a vanishing heat flux. In this section we give a criterion to identify an interval E = [em, CM) such that the probability for a particle to have energy outside E can be neglected. The restriction of the energy range to a finite interval bears consequences for Eq.(4.8), because it leads to a restriction either on the outgoing energy e', or on the angle between the incoming directions Cl • f2», if the outgoing energies are fixed. This is a well-known fact: • Given the incoming energies e, e* and the angle between the incoming directions fi • fi*, the outgoing energies are confined inside the interval E>e>

E. 2

gR E_

gR

T' 2" + T

(4.28)

symmetrically with respect to its extrema.

For the construction of our model, it is useful to look at the same restriction from a different point of view. • Given the incoming energies e and e*, and an outgoing energy e' smaller than the largest energy the particle can possibly reach, E = e + e» (e^ is then determined by the conservation of energy, e'„ = E — e'), there is a limited range for the angle between the

Discretization

of the Boltzmann

Equation and the Semicontinuous

Model

69

pre-collisional velocity directions which yields the post-collisional energy e'. This can be expressed as: *

-

P'P'

2

(n-n,) <—. ee* It is then useful to define the domain

Sft = ( n , : ( f i - 0 , ) 2 < — } , ^

(4.29)

ee* J

which represents either the entire surface of the unit sphere S2, if e'e* > ee* or, taking fl as polar axis, the part between two parallels symmetric with respect to the equator, as shown in Fig. 1. In order to understand the consequences of working in a finite energy interval, let us start with the basic steps of the derivation of the Boltzmann equation. The change during the time interval [t, t + dt] of the number of molecules in the element dx dv around (x, v) due to collision is J dvdtdx

— dvdtdxx

/

cr{l,g)

JR3XS2

g[f(t, x, v ' ) / ( t , x, v'*) - f(t, x, v ) / ( t , x, v.)] dg dv, .

(4.30)

Therefore computing the change due to collisions during the time interval [t, t + dt] of the number of molecules in the volume element dx around x, having velocity direction in the element d£l around fi and energy in J C E, one has / JV2ede = / Vtede Ji

Ji

x f

/ JR+

y/2eZde+ /

d«*

JS2

d g V ( 7 , 5 ) 5 [ / ( v ' ) / K ) - / ( v / 2 7 n ) / ( V 2 7 : r 2 * ) ] . (4.31)

In our approach, it is useful to consider the output velocities not as functions of g' as in (4.5), but as functions of an output energy e' and of the angle d between the planes containing the pre- and post-collisional velocities, as described in (4.78) and (4.79) of Section 1.6, so that we can discretize the outgoing energy e'. This also means that in the scattering cross section, 7 will be considered as a known function of e' and 1? through Eq.(4.80), given in Section 1.6.

70

Lecture Notes on the Discretization

of the Boltzmann

Equation

Fig. 4.2 Collision dynamics. Dashed arrows are in the plane of the figure which represents the pre-collisional plane. Full arrows refer to the post-collisional quantities which are in the plane which intersects the plane of the figure along R. and forms with it an angle i9.

Referring to Fig. 2 (and using Eq.(4.77) of Section 1.6), one can change variable in the surface integral over g' expressing dg' as dg' = sin
(4.32)

so that (4.31) can be rewritten as / Jy/2ede = I de I Jl

Jl

x [ JD,,

de' f JO

de*

JR+ 2T

dtl* JS2

Q

/££-

<W-V^(7,0)[/(v')/(v:) -

f(yfen)f(y/2£n.)], (4.33)

where D e ' is defined in (4.28) and depends on e, e* and on Q • f2». In view of the velocity discretization, it is convenient to exchange the integrations over fi* and over e' in order to group together the integrals over all the energy variables. In order to do that one has switch from the the post-collisional energies De>, to the pre-collisional velocity directions

Discretization

of the Boltzmann

Equation and the Semicontinuous

Model

71

S^. This allows us to rewrite Eq.(4.33) as / jV2ede= Ji

de Ji

de* /

JR+

J(e,e»,e',Q)de',

(4.34)

JO

where ~

A f

/•27T

f f

J{e,e*,e',n)=

f

dft. / JSf,

Jo

8 fee~

dti

V

,* A

u(e,e„e,,n.fl„tf)

ii(e,e*,f2-n»)

x [f(e', n ' ( e , e „ e', tl, A., tf))/(e'„ n ' , ( e , e „ e', fi, n „ t f ) ) -/(e,n)/(e.,n,)]. (4.35) If one only considers collisions with energies in the interval E — [e m , CM), one can deduce a kinetic model of the same form as in (4.1) but with the collision operator replaced by f de. [ J ( e , e „ e ' , n ) d e / ,

Qb(f,f)=

JE

(4.36)

JE'

where E' = E n [0, e + e*]. For this kinetic model, the conservation equations (4.13) still hold, and the Maxwellians can still be written as in (4.16), but the relations among the Maxwellian parameters and the macroscopic observables, computed over the finite domain E x S2, is not as simple as in (4.17). The Euler equations can then be written in the form (4.26) with 77^, £„• and eq given by formulae similar to (4.24), (4.25) and (4.27), in which M+ is replaced by E. However, this replacement is not painless because these terms do not vanish any longer and give rise to spurious terms in the Euler equations. In order to see that, take em = 0 first, and proceed in terms of the dimensionless variables introduced in (4.18) and (4.22). A lengthy calculation gives M°> eM) = e*(v,eM)

=

T> is \T> is \

'

(4-37>

~.2I~ T

,

(4.38)

,

, , . (4.39)

x _ 2[7Vg(eM) - e(eM)Mq(eM)} (n~ eg(0, eM) p | ^

72

Lecture Notes on the Discretization

of the Boltzmann

Equation

where eu = y/2e~M/U Mr(e) = 2e~re [(r + 2) sinh 2 re — (r + l)resinh re'cosh re - r 2 ^ ] , M*(e) = e-Tl2'2[(r

+ 3 + r2!2) sinh re - (r + 3)recoshre],

A/"g(e) = 2 e - " 2 { [ r 2 + 4r - 2 - (r + 2)r 2 e 2 ] sinh 2 re — (r 2 + 3r — 4)re"sinhre'coshre — (r + 2)r 2 e 2 } , •M?(2)

= e_re

^2[r{f + 5) — r 2 e^](sinhre'— re cosh re) ,

P w (e) = £{e) - 2 e - r 5 2 / 2 s i n h r e , T>q(e) = r£(e) — 2 e _ r e / 2 [(r — 1) sinh re'+ re'cosh r e ] , ar/2

£(e)

^2r"e r / 2

Erf /•vT(

(1+2)

Erf

5(1-2)

dtw,

and Erf( •) is the error function. The zero in the argument of rj^, s^, and sq in (4.38)-(4.39) is due to our choice em — y/2em/U — 0 as the lowest end of the dimensionless interval [0, ej\/). The terms in (4.38)-(4.39) do not vanish, hence the pressure tensor is not isotropic, and the heat flux does not vanish either. However, the two spurious terms thus introduced in the Euler equation go exponentially to zero, as ZM grows. The introduction of a lower bound, say em > 0, gives further contributions to the spurious terms which go to zero as G.rL j when e m tends to zero. In this case r;x, sn and eq can be written as Vn(em,eM)

=

(4.40)

2{-A/^(eM)+Mr(e m )-[g(ey)-g(e m )][^^ [V„{eM)-Vv(em)][Vq{eM)-'Dq(em)\ £n(em,eM)=

(4.41)

2{A/"7r(eM)+.A/'7r(em) - [£(eM)-£(em)}

\Mv{eM)+M7,{em)\

+AC(e m , e M ) }

[Dq{eM)-Vq{Zm)Y £q(em,eM)= 2{-A/" g (e M )+.A/'q(e m )-[£(eM)-^ [Dq{ZM)-Vq{Zm)Y

(4-42)

Discretization

of the Boltzmann

Equation and the Semicontinuous

Model

73

where AC(e m , e M ) = e _ r ( 5 - + ^ ) / 2 { - 4 ( r + 2) sinhre m s i n h r e M + (r + l)r[(e m + eM) s i n h r ^ + e M ) - (eja - e m ) sinhr(e M - e m )] + r2[(em + eM)2coshr(eM

-em)

- (eM - e m ) 2 c o s h r ( e M + em)]},

and Mg(em: eM) - e - r ( e ™ + ^ ) / 2 x {-4(r 2 + 4r - 2) sinhre m s i n h r e M + 4(r + 2)r 2 e m eM coshre m coshreu + (r 2 + 3r — 4)r x [(e"m + e M ) sinhr(e m + e y ) - (eju - e m ) sinhr(?M - e m )] + r 3 [(e m + e M ) 2 coshr(eM - e m ) - (e M - e m ) 2 coshr(eM + e m )]} . These quantities characterize the magnitude of the spurious terms introduced by the restriction of the velocity space to a sphere or a hollow sphere, and they provide us with a tool to control such terms. On these grounds, one may adopt the following procedure to select e m and eu'1. Identify the range of internal energy and drift velocity characteristic of the flow to be described (and therefore the range of dimensionless parameter r); 2. Set a desired accuracy £ to limit the spurious terms £„. (or % ) and eq appearing in the Euler equations (4.26) or in the definition of heat flux and pressure tensor, and define e m and EM such that eb = £M + £m <£ (say, E~M = em = e/3); 3. Set eM so that en (or 7yw) and sq in Eqs.(4.38)-(4.39) have absolute value smaller than EM- For instance, for a given r Eq.(4.39) gives e q = £q{0,eM), which (locally) defines e « = ej^(0,£ q ). Therefore we can set ?M =eM(eM)

= max{eM(0,e M ),e^(0,eM)};

(4.43)

4. Having set e^f according to (4.43), consider ~em so that en (or 77^) and eq in Eqs.(4.40)-(4.42) have absolute value smaller than £t = Em+EM- For instance, Eq.(4.42) determines the values of e m which yield £,(em,CM(EM)) = em + £ M , that is em = e\{Em + E~M,£M)Therefore, if computationally convenient, one can set ~

def

em = em(Em,EM)

= mixi{e^l(£m + £M,£M),e^n(£m+'£M,£M)}

• (4.44)

74

Lecture Notes on the Discretization

£ = 10- J

£ = 10" 2 £M = ^ 1 0 - 2 2.265 5.326 15.72

EM = ^ l O - 1

r=10 r=l r=0.1 Table 4.1

2.025 4.603 13.53

of the Boltzmann

£=10"3

£M = £102.457 5.908 17.49

3

Equation

£ = 10~ 4 £M = ^10-4 2.622 6.410 19.03

Values of e^ (r,SM = j ) defined in (1.51) at different values of the expected

macroscopic parameter r =

r=10 r=l r=0.1

£ £ , and overall accuracy e.

e = 10- 1 £6 = f 10- 1 0.406 1.136 2.606

£ = 10- 2 sb = | 1 0 " 2 0.208 0.230 0.684

£ = 10-3

eb = flO" 0.103 0.104 0.303

3

£ = 10~ 4 £6 = f l O " 4 0.0482 0.0466 0.136

Table 4.2 Values of e m ( r , e m = | , £ M = f ) defined in (1.52) at different the expected macroscopic parameter r = £ g , and of the overall accuracy e m + ^M = f s) and loit/i ej^f given in Table 1. Values above 1 correspond cases in which the contribution £;, deriving by introducing any non vanishing admissible e m < 1 is always smaller than required.

values of e (ej, = to those physically

This procedure is used to plot Figure 3 and to give Tables 1 and 2. In particular, Figure 3 gives e„ and eq as a function of eju at different r, Table 1 gives 5 M ( | ) and Table 2 e m (§, §) at fixed values of r and overall accuracy e. It should be noticed that for physical reasons e m should be chosen smaller than U2/2 and eni larger than U2/2, i.e. em < 1 and BM > 1 (recall that em = U2em/2 and eM = C 2 e|f/2). In Table 2 there are, however, some values of r and £ for which the contribution associated with any em < 1 is always less than £/3. Furthermore, except for the very special cases where eq is close to changing sign (say, |£ g | < 1 0 - 6 ) , e„ is always an order of magnitude smaller than eq. Therefore, e m and ~SM are essentially determined by e-^ and e ^ . As a consequence, if the magnitude of the spurious term in the energy equation of (4.26) is not a concern, and one only wants the magnitude of the spurious term in the momentum equation to be smaller than a certain a > 0, even

Discretization

of the Boltzmann

1

T

:

Equation and the Semicontinuous

1

!

1

Model

-

0.1

^

0.01

:\

0.001

n

-

le-05

-

-

le-06

-

le-07

-

0.0001

le-08

75

-. \

r= l

1 r = 10

>v

-.

r = 0.1

-i

\

\ 1

i

1

i

1

1

Fig. 4.3 Plots of STT and of eq as a function of e « at different values of the expected macroscopic parameter r = p2£ . The two plots give an indication of the value of the parameter 7>M to be chosen, in the discretization, so that the induced magnitude of the spurious terms is small enough.

larger values of em and smaller values of ej^f can be used. Roughly, one can use the values of e m and ejvf reported in Tables 1 and 2, which correspond to 10a instead of a. For instance, the choice r = 10, e ^ = 2.265 gives rise to a spurious term in the momentum equation of order 1 0 - 3 instead of

io- 2 .

76

4.4

Lecture Notes on the Discretization

of the Boltzmann

Equation

Energy Discretization and Kinetic Model

Once the interval E = [e m , ew) has been identified, so that the probability for a particle to have energy outside E is negligible, we adopt a discretization procedure which divides E in n + 1 subintervals It of equal width A: Mi = [em + *A, e m + (i + 1)A)],

» = 0,..., n,

(4.45)

with A =

eM

~y . n+ 1

(4.46) y ^

Denoting the mid-point energy of the interval I ei = em+(i

+ -jA,

(4.47)

it is trivial to check that the following holds: Proposition 4.1 The discretization has the following properties • Vi,j,h

s.t. ei + ej—eh&E,

• If em = £A

3 k = i + j — h S [0,n] s.t. e; +

£ € IV, £/ien Vi,j s.t. e; + e^ < eu ,

ej + e^ =

SUpJi+j+*.

It needs to be mentioned that the condition em = £A does not represent a real restriction on the choice of E, since E can be easily adjusted to satisfy this condition. Remark 4.1 The proposition assures that: a) For any couple of input energies belonging to the discretization, if one of the output energies also belongs to the discretization, then either the other output energy belongs to the discretization, or it is outside E, but, as already stated, the occurrence of this last case has negligible probability. b) If em = £A, then either [e m ,ej + ej) contains E, or it is exactly equal to the union of i + j +1 + 1 subintervals i+j+e

[em,ei + ej)=

[j h=0

Ih.

(4.48)

Discretization

of the Boltzmann

Equation and the Semicontinuous

Model

77

This is useful to properly handle the collision operator. The following step consists of considering any function of velocity as a function of energy and velocity direction, and in approximating its energy dependence in E by a stepwise interpolation defined on the partition l j . This means that we take n

F(v) = F(e, n ) * £

Xi(e)Fi(fl),

(4.49)

i=0

where Fi(Q) = F(ei,£l) and Xi(e) is the characteristic function of the interval It. It is useful to note that

J

Xj(e)de

= A6ij,

(4.50)

where 6ij is the Kronecker delta. Hence, one can approximate the integral of a function denned in E as f F(e) de*

[ Y, Xi (e)Fj de = &j^Fj,

JE

JE

J=O

(4.51)

j=o

with / F(e) de = AFj .

(4.52)

Jij

The quadrature rule (4.51) is nothing but the mid-point rule which gives an error that goes to zero as A 2 . If we approximate the energy dependence of the Boltzmann one-particle distribution function using the stepwise interpolation, we can write (4.34) as

/ J\/2ede« 52 Jli

/ Xh(e)de /

h~j=0

Jli

jR

+

Xj( e *) d e * /

J(eh,ej,e',Cl)de',

JO

(4.53)

78

Lecture Notes on the Discretization

of the Boltzmann

Equation

where we have chosen I = I{. Using (4.50), (4.51) we have / Jsf2e.de K, ^

Xj(e*)de*

/ Xh(e)de

J{eh,ehe'\(l)de'

^ re+ ~ = & J2 ^.ej-.c'.njde', j=0 Je"> 2

(4.54)

where e+ = min{e/i + ej, eu) and e + = min{ei + ej, eu\Recalling Remark 4.1, and because of Proposition 4.1, the interval of integration over e' either results equal to E, or is the union of i + j + £ +1 subintervals of the partition, that is e+=supl/j+,

h+=min{i+j+

£,n}.

(4.55)

Discretizing also e' in the same fashion, gives n

/»

/ J\FTede « A 2 ^ J1*

pe+

/

n

J^Xh(e') de'Jfa, ej, eh, ft)

j=0 J'*™ h=0 n

h+

= A3^^J(ei,eJ-,eA,n).

(4.56)

j=0 h=0

For consistency, one has to neglect the collisions with ek = ei + ej — e^ outside E. This allows us to write the sum over the index h related to an output particle in a more symmetrical way n

/

n

jV^denA3^

J2 J{euej,eh,(l). (4.57) i=o h,k=o h+k=i+j Using the same procedure on the left hand side and equating gives dfi - 4 A 2 ^ ^ i J (4.58) -'

j=o 3=0

X

^ 0 h+k=i+j h

J

°

JSet(et,ei,eh)

Rij(fl

•

flt)

Discretization

of the Boltzmann

Equation and the Semicontinuous

Model

79

where

/< = /<(n) = /(V§i;n),

/A = /h(n;jfc(n,n,,tf)) = /( v ^:n , (e 4 > e > > e h > n, «.,<>)), /:fc = A(A^(n,n.,i?)) = /(>^^n , .(e JI e i> e h ,n > n.,t?)), and where a ^ , Rtj, Q£k and fi^* are respectively the differential cross section (see also (4.80)), the collision momentum, and the post-collisional directions, which will be given in (4.79) for e = e», e* = ej, e' = eh and < = ek.

Remark 4.2 The integrand in (4.58) presents a denominator which vanishes in the case of head-on collisions, e4 = ej, fl = —O*, i.e. for R = 0. Thus, we have to check whether the integrand has a singularity, in that circumstance. Referring to Fig. 2 and choosing ft as polar axis in the integration over S^, one has that y/2iA

#«("•«.)

dnfidfldr

sinfcosf^

^ 2 (i + n-n.)

y^?

^ fi

JO

^

2

where T is the longitude and /? the colatitude (cos/3 = O • O*), because ej = ej. Therefore there is no singularity as /? goes to ir.

4.5

Conservation and Euler Equations for the Discretized Model

Before deducing the conservation equations and the Euler equations for the discretized model (4.58) it is useful to introduce the following preliminary definitions Definition 4.1 Given a regular function of energy and velocity direction <j>(e,Q), its restriction to the sequence of energies e, defines the vector function

#(«) = ($ 0 (n),..., *„(«))

with

* 4 (n) = <j>(eu n ) , i = o,..., n.

80

Lecture Notes on the Discretization

of the Boltzmann

Equation

We say that $ is a collisional invariant if n

< * , J >d=/ ^ i=0

.

v / 2 ^ / ^i{tl)Ji{tl)dSl

= 0,

YA > 0,

(4.59)

•'Sa

where J = (Jo, • • • ,Jn) is the collision operator defined by the right hand side of Eq. (4.58). According to the discussion of Section 4, the following can be proved: T h e o r e m 4.2

The following statements are equivalent:

a) & is a collisional invariant; b) 3>i(fi) =0,(1 + y / iib ( j • SI — c^Ci, i = 0 , . . . , n, Si G S2 for all values of ad, hd and Cd independent ofi and CI. Proof: The proof consists of proving the equivalence of both b) and a) to the following intermediate statement c)

Vi,j, h, k = 0 , . . . , n and ei + ej=eh

+ ek and

WSl, Cl*,Cl',Cl'r € S2 such that y/el SI + ^/ej ft* = yfe~h SI' + je* Sl'r

= • $4(n) + ^(n.) = *!.(«')+ **(«'.) a) <& c) According to (4.58) Eq.(4.59) can be written as .2TT

dSldSl*

< i,j,h,k=0

/S 2 xS f t

•

h+k=i+j+j

aij{Sl -Sl^d) /€i6i

RijiSl-Sl*)

, , (fhf*k -

fif*j) * i ( 0 ) ,

where S^ = S ^ e ^ e ^ e / j ) . The integration domain S2 x S^, the summation and the term in the square brackets are invariants with respect to the exchanges v /£7f2<—>^/ejn,,

y/e~h~Sl'<—>v/ek$V„.

Then, separating the gain from the loss contributions, i.e. writing < *,J >=< *,G > - < *,L >,

Discretization

of the Boltzmann

Equation and the Semicontinuous

Model

81

one obtains

<

*,L> = 4A2 Yl / • • u i nJ0 i,j,h,k=0 ' h+k=i+j

-»°J

M

I

dtidd* JS2X

.ffij(fl-n.,tf) ^(nj/^n.) ^•(n-n,)

[$i(fi) + $ j (a) (4.60)

<*,G>=4A^ t

r«/s

h-\-k=i-\-j

/CjC

'^

.tr 0 -(n-n.,fl)

(ill (ill* S

A

/h(n')/fc(nt)

*i(n) + * j ( n , ) . (4.61)

The second step is to make the substitutions i—*h, j —* k, fi —> fV , and f2„ —> O^,, in Eq.(4.61). Then, recalling that in [13] it was proved that for fixed incoming energies ej and e,, and outgoing energies e^ and e^ compatible with conservation of energy, and for a fixed angle i? between the pre- and postcollisional plane, the Jacobian of the transformation ($V, d'A —> (tl, $7* J given by the conservation of momentum is

(n'.ni) >(n,n,)

e^efc

This allows us to transform the integral over (fl',Cl't) integral over (fi, fi«) € S2 x S^, so that we can write

£ S2 x S^ into an

82

Lecture Notes on the Discretization

< $ , J > = 4A2 Yl

/

-• -• u ,.=0 Jo i,},h,k=0 i+j h+k=i+j

of the Boltzmann

dd I JS22xS„ x

Equation

dddn*

x [*h(n') + $fc(fi'j - *i(n) - 4,(n.)". Because the cr^fe are positive in the domain of integration for all admissible collisions, we obtain:

<#,j> = o

<=>

*h(n,) + *fc(n:)-* i (n)-* i (n.) = o,

which is the desired equivalence. c) «=>• b) We denote respectively by Mb and Mc the linear spaces of the vector functions *(fi) = ( $ o ( 0 ) , . . . ,$n(£l)) verifying the conditions b) and c). It is evident from the definitions of b) and c) that Mb C Mc, that is the model preserves mass, momentum and energy. We need to prove that Mc Q Mb meaning that there are no spurious invariants. In order to do that we express Mc (and consequently Mb) as the direct sum Mc = M* @ M'c so that: * ( f l ) G Mc

$ = $* + *

where * * € M*c , * G M'c

with (4.62)

/ *i{n)dn = o,

(4.63)

for all i = 0 , . . . , n. Consequently, one has Mb = Ml © M'b where

* * eM*b * ( « ) G M'b

$* —o-d- cdei tf J = y/Tttl • bd = y/elC(Q)

i=

0,...,n, i = 0,..., n

Discretization

of the Boltzmann

Equation and the Semicontinuous

Model

83

The proof of the implication A4'c C /A'b is similar to the proof concerning the planar semicontinuous model given in [10] (pp. 119-120) for two velocity moduli only, and extended in [11] (pp. 74-76) to any number of velocity moduli. Following the line of argument given there one can prove that for any \&(fi) verifying (4.63) and the following condition c')

Vi,j,h,k

= 0,. ..,rc

ei + ej = eh + ek

and

V f i , n . , l V , n ' » G S2 n i l

s.t.

and

*Jel& + ^/ejn» = y/ehCl' + y/ek£l*

=> $<(n) + *j(n,) = $h(n') + *fc(n'.) where II is any collisional plane one has ^* = y/TiC(n)

i = 0,...,n.

(4.64)

This means that the linear space M.'c, defined by c') is such that M!d C M.'b- Taking into account the arbitrarity of the collisional plane II and that c) =£• c') (that is, M'c C M'c,) it necessarily follows that M!c C J\A'b. It remains to verify that JA* C JAb. Taking into account that the elements of M* are independent of Q, from c) it follows that §*€MJ

=>

** + $*j = $*h + $£

where

i + j = h + k. (4.65)

Equation (4.65) yields n — 1 linearly independent relations between the n + 1 components of $*. There exist then a
$* = ad - c^ei,

that is, A-l* C JAl, which ends the proof of the equivalence of b) and c). £

R e m a r k 4.3 This theorem not only guarantees the conservation of mass, momentum and energy for the discretized model (4.58), but also excludes the existence of spurious collisional invariants with no physical meaning.

84

Lecture Notes on the Discretization

of the Boltzmann

Equation

The space M. of collisional invariants is then spanned by the basis

(*<0>(n) = ( i , . . . , i )

corresponding to conservation of mass

* (2) (n) = (v^,-..,V^)n-ev &3)(n) = (y/e5,...,^)n-eXi

corresponding to conservation of momentum

*( 4 )(fi) = ( e o , . . . , e n )

corresponding to conservation of energy

The following theorem can then be proved following classical methods

Theorem 4.3

The kinetic model (4-58) has the following properties:

• It preserves mass, momentum and energy, defined by

Ud

n

*

.•-=o i=0

*'S2

2mA Pd

1

n

^

.

J2ei t=0

i=o

n/i(n)dn,

(4.66)

^S'

J

^

• In the non-homogeneous case it yields the usual conservation equations (

^

+ V-(pdVd)

= 0,

d

Q-t(pdUd) + v • (n d + pdvd ® u d ) = 0,

!M*«3i+v.

qd + n d u d + (ed + \pdud) u d

= 0,

(4.67)

Discretization

of the Boltzmann

Equation and the Semicontinuous

Model

85

{V^lCl

V)fi(Q)dQ,

where " Ud = mA^2V2e~ i=0

f ,

'S2

- U) ® {y/2eSl -

A n r qd=^-^V/2e7/ lv^eJn-U|2(V^eJn-U)/j(n)dn; 2 -tea j = o (4.68) • 77ie Maxwellian equilibrium state is defined by the following equivalent conditions a) (log/o,... , l o g / n ) belongs to the space of collisional invariants; b) fi = Adexp[y/BdeiBd • tl - Cdei], Ad > 0; c) J[f, f] = 0 where f = ( / 0 , . . . , /„); • In the spatially homogeneous case the Boltzmann

H-functional

H = jJT V2e~ f MB) log fi(n) dti i=0

(4.69)

''S2

is such that ^ < 0 where the equality sign holds if and only if the system is in the Maxwellian equilibrium state. It can be noticed that the macroscopic quantities defined in (4.66) and (4.68) related to the discretization are the approximations, through our approximation procedure, of the corresponding quantities obtained from the continuous Boltzmann equation. The relation between the Maxwellian parameters and the definition of the macroscopic observables pd, JJd and £d can be obtained substituting in Eq.(4.66) the Maxwellian distribution densities. This leads to: pd = 4irmAdAJ

— S^ e~CdCi sinh V Bd f—'

y/Bdei,

i=0

Pd.Ud =

d

y^e~ c < t e < (y/Bdei cosh. y/Bdet

- sinh \fBdei J ,

i=0

i

nrn

Sd + ^PdUJ = 4TTmAdAJ—B

] T eie~Cdei sinh y/Bd~e~, - «

(4.70)

86

Lecture Notes on the Discretization

of the Boltzmann

Equation

where the hats denote unit vectors. As for discrete models, the map F : (Pd, Ud, U d , £d) —• (Ad, Bd, Bd, Cd) represented by (4.70) is one-to-one, even though, differently from the continuous case, the inversion cannot be in general performed in terms of elementary functions. Here the subscript d stresses precisely the fact that the macroscopic observables are computed from the discretized model, and that the Maxwellian parameters are not those of Eq.(4.17) but are related to such a discretized kinetic model. In particular, given Bd and Cd, the quantities Ud and £d/pd are expressed by y^e 2

Ud

Cdei

(^Bdeicosh

y/Bdei - sinh

^2 e~CdCi sinh

2^ 2£rf , Pd

TT u 2

r d

y/Bdei)

j=o

_

—

eie~CdCi sinh

y/B^i

VBdet

»=o

7T

J2 e~C"ei sinh

y/B^i

i=0

If the dimensionless values Bd = y^-Ud, and Cd — CdUj are introduced, for any given r they are determined by the system Y,e-CdS*/2[Bdei

coshB d ii - (Bd + 1) sinhB d ii] = 0,

i=0

Y^e-Cde\l1

(4.72) U _

1

_ l\

sinh5dg.

= 0

.

However, for not too small values of n, we have Bd « Cd w r, as is the case for continuous models. Moreover, using the values obtained from (4.72), or setting Bd — Cd = r, makes a very little difference -or no difference at all- in the identification of the discretization parameter n through the procedure described below. If one writes the Euler equations related to the kinetic model (4.58), equations similar to (4.26) are found, where the macroscopic observables

Discretization

of the Boltzmann

Equation and the Semicontinuous

Model

87

are replaced by their discretized version, and where en and eq are expressed by J~] e

Cd< ei+ej)

-

[(Bdei

+ 2)siSj - Bdy/e^e]CiCj - \/Bdei

asj]

1 2

Cdei

Y^e-

{^/BdeiCi

-Si)

i=0

(4.73)

^

e

-C

r f

(e

i + e

,)

( e

._

e

.)^

c

.s.

-d-{'j=0

-Bd^B~d-\, Cdei

Y^e-

{^BdeiCi

(4.74)

- Si)

t=0

where c, = cosh y/Bdei and s; = sinh \/Bdei. With respect to the spurious terms £„• and £g encountered in Section 3, when limiting the range of allowed energies, e^ and e^ present a new contribution related to the discretization of the interval E in n + 1 subintervals and to the introduction of the mid-point rule which brings an error of order A 2 . Actually, the constant of proportionality could be explicitly written, but it is useless, since what matters is the overall magnitude of the spurious terms, which is expressed by (4.73) and (4.74), rather than some information on the last discretization step. Once EM and em have been chosen, together with the desired accuracy sn for this discretization, and the discretization interval has been identified by the values 5 M and e m given by (4.43) and (4.44), respectively, one can set n (which will be of order l/y/f^) so that the overall discretization terms denned in Eqs.(4.73) and (4.74) are both smaller than e = SM + £m + £n, that is ed(em,eM,n)'i=/max{|4|,kq|} < £ • The results of such a procedure are reported in Table 3 where the value of n determined for different values of r and accuracy e with £M — £m = £n = f is reported.

88

Lecture Notes on the Discretization

r=10

r=l r=0.1

of the Boltzmann

Equation

£ = 10- 1

£ = 10- 2

e = 10- 3

£ = lO" 4

2 7 5

14 10 9

52 250 263

238 1451 1531

Table 4.3 Values of n at different values of r, such that the spurious terms denned in (1.81) and (1.82) are definitely smaller than the given accuracy e. The values of e/^ and CTTI a r e those obtained in Tables 1 and 2, corresponding to &M = % = £n = § • In those cases in which any value of em < 0.5 gave a contribution smaller than e/3 the value em = 0.5 has been fixed.

r=10

r=l r=0.1

£ = 10-!

£ = lO"2

£ = lO"3

£ = 10- 4

5 5 5

17 9 9

70 391 411

2208 2330

344

Table 4.4 Values of n at different values of r, such that the spurious terms defined in (1.81) and (1.82) are definitely smaller than the given accuracy e. T h e values of eM are those obtained in Table 1, while em is always set to zero.

The number of discrete energies needed to keep the spurious terms below a moderate magnitude is quite small, but rapidly grows if greater accuracy is required. In Table 4, the same thing is repeated with the values of ejw given in Table 1, and with e m = 0. In this way the effects of the presence of a non-vanishing em are evidenced. In our opinion, to take e m ^ 0 is worthwhile only for very small e, while the faster decay of the contribution related to the choice of ? M , makes the possibility of setting £n > £M might be of interest. The values reported in Tables 3 and 4 are computed starting from large values of n, which, of course, yield an overall magnitude of the spurious terms £d « e m + £M, and reporting the value for which the value e is achieved. However, it should be noticed that the terms due to the three steps of the discretization often do not have the same sign. Therefore, for given em and e ^ , it is possible to find much smaller values of n for which the overall magnitude ed practically vanishes. This behavior is evident in Fig. 4 where the cusps in the log scales locate the change of sign. Typically, £d is negative for very large values of n. Decreasing n, £d becomes positive first, and then negative again at very small values of n. In this last case,

Discretization

of the Boltzmann

Equation and the Semicontinuous

Model

89

l

0.1

0.01 0,0

0.001

0.0001

le-05

1

10

100

1000

10000

n Fig. 4.4 Plot of sd as a function of n for e m = 0, e~M — 17.49, and r = 0.1. T h e non regular minima correspond to change of sign in ed. Notice the asymptotic behaviour towards the value £(, = EM = 5 I O - 3 determined when R+ is substituted with [0,17.49].

because of the poor resolution, the change of sign may generate in the logscale plot an abrupt change of slope. An example of smaller values of n yielding a given overall magnitude of s is reported in Table 5. The difference is particularly relevant for smaller values of the spurious terms. We conclude that very few discrete energies are needed to keep the magnitude of the spurious terms moderately low (e « 1 0 - 2 ) . Furthermore, even fewer energies are needed to control the spurious term in the momentum equation, since its magnitude is one order smaller, e.g. em = 0, SM = 16 and n = 9 would give e « 1 0 - 3 . From the viewpoint of application, it seems that the method is more efficient for higher values of r, i.e. when the kinetic energy is much larger than the internal energy. Some flexibility can be obtained by adjusting the various contributions to the overall error e (e.g. changing eju)- However, already from Table 1 it can be seen that the extrema of the energy interval depend weakly on slight changes of EM and £(,. In addition, changing slightly the energy interval influences the numerical complexity only through the possible increase in the number n of energy levels. The important thing is to keep this number low. For this reason, it seems better to allow a larger

90

Lecture Notes on the Discretization

r=10 r=l r=0.1

g^lQ-1 (0, 2.1, 5) (0, 4.7, 5) (0, 14, 5)

g = lQ- 2 (0.39, 2.3, 7) (0.1, 5.4, 9) (0, 16, 9)

of the Boltzmann

£ = 10~ 3 (0.3, 2.46, 15) (0.35, 5.92, 22) (0.71, 17.5, 13)

Equation

£ = 10~ 4 (0.21, 2.63, 33) (0.27, 6.45, 72) (0.5, 19.1, 208)

Table 4.5 Examples of values of (e m ,ejvf,n) with n smaller than the value reported in Table 4 and which yield the desired overall magnitude of the spurious terms.

error contribution to the error directly related to the discretization than to the one related to the identification of the energy interval. 4.6

E n e r g y Formulation of the Collision Dynamics

To conclude the description of our model, we must express the collision dynamics in terms of the particles' energy per unit mass e, and velocity direction. For the sake of simplicity, we drop the index referring to the energy discretization. The pre- and post-collisional energies denoted respectively by (e, e») and (e',e,), are related through energy conservation (4.7), while the pre- and post-collisional directions, denoted respectively by (fi,fi*) and (Ji',n(,), are related through the conservation of momentum, and are given, up to a scattering angle, by

n' = - W v ^ + y/rM* + VF=sg), 2Ve'

where g' € S2, and 5 is defined in (4.8). To clarify the relation between the incoming and the outgoing velocity direction, refer to Figure 2 and decompose the direction of the pre- and post-collisional relative velocities g and g' as g = cos y>R + sin ipeR , g' = cos
(4-75)

f'e'R,

where the unit vectors e# and e'R are orthogonal to R = H/R and belong respectively to the plane containing the pre- and post-collisional directions.

Discretization

of the Boltzmann

Equation and the Semicontinuous

Model

91

The expression of the angles

2(e, - e) gR

g'R

VE2 - S2d

4ee» - S2 E2-S2

sirup =

cosy

E - 2e

(4.76)

2(e'-e'J gR

2e'-E 2

VE

- S2

4e'ei - S2 E2-S2 '

sirup

(4.77)

while the unit vector e ^ can, for instance, be defined by {2e + S)y/2e~M* - (2e„ + S)-\/2eft „ ft x ft, . . cos v + — fl\/4ee* ^ |ft x 7i—ftj sin v if ft ^ ±ft»; e x - (ft • e x )ft |ft x ex ey - (ft • e y )ft Ift x e„|

ft x ex . costf+ — — sintf |ft x e^ cosi? +

ft x ey

« ifft =

LI? if ft =

Ift x e„

-

„ ft»7tex; ft»^e!/. (4.78)

The angle d G [0,2n) appearing in (4.78) is the angle formed by the preand post-collisional plane and is not determined by the conservation laws. In conclusion, the outgoing velocity directions can be written as

ft' =

v /4e

,

e; - S2 ^

VWR

2e' + S

GR+

V^R2

(v^ft + v ^ ^ * ) . (4.79)

ft!

y/4e'e'« - S2 ^ /2ZR

•*R

2e'„ + S +

R2

(Vift + v^ft*)'

Prom Eq.(4.5) and Figure 2, it is evident that, as g' runs over S2, the post-collisional velocities v ' and v^, span the surface of the sphere centered

92

Lecture Notes on the Discretization

of the Boltzmann

Equation

in R / 2 with diameter equal to the modulus of the relative velocity g. Furthermore, for any g' the post-collisional velocities point at two antipodal points. More in detail, if the polar axis of the unit sphere is set along R, one of the post-collisional velocities runs, with varying $, over a given northern parallel and the other over the relative southern parallel (at the antipodes). The other angle defining g', the colatitude tp', determines through (4.77) the energy of the outgoing particle. As is well known, the scattering cross section is a function of the modulus of the relative velocity and of the deflection angle a € [0,7r] formed by the pre- and post-collisional relative velocities, which can be written as 7 = cos a = g • g' = cos ip cos ip' + sin tp sin
(4.80)

_ 1 ~ E2 - S2 (e. - e)(e' - e'J + 4ee, J[1 - (ft • ft,)2]

cosi9 ,

Note that 7 is an even function of ft • ft* and of 1?, and observe that 7(e„e,e'.,n,-n,0)=7(e,e.le,,n-n.,0),

(4.81)

7 ( e ' X , e , f t ' - f t ; , - t f ) = 7(e,e,,e',ft-ft»,tf), which means that the deflection angle is symmetric under the exchange of the roles of the field and test particles. These properties were used in the proof of Theorem 1.2. 4.7

Concluding Remarks

This chapter has presented a kinetic model based on a discretization procedure of the Boltzmann equation written in the energy formulation, in which particles take a discrete number of equidistributed energies, and which has the property to automatically preserve mass, momentum, and energy without giving rise to spurious collision invariants. One of the advantages of the discretization of the velocity space in a particular sequence of spherical shells is that for given pre-collisional energies, conservation of energy is satisfied naturally. Then, conservation of momentum for fixed pre-collisional velocities and post-collisional energies

Discretization

of the Boltzmann

Equation and the Semicontinuous

Model

93

restricts the post-collisional velocities to given (symmetric) parallels of the sphere with polar axis along the collision momentum R, with center in R / 2 , and with diameter equal to the modulus of the relative velocity. In place of the five-fold collision integral over an infinite domain, the discretized model is characterized by a collision term which consists of a sum of integrals over finite domains, namely the cartesian product of a unit circle and the portion of the unit sphere between two parallels symmetric with respect to the equatorial plane perpendicular to the velocity of the field particle, it is known [21] that the quadrature rules for periodic functions or for functions over spheres have a faster convergence. Therefore the collision step of the splitting algorithm can be performed with good precision and small computational effort. Deriving the model via a controllable approximation procedure makes it possible to obtain estimates on the "distance" between the discretized collision operator and the continuous Boltzmann equation, i.e. consistency of the quadrature rule. Finally, the fluid dynamic limit related to the discretized Boltzmann equation tends rapidly towards the usual Euler equations with an isotropic pressure tensor and vanishing heat flux, when the number of energies tends to infinity and the discretization interval tends to M+. Actually, the evaluation of the magnitude of the spurious terms yields itself a procedure to set the discretized energies. This procedure could be developed refining the interpolation step described in Section 4. In fact, in the model described in this chapter the distribution function is approximated by a piecewise constant function leading to a quadrature rule with a rate of convergence of the order of 1/n 2 . This is a rather rough approximation that can be improved using for instance, basic interpolants that span more than one energy interval. A possibility is to use cubic splines \i (e) with support also in the neighbouring intervals, i.e. in Ij_i UI* U l j + i . The structure of the model would still be similar to (4.58) with two extra summations related to the integration of Xi( e ) o v e r the intervals Ij±i. The numerical complexity would then remain the same, but the fact that the distribution function would be approximated with a cubic spline (i.e. a piecewise C 2 function rather than a piecewise constant function) would lead to a faster rate of convergence and therefore to the possibility of using fewer nodes to achieve the same accuracy. However, the exact form of the discretized collision operator depends on how the interpolation at the extrema of the finite energy

94

Lecture Notes on the Discretization

of the Boltzmann

Equation

interval is performed. As stress throughout the chapter this step has to be done with care, in order to preserve the conservation laws and the other properties that characterize the Boltzmann equation. 4.8

References

[1] H. Neunzert, J. Struckmeier, Particle methods for the Boltzmann equation, Acta Numerica, (1995), 417. [2] V.V. Aristov, F.G. Tcheremissine, The conservative splitting method for solving Boltzmann's equation, USSR Comp. Math. Phys., 21 (1980), 208. [3] C. Buet, A discrete-velocity scheme for the Boltzmann operator of rarefied gas-dynamics, in Proc. of 19 t/l Rarefied Gas Dynamics Symposium, J. Harvey and G. Lord, eds., Oxford University Press, (1995), 878. [4] C. Buet, A discrete-velocity scheme for the Boltzmann operator of rarefied gas-dynamics, Transp. Theory Statist. Phys., 25 (1996), 33. [5] C. Buet, Conservative and entropy schemes for the Boltzmann collision operator of polyatomic gases, Math. Models Methods Appl. Set., 7 (1997), 165. [6] A.V. Bobylev, A. Palcewski, J. Schneider, Discretization of the Boltzmann equation and discrete velocity models, in Proc. of 19*'' Rarefied Gas Dynamics Symposium, J. Harvey and G. Lord, eds., Oxford University Press, (1995), 857. [7] A.V. Bobylev, A. Palcewski, J. Schneider, On approximation of the Boltzmann equation by discrete velocity models, Compt. Rend. Acad. Sci, serie I, (1995), 639. [8] A. Palcewski, J. Schneider, A.V. Bobylev, A consistency result for discrete-velocity schemes of the Boltzmann equation, SIAM J. Num. Anal., 34 (1995), 1865. [9] F. Rogier, J. Schneider, A direct method for solving the Boltzmann equation, Transp. Theory Statist. Phys., 23 (1994), 313. [10] N. Bellomo, E Longo, A new discretized model in nonlinear kinetic theory: The semicontinuous Boltzmann equation, Math. Models Methods Appl. Sci., 1 (1991), 113. [11] E. Longo, L. Preziosi, N. Bellomo, The semicontinuous Boltzmann equation: Towards a model for fluid dynamic applications, Math. Models Methods Appl. Sci., 3 (1993), 65.

Discretization

of the Boltzmann

Equation and the Semicontinuous

Model

95

[12] L. Preziosi, The semicontinuous Boltzmann equation for gas mixtures, Math. Models Methods Appl. Sci., 3 (1993), 665. [13] L. Preziosi, E. Longo, On a conservative polar discretization of the Boltzmann equation, Japan J. Ind. Appl. Math., 14 (1997), 1. [14] L. Preziosi, L. Rondoni, Conservative energy discretization of Boltzmann operator, Quarterly Appl. Math. 57 (4) (1999), 699. [15] W. Roller, F. Hanser, F. Schiirrer, A semicontinuous extended kinetic model, J. Phys. A 33 (2000), 3417. [16] W. Roller, A semicontinuous kinetic model for bimolecular chemical reactions, J. Phys. A 33 (2000), 6081. [17] W. Roller, F. Schiirrer, P n approximation of the nonlinear semicontinuous Boltzmann equation, Trans. Th. Stat. Phys., 30 (2001), 471. [18] R. Temam, Sur la stabilite et la convergence de la methode des pas fractionaires, Ann. Math. Pura Appl, 79 (1968), 191. 19] L. Desvillettes, S. Mischler, About the splitting algorithm for Boltzmann and BGR equations, Math. Models Methods Appl. Sci., 6 (1996), 1079. 20] R.J. Di Perna, P.L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math., 130 (1989), 321. 21] P.J. Davis, P. Rabinowitz, Methods of Numerical Integration, Academic Press (1984).

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Chapter 5

Semi-continuous Extended Kinetic Theory W. Roller Institute fur Theoretische Physik, Tech, Univ. Graz, Austria

5.1

Introduction

The standard formulation of the Boltzmann equation [3] describes the evolution of gas particles without internal degrees of freedom. In this case, the dynamics of the gas can visualized as the interplay of free streaming of structureless particles and elastic binary collisions. Internal energy levels of atoms and molecules, however, play an important role in many applications where inelastic collisions have to be considered. An example is the interaction of gas particles with photons studied in the field of Radiation Gas Dynamics [16]. Photons can excite the electrons of the gas particles or induce vibrations and rotations leaving the particles in an excited internal state. Through super-elastic binary collisions of the excited particles, e.g., their internal energy can be released in kinetic energy. Another example is the study of chemical reactions, where inelastic collisions induce changes in the chemical composition of a gas mixture. These chemical reactions are often accompanied by vibrational and rotational excitations of the involved molecules, states that can be treated as internal energy levels of the gas particles. The extended kinetic theory deals with the dynamics of a rarefied gas mixture when the effects of such non-conservative interactions are considered together with the usual elastic scattering mechanisms. It is the combination of nonlinearity and nonconservativity (of kinetic energy) due to absorptions, creations, chemical and nuclear reactions that gives rise to the most interesting dynamical behaviours (cf.[22]). 97

98

Lecture Notes on the Discretization

of the Boltzmann

Equation

The study of these processes is complicated by the complexity of the nonlinear collision terms that oppose a rigorous analytic treatment of the extended Boltzmann equations. Thus, in order to obtain numerically trackable models, the integral operators of the collision terms have to be simplified, i.e. discretized. Semi-continuous models [2; 17] are obtained by discretizing the speed variable leaving a continuum of possible directions. They are closer to physical reality than discrete velocity models (DVM), where only a finite set of different velocities is accounted for [19]. The great advantage of the semi-continuous approach is that collision terms typically contain only sums of integrals over compact domains (parts of the unit sphere). In numerical implementations, these integrals can be treated, e.g., by resorting to an expansion of the distribution functions in terms of spherical harmonics. The aim of this chapter is to establish a consistent semi-continuous kinetic model for the description of non-conservative phenomena. Much work has already been done to formulate the relevant continuous kinetic equations. The simplest model in Radiation Gas Dynamics describes the dynamics of a gas comprised of two-level atoms interacting with the radiation field of monochromatic photons. Extended kinetic equations for this model are formulated and studied in [21]. The interaction of the photons with the gas particles is described by means of Einstein coefficients. Inelastic scattering processes are also included in the formalism of Boltzmann like transport equations. In [7], a third monatomic species of gas particles is included in the kinetic model. This model, however, takes into account only inelastic excitation and de-excitation phenomena whereas elastic collision terms are neglected. In a Radiation Gas Dynamics application such terms are essential to drive the gas towards kinetic equilibrium. Gas kinetic approaches based on the Boltzmann equation have a long tradition in the study of chemical reactions [18]. However, a rigorous Htheorem for the relevant kinetic equations has been derived quite recently in [20]. The formalism of [8] includes, in addition, internal energy levels of the gas particles as well as the interaction with a photon field. Based on the above mentioned work, we introduce a semi-continuous model that includes (in) elastic scattering and reactive collisions well as the interaction with a radiation field. In order to simplify notation, we will not consider the most general model [8]. Instead we will confine ourselves to formulating semi-continuous equations for a five-component gas mixture comprised of the species A,B,C,C* and D, where C* is an excited state

Semi-continuous

Extended Kinetic

Theory

99

of C. All species can have different masses and the mixture can undergo chemical reactions of the form A + B ;=± C* + D. Moreover, monochromatic photons p can trigger excitations of C via C +p -> C*. From a technical point of view, the major new aspect as compared to the original model of Preziosi and Longo [17] is the introduction of different masses of the involved species. This complicates the expressions for the conservation of momentum and energy. However, since only the speed variable is discretized, the high flexibility of the scheme allows us to deal with arbitrary mass ratios [12]. To this end, it is necessary to introduce a separate set of allowed speeds for each species. Inelastic and reactive collisions are accounted for by suitably adopting the domains of integration in the collision operators [11]. This chapter is organized as follows: After introducing the continuous kinetic model in Sec. 5.2, we establish the appropriate semi-continuous kinetic equations in Sec. 5.3. By investigating the collision terms, conservation laws and an if-theorem are shown. For a numerical implementation of the semi-continuous equations, the remaining integrals have to be treated. In Sec. 5.4 this is done by resorting to a P/v approximation. The relaxational behaviour of the semi-continuous model is studied in Sec. 5.5. A comparison with exact solutions of the continuous Boltzmann equation shows the quality of the semi-continuous approach. In Sec. 5.6 we present some applications of the model. First, the evolution of the distribution functions of a three-component gas mixture A, C, C* under the impact of a strong laser pulse is investigated. Then deviations from the equilibrium distribution in fast chemical reactions are studied. Examples of the evolution of the densities and temperatures of the involved species are given.

5.2

Continuous Kinetic Equations

In this section we review the continuous kinetic description of a gas mixture comprised of the species A, B, C, C* and D. The kinetic model is formulated in terms of the distribution functions fN(v,x,t) for each species N = A, B, C,C*,D depending on velocity v, space x and time t. Apart from the free streaming of the molecules, the evolution of the distribution functions is affected by • elastic binary collisions N + M ^± N + M

100

Lecture Notes on the Discretization of the Boltzmann Equation

• inelastic binary collisions C + N ^ C* + N • chemical reactions A + B ^ C* + D • interaction with monochromatic photons p, i.e. — absorption and spontaneous emission C + p ;=± C* — stimulated emission C*+p-*C+p + p. By inelastic scattering and chemical reactions, kinetic energy is transferred to internal energy. Denoting the internal energy of N by EN, the difference in internal energy of C* and D compared to A and B is given by AEchem = Ec' + ED - EA - EB > 0. The energy gap between C* and C will be denoted by AEint = EG* - Ec > 0. For a general binary collision of gas particles N + M^±N'

+ M',

(5.1)

the conservation of mass M, total energy E and momentum R reads M =

m + m» = m' + m'*

(5.2a)

E

=

p2/(2m)+p2J(2mt,)=p'2/(2m')+p'?/(2m'lt)±AE

(5.2b)

R

=

p£l+p*Si*

(5.2c)

=p'Cl'+p'tfl't

,

where m,m*,m' and m'„ stand for the masses of N,M,N' and M', respectively. Equivalently, p,p*,.. • denote the moduli and the unit vectors Cl, fit, • • • the direction of the momenta. The energy variation AE is zero for elastic scattering, AE = AEint for inelastic scattering and AE = AEchem for reactive collisions. Primed symbols generally refer to post-collisional quantities. The integro-differential equations [20] governing the evolution of the phase densities fN are of the form

-dT

+ v

--^-

Z,

J

+

J

>

(5-3)

M=A,B,C,C',D

where the term JNM describes elastic binary collisions N + M ^± N + M, and JN contains the contributions of non-conservative interactions to the evolution of species N. The last term can be decomposed into JN

=

JN+

jN+

jN

+ TlN ,

(5.4)

Semi-continuous

where JN

Extended Kinetic

Theory

101

describes the effect of reactive interactions on iV = A

v

v

A,B,C*,D.

v

The terms J , J account for the effects of collisional excitation and de-excitation respectively on species N = A,B,C,C* ,D. The interaction with photons is covered by 1ZN and appears only for N = C,C*. Species C and C* interact with the radiation field according to the photon-particle interaction term. Using the Einstein coefficients a for spontaneous emission and /? for absorption and stimulated emission, this term can be written as c c c c t

n \i) = -n (i) = J^dft, (pi(n )f (v)-(a + f3i(ti*))f \v)) .

(5.5) Here, I denotes the specific intensity of photons with energy AEint- The evolution equation for the specific intensity I(t, x, Cl) is given by [16]

^t+cCl~=qL(x,t)-cAEint

J^dv

(plfc(v)-(a

+ f31)fc'(v))

,

(5.6) where c is the speed of light and q^ (x, i) represents the external light source. Next we consider the explicit expressions for the particle-particle collision terms. To this end, we decompose the pre-collisional relative velocity g = v*— u a s g = gh with \fi\ = 1 and the post-collisional relative velocity g' = v'„ — v' as g' = g'h' with \n'\ = 1. For elastic scattering, the relative speeds before and after the collision are equal, g' = g. The elastic collision terms can be cast in the form JNM

= Jdv. R

3

Jdn'gcTNM(g,X)[fN(v')fM(v'f)-fN(v)fM(v*)] S

, (5.7)

2

with the elastic cross section aNM depending on the relative speed g and the cosine x = cos ip = h • n of the angle of deflection ip. The post-collisional velocities are given by v' = i

(R - mMg n)

and

v[ = ^-(R

+ mNg n') .

(5.8)

For non-conservative collisions with energy gap AE, we introduce the speeds gNM via g2NM = 2AE//J,NM, where nNM = mNmM/(mN + mM) stands for the reduced mass. Then the relative speed g' after a non-conservative

102

Lecture Notes on the Discretization

of the Boltzmann

Equation

interaction reads

AM' =

fe(?2±4).

7

M

(5-9)

The minus sign refers to the endothermic and the plus sign to the exothermic direction of the interaction, respectively. Of course, for inelastic (nonreactive) scattering the fraction of the reduced masses is equal to one. The A

V

terms JN,JN, and JN accounting for non-conservative collisions (5.1) with the cross sections G^M (#> x) n a v e the expression

J$MM' = Jdv.Jdn'gv%'£'(g,X) s2 " / ,,NM \

3

(^W) / w V)/ M 'K)-/>)/ M (**)

and it is understood that a^M (#> x) = 0 when g < g^M in the endothermic reaction. The gain terms result from microreversibility conditions [14]

(»N'M')292^M.(9,X)

= (^M)V)2<«V,X) •

(5.H)

The post-collisional velocities, depending now on the relative speed after collision g', are given by v' = ±(R-

mM'g'h') ,

„', = ! ( «

+ mN'g'h') .

(5.12)

This model can be seen as a special case of the more general kinetic model presented in [8], with the additional external source of photons qL{x,t). In [8], the conservation of mass, momentum and total energy as well as an H-theorem are derived. Moreover, Maxwellians are found as equilibrium solutions with equilibrium particle densities within the same species distributed according to the Boltzmann factor. This entails Planck's law as the equilibrium intensity distribution. In chemical equilibrium, the mass action law is recovered. 5.3

Semi-continuous Kinetic Equations

In order to discretize the above sketched model, we apply a generalized form of the procedure [17] yielding a semi-continuous extended kinetic model [11;

Semi-continuous

Extended Kinetic

Theory

103

12]. As a first step of the discretization procedure, we restrict the range of the particle's kinetic energies to the interval [Em,EM), 0 < Em < EM < oo. All particles with kinetic energies outside of this interval will be neglected. Second, we introduce an arithmetical sequence of energies Ei = Em+ with 5 = (EM — Em)/(n groups)

(i + \)

S,

i = 0,l,...,n,

(5.13)

+ 1) being the centres of the subintervals (energy

Ii=[Ei--,Ei

+ £],

»= 0,l,...,n.

(5.14)

Since we allow for different masses m " of the gas particles, the energies Ei correspond to a separate set of speed knots

V

? = )l^{E™+{i+~2)6)>

(5 15)

"

for each species. The associated momenta are given by pf = mN vf. As a third step, the energy gaps AEch,em and AEint have to be approximated multiples of S. Thus we set AEchem = Qchem 6 and AEint = q%nt S with Qchem, Qint € { 1 , 2 , . . . , 2n - 1}. By labelling the energy groups of N,M,N',M' by i,j,h,k, respectively, for elastic collisions, detailed energy conservation is expressed as i + j = h + k. Moreover, reactive (inelastic) collisions imply i + j = h + k± qchem (i+j = h + k±qint). Now we formulate kinetic equations for particles that are constrained to move with the speeds v^Q. To this end, we introduce the set of distribution functions / = {f?{£l,x,t) = fN(v^Cl,x,t)}, i = 0 , . . . ,n, N = A,B,C,C*,D. The evolution equations for / are obtained by integrating Eq. (5.3) over each energy group. When appearing as an integrand, any function of kinetic energy (and thus of speed v) is approximated by a piecewise constant interpolant defined over the above stated discretization. The resulting semi-continuous equations are of the form M

where J/

VM

denotes the impact of elastic collisions between species N and

104

Lecture Notes on the Discretization

of the Boltzmann

Equation

M on the evolution of the i-th energy group of TV. Similarly, J^ contains the impact of non-conservative interactions on the i-th energy group of species N.

5.3.1

Collision

terms

For the derivation of the collision terms we proceed following the lines of [11; 12]. The detailed conservation of momentum, i.e. Eq. (5.2c), implies a restriction on the set of possible collisions. In fact, by taking the square of Eq. (5.2c), we obtain using |ft| — 1 p2 + pi + 2ppM

• ft* = p'2 + p'2 + 2p'p*ft' • ftl .

(5.17)

As a consequence of —1 < ft • ft, < 1, we have

2^

(V-P1.)2

- V -rf) < A • ft. < ^

((P'+P'S -P2

-PI)

for colliding particles of momenta pft and p*ft». For a fixed ft, the set of all ft* satisfying the above condition will be denoted by £>», i.e. -D*(ft,p,p»,p',p'J =f {all allowedft, given ft, p,p*,p',p'*}

.

(5.19)

Three examples of such sets are depicted in Fig. 5.1. Moreover, it is important for the discretization that the angle of deflection a and thus ft and ftt can be expressed as functions of the variables (u,i;*,i;',ft,ft*,$), where i? € [0, 27T) [17]. Also the surface element dn' appearing in the collision terms of the continuous kinetic equations can be cast in a convenient form [12], i.e. M2v' M

' = ^VRdv'M-

{5 20)

'

By following the considerations outlined in section 4 of [17], we are now able to derive semi-continuous versions of the collision terms. We integrate Eq. (5.7) over the energy intervals I, with the appropriate measure y/2E/m3dE = v2dv. All functions of kinetic energy are approximated by piecewise constant interpolants. For the collision N + M ^± N + M, the

Semi-continuous

Extended Kinetic

(a)

Theory

105

(c)

« \

n n,

D

5T—i

D.

Fig. 5.1 Domains of O , as implied by the conservation of momentum for elastic scattering of particles of equal mass (a), collisional de-excitation (b), and collisional excitation

to-

resulting elastic collision term reads (cf. [12]) 2n

62

did j

°

h i+

I

<M*

D.(£l,mNvy,mMvV,mNv£,mMv>f)

jlh°+k0

x { « n • Ct„d)}NM [/fcM(«'j/f(fi') - / f (ft,)/f (n)] . (5.21) In this case, the unit vectors in direction of the pre- and post-collisional velocities are given by

e

•*'

(5.22)

With the surface element dn' as given by Eq. (5.20), we find for the kernel

{A^-^mNM

=

M2

^^NM(9^)-

(5.23)

Microscopic reversibility implies several symmetries of this kernel whose actual form can be found in [12]. Now, the semi-continuous collision terms for the bimolecular chemical reaction and the inelastic collisions are presented. For brevity, we consider the general non-conservative interaction N + M ^ N' + M'. The

106

Lecture Notes on the Discretization

of the Boltzmann

Equation

discretization of Eq. (5.10) yields

-

2

2TT

w™'w=JhFT,
° M-*=i+W
x{^/(n.ft.,t?)}^,

/*>

/

d«.

D.(fl,i«lv«>"«f,m'''«h''',ro*''viM')

3

( j W ) /^'(n:)/r(n')-/f(n,)/^(n) (5.24)

where g = qchem for chemical reactions and
^M(R-mM'9'fl')>

^

=

^wi(R

+ mN g,fi

'

')-

(5 25)

-

The kernels A are of the form M?/(ft •«.,,?)}##'

= ^ K F < M ' ( 9 , ^

•

(5-26)

In the case of inelastic collisions N + C ^ N + C*, Eq. (5.24) simplifies since the mass ratio cancels out. For collisional excitation (indicated by A), we find for interaction with N = A,B,D

m

WIT

S--2

«,

•>

m

~?0

^la JO h+k=i+j-qint

f

JD,(U,mNv^,mcvf,mNvNtmcvc^

x{i?/(ft.n,,tf)} J V [/f(nl)/^(n')-/f(n.)/f(n)] , (5.27) whereas collisional de-excitation (indicated by V) is described by the terms v

r2 A2

n

""

"

^[/]=4TE^

E

r2n r*'"

/««/

r

dCi

*

h+k=i+j+qint

x{i?/(A-n.,tf)} J V [/f(n , j/^(n')-/f(ft.)/^(n)] • (5-28)

Semi-continuous Extended Kinetic Theory

107

The impact of inelastic collisions on species C and C* is, of course, more pronounced. It is described by a sum of the terms

J*CNlf] = J L Yv" V j=0

dn.

/ d$ /

M^o - / ° h+k=i+j-qi„t

iD.(n,m^f,m«»>^hVs»;')

x{A^(n-Ci^)}N[^(n't)f^(ft')-f^(Cit)ff(Ci)\,

(5.29)

for C and equivalently c N

n

j2

n

/•27T

Ji ' \f\ = ^ h v E < E

M

dCi

*

3

h+k=i+j+qint

x{A^(Ci-Ci^)}N[f^n[)f^n')-f^(Ci,)ff\n)] for C*, where N = 5.3.2

, (5.30)

A,B,C,C*,D.

Macroscopic

quantities

In the semi-continuous formulation, the macroscopic quantities of the gas are obtained by summing over all energy groups It and integrating over the directions Cl [17]. For each gas species, the particle density, the mean velocity, the momentum flux and the kinetic energy flux are respectively given by nN '"

uN

JS

j=0

= 3=0

s

rN

—

u

n

E (v7)3 / " ®ft/ T ( f t ) d " > 3=0

QN = l E C O 4 / " ^ " ) ^ -

( 5 - 31c ) ^- 3 1 d )

The total energy density e of the gas mixture reads e= ^ M

(kM + nM EM)

,

(5.32)

108

Lecture Notes on the Discretization

of the Boltzmann

Equation

where kN is the kinetic energy density defined as kN = (1/2) t r K ^ . The energy density e„, the energy flux Q„, and the energy density SL of the light source are respectively defined by

e„ = -c y/ 2 /(n)dn,

Q „ =y/ 2 n/(n)dn,

S

S

sL = —c qL. (5.33)

In order to derive macroscopic equations, we integrate the semi-continuous model equations (5.16) over all solid angles fl and sum over all energy groups Jj. The integrals over Cl and £l* are well suited for the loss terms. For the integration of the gain terms, we apply the important relation

fl(ft',ft'.) = d(ti • n'j = vv*_ 3(A,n,) 5(ft-n») p'K that can be found by differentiating Eq. (5.17) and following the lines of the proof of proposition 2.4 in [17]. Thus, instead of integrating the gain terms over ft and ft*, we can integrate them over fl and fi„ using the Jacobian (5.34). By summing over all species, we are now able to recover the discrete version of the macroscopic equations for the gas mixture,

!£»" + £ - I > v = 0. °T N d t

-^Ti

m

N dX

N

| £ d

°X

V „ "

+ ^

( 5 - 35a )

+

N

*

•(£

£

K

" = 0 ,

(QN + ENnV)

(5.35c)

+ Q„)

=

U

,

(5.35d)

N

where the sums are extended over N = A,B, C, C* and D. They are equivalent to the macroscopic equations of the continuous kinetic model [8]. The first three equations reflect the conservation of particles, mass, and total momentum. As an impact of the chemical reactions, the individual densities nN do not remain constant. Their total sum n, however, is preserved. Equation (5.35d) expresses the balance of energy which is affected by the external source term qL-

Semi-continuous

5.3.3

Properties

Extended Kinetic

of the collision

Theory

109

terms

The derivation of the conservation equations (5.35a)-(5.35d) and of equilibria involves symmetry relations of the collision terms. Adopting the notation of [11], for a set of arbitrary functions

The main properties of the continuous kinetic model [8] also hold for the semi-continuous approach, as follows. First, ( ( ^ J ^ ^ ^ J ^ N

+^ J ™ )

(5-36)

M

vanishes for tp? = 1; f = m f (i>f) 2 /2 + N E . This correspond to the conservation equations stated in Eqs. (5.35a)(5.35d). Furthermore, we obtain a semi-continuous form of the spacehomogeneous H-theorem [20] by setting ipf = log (f^f{mN)3). It reads 2 | =«log/,J»<0.

(5.37)

This H-function is zero (collisional equilibrium of the gas mixture) if and only if / ^ (ft) = AN exp [mN {v?b • ft - c ( w f) 2 )] ,

(5.38)

with the constants AN, 6, and c = l/(2fc B T). The constants AN are related to the densities nN of the components [12]. In chemical equilibrium we thus obtain for b = 0 nc*nD frncmD\3/2 ( AEchem\ - ^ T = \ ^ B ) ^p [—j^r-) ,

(5-39)

which is the mass action law [12] and for general b nc'

^=

eXP

(

AEint\

l~Wj-

(5 40)

"

The latter equation implies that the densities of ground state and excited species are linked by the Boltzmann factor.

110

Lecture Notes on the Discretization

of the Boltzmann

Equation

In the absence of an external light source, the intensity I corresponds to the self-consistent radiation of the gas mixture. In equilibrium, radiationinduced transitions must vanish, i.e. TZC = Up — 0. By substituting Eq. (5.38) and Eq. (5.40) into Eq. (5.5), we obtain exp(AEint/(kBT))

- 1'

which is exactly Planck's law of radiation. A proof of these relationships follows the lines of the proofs in Sec. 4 of [11], and applies classical arguments of kinetic theory [3]. Their application is possible due to the special form of the discretization of the speed variable [11; 17]. The microreversibility conditions are of crucial importance for the properties of the collision operators. For elastic collisions they imply (1, JNM) = 0 as well as (
(1,JA)

= (1,JB)

= -(1,JC*)

=

~(l,JD)

for the collision operator relevant to chemical reactions. Thus, the semicontinuous kinetic equations share the fundamental properties of the original continuous kinetic model [8].

5.4

Treatment of the Remaining Continuous Variables

In the above treatment, we have discretized the speed variables, i.e. the moduli of the velocities of the gas particles. The distribution functions fN still depend on three continuous variables, namely the position x, the direction fi of the velocity and the time t. For numerical calculations, we have to treat these variables. The simplest case is to neglect dependence of fN on x and Cl. Then we describe the temporal evolution of a space-homogeneous and isotropic gas mixture. Naturally, space-dependent problems are more challenging, and we will confine ourselves to one-dimensional geometries. 5.4.1

Space-homogeneous

formulation

As already stated above, the simplest case is that of a spatially homogeneous and isotropic gas mixture. In this case, the evolution function is no longer dependent on x and Cl. To derive the space-homogeneous isotropic

Semi-continuous

Extended Kinetic

Theory

111

equations, we resort to the so-called PQ approximation by making the ansatz / f (

"} = 4

^^

(5 41)

•

-

By integrating the above equation over all directions Ct, we obtain

n?=v? f f»{fl)dfl.

(5.42)

^ f = Qf + £ ^ M -

(5.43)

7s 2 We infer from Eq. (5.31a) that the quantities nf correspond to the number densities of particles N within energy group i. Evolution equations for nf are obtained by integrating Eqs. (5.16) with respect to tl. This yields

M

This is a set of coupled ordinary differential equations. The integrated elastic collision terms read

^M-J^rt

(mNM<^-{i%}™n?nf),

t j=0

h,k=0

h+k=i+j

with the integrated elastic cross sections given by

m

Juo

JX

JUQ

du

The domain of integration over u = Cl • Cl*, i.e. (uo,ui), coincides with the bound of the product fi •fi*as given in Eq. (5.18). For species A, the integrated collision term relevant to reactive interactions is given by

& = ^&^± j=0

t *?(*3$»f «?-*»?).

h,k=a h+k—i+j—q0hcm

\

h

k

'

where Ji = (fiAB/(iCD) . For species B, the above formula applies with the substitutions A <-» B, C <-> D and JCff -> /C*/1. Similar expressions can be found for the terms Qf * and Qf. In fact,

^-SSOTS: j=0

£ c(^-^-r-f)

h,k=o

h+k=i+j+qcher„

x

l

3

'

112

Lecture Notes on the Discretization

of the Boltzmann

Equation

and, again, Qf is obtained by exchanging A <-» B, C «-> D and K.l^k -» /Cj^. The quantities /C are evaluated with the endothermic cross section ^

= 2.M2 /

-£-aTBD(9)du,

(5.44)

and, of course, microreversibility can be exploited to formulate the integrated collision terms. Equivalent formulae apply for the terms relevant to inelastic scattering. The mass ratio Ji is then to be replaced by one and the up-scattering kernels read {K%}N

5.4.2

= 2n(mN

One-dimensional

+ mc)>

P

-f-

Juo

9 "•

ZN(g)du.

problems

The effort of treating one-dimensional scenarios is much greater than that of space-homogeneous problems. Now the distribution functions fN depend on one spatial variable z, say. While the collision term of Eq. (5.16) is local in space, the streaming part of the Boltzmann equations links neighbouring spatial points z and z + dz. This implies that, if we want to account for streaming, we cannot simply integrate over all angles Cl. We can either discretize Cl by allowing particles to fly only along a finite set of directions. Or we expand the distribution function for each speed in a truncated series of spherical harmonics. In this chapter we will adopt the second strategy. Moreover, we assume azimuthal symmetry which implies that we have to expand the projection of Cl on the z-axis in terms of Legendre polynomials Pi. Using spherical coordinates Cl = (sin 0 cos ^>, sin 0 sin 0, cos#), we make the ansatz N

f?(£l,x,t)

= ftN(cos0,z,t)

= 5>£(z,t)P*(cos0) ,

(5.45)

(=0

where a^z, t) denotes the Z-th space-time dependent coefficient of the Legendre expansion of ff*. In order to evaluate the streaming term of Eq. (5.16), we insert the ansatz, Eq. (5.45), into the LHS of the Boltzmann Equation (5.16), multiply the result by P\(cosd) and integrate over the angles 6 and <j> using the

Semi-continuous

Extended Kinetic

Theory

113

measure sin OdOdcj): } 3 2-K I sin 6d6 PX(cos 0){-Y,

N

tfiPi(cos

N 8 6) + v? -g- cos(0) £ aftfi(cos 6)).

1=0

Q

1=0

The integrals can, in principle, be evaluated to all orders N. A common approximation is N = 3, i.e. the P3 approximation where all Legendre moments higher than three are neglected. In this case we are led to the following set of equations for A = 0,1,2,3: gi2ao,i

+ v

hi

i ^ 3

= <Jo,i + l^

J

o,i

>

(5.46a)

'

(5.46b)

M

di3ai,i

+ v

i 3 J 3 \ao,i+

5 a 2,iJ = Ji,i + 2^,Ji,i M

^ 5 a 2 - i + Ui a i s l r ^ + r M - ^ + z - ^ '

(5 46c)

-

M

^7a3,i

+ v

i a^7 5

2

'*

^s.i + Z ^ J 3,i

•

(5.46d)

M

The calculation of the collision terms J^t and J j ^ is rather cumbersome. The latter terms depend on {a^0,..., a^n, a^0,..., a^n} whereas the former terms can depend on the whole set { a ^ 0 , . . . , afn}. The collision terms are inferred from Eq. (5.16) by the same procedure as the streaming terms, i.e. by using the ansatz, Eq. (5.45), multiplying by P\(cos9) and integrating over (j> and 6 using the measure sinOd0d(f>. To some extent, the integrals of the loss terms can be simplified by hand. This is possible because in the loss terms the integration extended over angles that appear directly in the integrand. Due to the transformation to pre-collisional velocities, the evaluation of the gain terms is much more complicated. However, both gain and loss terms can be obtained by using a symbolic processor like MAPLE and performing one remaining integral numerically. The appropriate mappings and expressions can be found in [13]. The resulting equations are still sets of coupled partial differential equations in z and t. They can be solved by resorting to the fractional step method [1]. This approach splits every time step of the evolution of the gas mixture in a free streaming part (no collisions) and a collision part (no streaming).

114

5.5

Lecture Notes on the Discretization

of the Boltzmann

Equation

Relaxational Behaviour

Due to the multitude of allowed collisions, a semi-continuous kinetic model is supposed to be closer to the full continuous kinetic description than discrete velocity models [15], where a finite set of allowed velocities v restricts the number of possible collision processes drastically. Per constructionem, semi-continuous kinetic equations are still of integro-differential type. However, the integration is restricted to compact domains (parts of the unit sphere). Due to their practical usefulness and formal agreement with the continuous theory, it is reasonable to ask about the inaccuracies induced by the derivation of the semi-continuous models. Partially, this has been done in [17] where the fluid dynamic behaviour of the model is investigated. By using local Maxwellians, one obtains estimates of the spurious terms introduced by their discretization procedure. In this section we will investigate to what extent the local relaxation of the semi-continuous model coincides with that of the full continuous Boltzmann equation. For the nonlinear Boltzmann equation, analytical solutions exist only for particular interaction models [5]. An exact solution, the so-called BKW mode, is known for Maxwell molecules (MM) whose cross section a is inversely proportional to the relative speed g. A further analytical solution exists for the case of very hard particles (VH), with a ~ g. This solution, however, is only valid for a two-dimensional velocity space. Thus, we can only compare the three-dimensional semi-continuous model with the BKW mode. The results of this comparison are outlined in Sec. 5.5.2.

5.5.1

Kinetic

equations

and BKW

mode

In order to compare the semi-continuous formulation with the BKW mode, we consider a gas comprised of one single species, A say, of a monatomic gas. Since we treat only one species in this section we drop the superscript N referring to the species. The phase density fi(fl, x, t) of A evolves according to the semi-continuous Boltzmann equation

%+**•%

= * •

(5-47)

Semi-continuous

Extended Kinetic

Theory

115

In the case of isotropic elastic scattering, the nonlinear collision terms are of the form 2v

J

V

< = T£ '- £ j=0

h,fc=o

/

dCl

* / d04*(n-ft,)[/fc(n')/*(ftl) - /<(«)/,-(«,)]

•/r>*

£

with the kernels expressed in terms of the differential cross section a(g) and the total momentum R as A^(Ct • fi*) = 4a(g)/R. The domain of integration is given by Eq. (5.18) that simplifies to D* = {Cl* : \Cl •fi*|< VhVk/(ViVj)}.

The analytic solutions of the BKW mode assume spatially homogeneous and isotropic conditions. Therefore, we perform a PQ approximation by means of the ansatz Eq. (5.41). The evolution equations for nt are obtained by integrating Eq. (5.47) with respect to Cl:

^ =y £ £ j=0

(#»*»*-#»"*).

M8)

h,k=0

h+k=i+j

with the integrated cross sections Itf=2n

f

AlJk(y)dy.

(5.49)

J —X

The domains of integration (—x,x) coincide with the bounds D* of the scalar product Cl • fi*. The microreversibility implies for the integrated cross sections lH = ^ - 1%. (5.50) hk vhvk l] For Maxwell molecules, the kernel A^Nu) is a symmetric function of its argument u = Cl • Cl*. We find the expression A%(u) =

K

J(vt+v*)*-4v?v]u*' where the positive parameter K controls the strength of the interaction. By inserting this formula into Eq. (5.49), we obtain 27TK f arctan ( iffiffii ) fclf o r vhvk < "»«; /** = _ >' 2" < ~ • ViVj y arctan ( j ^ I ^ J for vhvk > ViVj

(5-51)

116

Lecture Notes on the Discretization

of the Boltzmann

Equation

1

'

iThW 1 •

! i I •'

x=4 x=5 x=6 x=8 x=10 x=12 x=14

//'/

'/// 0.04

0.06

0.02

0.04

time

0.06

0.08

0.1

time

Fig. 5.2 Relaxational behaviour of Maxwell molecules at different kinetic energies. The curves calculated with the semi-continuous model coincide with those of the analytical solution. The initial distribution corresponds to the BKW mode at 77 = —0.4.

On the other hand, denoting the kinetic energy by x = v2/2, the BKW mode [5] reads in three dimensions

f(x,t) =

2y/x V^(l+7i)

3/2

expr ~" V

1+

7i

! + 7 i 7 l " " l + 7i V2

l + 7i

)}

and the temporal evolution of 71 is given by 7 l (f)

=j?e_A2t,

-0.4
(5.52)

The quantity A2 is a nonlinear eigenvalue of the collision operator. Its value is linked with the strength of the interaction measured by K. For isotropic scattering, A2 is given by A2 5.5.2

27T K

T 4

(5.53)

Discussion

By solving numerically Eq. (5.48) with the integrated kernels (5.51), we obtain the relaxation of a gas of Maxwell molecules. The kinetic energy is resolved using 175 energy groups. The strength of interaction is taken to be K = 100 and the initial data correspond to 77 = —0.4. Figure 5.2 shows the relaxation for different kinetic energies x. For very low and very high energies (x < 1 or x > 4), the particle density increases whereas for average energies (1 < x < 4) it decreases during the evolution. For high energies, the relaxation becomes slower. It is remarkable that not

Semi-continuous

Extended Kinetic

52.76

Theory

117

•

•

fitted values Xg

52.72 52.68

-ii'-1 ji \!

Initial BKW Maxwellian

52.64 -:' i '• £

52.60

-§

I

52.52

52.44 52.40

—

>

\».

-ii :i •ii

i

1

0.8

'•>.

ii

52.48

1.2

-li '•'. ii \!

52-56 -ii

"5

*

exact value Xg - - -

*

;

•

'•>. -A \

+

*t •

\

•

•

+ •

•

*

•

* -

*

0.0

52.36 2

4

6

8 10 kinetic energy

12

14

16

18

Fig. 5.3 Eigenvalue A2 versus kinetic energy x of the Maxwell particles. The dashed line refers to the exact value of A2. For comparison, also the initial distribution of the BKW mode and the final Maxwellian are plotted.

0.01

0.02

0.03 time

0.04

0.05

0.06

0.01

0.02

0.03

0.04

0.05

0.06

time

Fig. 5.4 Temporal evolution of the adimensionalized entropy for Maxwell molecules (left) and hard spheres (right).

the slightest difference between the BKW mode and the relaxation of the semi-continuous model can be observed. The curves for both cases match. In order to obtain a quantitative result, we investigate the only parameter in the BKW mode, i.e. the nonlinear eigenvalue A2. For some chosen energies x, the parameter A2 is adapted by a least squares fit of the data of the semi-continuous model. In Fig. 5.3, the fitted data are given. They coincide with the exact value within less than 0.2% for average particle

118

Lecture Notes on the Discretization

of the Boltzmann

Equation

1.01

6 8 10 12 14 16 18 kinetic energy

0.99

6 8 10 12 14 16 18 kinetic energy

1.01

1.00

6 8 10 12 14 16 18 kinetic energy

0.99 6 8 10 12 14 16 18 kinetic energy 1.01

o 1.00

0.99 6 8 10 12 kinetic energy

14

16

18

6 8 10 12 14 16 18 kinetic energy

Fig. 5.5 Relaxational behaviour of different gas particles. The plots show the ratio distribution function / vs. Maxwellian M at various instants of time. In the course of time, all curves tend to f/M = 1. Note the slow relaxation of the tails in the case of Maxwell molecules (MM). An especially fast relaxation is observed for a hard sphere gas (HS) at x w 0.

speeds. We observe that the discrepancy increases for high speeds. One reason for this behaviour is the unphysical but necessary cut-off at a maximum kinetic energy EM- Particles propagating with kinetic energies near the allowed maximum x = 18 have a very limited possibility of scattering, because no particles with greater energies may result from the collision pro-

Semi-continuous

Extended Kinetic

Theory

119

cesses. Nevertheless, even here the discrepancy is less than one per cent. The accuracy of the relaxational behaviour of the semi-continuous model is in striking contrast to the results obtained with discrete velocity models (DVM). In fact, a comparison of a DVM with analytical solutions [19] reveals major discrepancies. The relaxation of the studied DVMs is five to seven times slower than predicted by the exact solutions. Apart from Maxwell molecules, also hard spheres (HS) with a(g) = const and very hard spheres (VH) with a(g) ~ g are interesting interaction models. For these models the integrated kernels 7 can be calculated analytically. For MM and HS, the evolution of the entropy is plotted in Fig. 5.4. In both cases, already nine energy groups approximate the relaxation very well. On the scale of the left plot, there is no difference between n = 50 and the entropy of the exact solution. To obtain the same accuracy, more energy groups are needed for the HS model. The types of relaxational behaviour induced by the various cross sections (MM, HS, VH) are illustrated in Fig. 5.5. The plots show the ratio semicontinuous distribution function versus discretized Maxwellian at various instants of time. The initial distribution corresponds to the BKW mode for TJ = —0.4. Local equilibrium is reached when this ratio equals unity for all kinetic energies. One observes striking differences between the HS and the MM dynamics. The tails (high kinetic energies) relax much faster in the HS case than for MM. On the other hand, the HS gas displays an interesting behaviour with a local minimum for low speeds that is neither observed for MM nor for VH.

5.6

Applications

In this section we show some applications of the semi-continuous extended kinetic model. First we consider a model in Radiation Gas dynamics without chemical reactions and present numerical results of space-homogeneous and spatially one-dimensional simulations. In the second part of this section we will turn our attention to fast chemical reactions.

5.6.1

LITA

By simplifying the semi-continuous extended kinetic model, i.e. setting the densities of B and D equal to zero, chemical reactions are excluded.

120

Lecture Notes on the Discretization

of the Boltzmann

Equation

Fig. 5.6 The geometry of the laser field induced by the two incident driver-beams. High intensities are shown in black. The region of interference coincides with the sample region of the experiment.

Consequently, we obtain kinetic equations describing the interaction of a three-component gas mixture with photons triggering transitions between the second and the third species. Such a situation is typical for a laser induced thermal acoustics (LITA) experiment [4], where photons (supplied by a strong coherent laser pulse) excite high frequency acoustic waves. The spatial intensity pattern of the photons results from a two-beam interference and is depicted in Fig. 5.6. In this scenario, the internal energy difference between C and C* can be orders of magnitudes greater than the mean kinetic energy of the gas particles. Therefore, deviations of the distribution functions from local Maxwellians due to de-excitation processes are to be expected. Recently, a kinetic approach to LITA experiments has been presented in the framework of discrete kinetic theory [10]. 5.6.1.1

Space-homogeneous problem

First we investigate a space-homogeneous problem related to LITA. The numerical calculations are done in Pb approximation. For the examples given below, the following parameters are chosen a = 105 s _ 1 , (3 = 107 m 2 J - 1 , m = 44 a.m.u., nc + nc* = 2.5 x 10 2 2 mr 3 , AEint = 0.27eV and for downscattering a (g) = 5 x 10~ 1 7 m 3 s _ 1 /p. For the dominant species A we consider two different densities, namely 2.5 x 10 23 m~ 3 (low density) and 2.5 x 10 25 m - 3 (high density). We investigate the following scenario. At time t = 0, the gas mixture is in thermal equilibrium with the radiation field at temperature T = 293K.

Semi-continuous

Extended Kinetic

121

Theory

10 ns 30 ns 50 ns 80 ns 120 ns 200 ns

12

I1I"

•

8

0.1

0.15 0.2 energy [eV]

0.15 0.: energy [eV]

Fig. 5.7 Evolution of species C. The left plot shows the first 8 ns where the laser pulse excites the particles and the number of C decreases. The right plot displays the re-apprearence of these particles due to inelastic collisions. Note the obvious deviation from a Maxwellian distribution.

Then a laser pulse supplies additional photons. Its intensity is given by the function IL(t) = I0 (t/rL) exp ( - ( £ / T L ) 2 ) with I0 = 100 W/m 2 and TL = 5 ns. A fraction of these photons is absorbed by species C yielding C*. The excited particles C* interact inelastically with particles A in collisional deexcitation events (cf. Fig. 5.8).

10 30 50 80 120 200

20

ns ns ns ns ns ns

10 ns 30 ns 50 ns 80 ns 120 ns 200 ns

^

£1

-15

f\ \ 5

-' 0

^ ^.^

~_ .^"_- •-Tr^r~^rr=r 0.05

0.1

0.15 0.2 energy |eV]

0.25

0.3

0

0.05

0.1

0.15 0.2 energy [eV]

0.25

0,3

Fig. 5.8 Evolution of species C* (left) and species A (right). Due to de-excitation processes the number density of species C* decreases in the course of time. The distribution of particles A is shifted towards higher energies as a consequence of the conversion of internal into kinetic energy.

The high energy released in such a process causes a distortion of the distribution functions. The actual deviation of the particle distribution functions from a Maxwellian depends critically on the assumed cross section a and is most pronounced for species C. Figure 5.7 shows the distribution function of particles C at different instants of time for Maxwell molecules

122

Lecture Notes on the Discretization

0.8

laser pulse — • MM: ratio C /C ~ HS: ratio C'/C —

0.7 OB 0.5 0.4 0.3 0.2 0.1

of the Boltzmann

f/

2.2

E2.1

i \\ i \

Equation

.•^J''"

MM , HS: total energy — MM: kinetic energy — • HS: kinetic energy —

/

t,o

/ 18

V

— 40

60

„ 80

1.7

wr

100

\l

/

.

120 time [ns]

time [ns]

MM , HS: kinetic energy — total energy —

8 10 time [ns]

12

14

16

18

0

2

4

6

8

10

12

14

16

18

time [ns]

Fig. 5.9 Temporal evolution of macroscopic quantities of the gas mixture for low density (top) and high density (bottom) of species A. The left column shows the excitation (ratio nc /nc) and the intensity of the laser pulse (in arbitrary units) whereas the right column displays the evolution of the kinetic and the total energy density of the gas mixture.

(a(g) = 5 x 10 m 3 s _ 1 / # ) a n d a low density of particles A (Fig. 5.8). The deformation can be seen for t = 30 ns in Fig. 5.7. The equivalent simulation for a hard sphere gas (diameter 3.5 A) does not show such significant deviations from mechanical equilibrium. The reason is that for hard sphere particles the collision frequency increases with relative speed and consequently the depletion of the high-energy tails is accelerated. On the other hand, the macroscopic quantities (excitation, energy, kinetic energy) of the gas mixture as depicted in Fig. 5.9 are relatively unaffected by the choice of the elastic cross section. The curves calculated for hard sphere molecules (HS) virtually coincide with those obtained for Maxwell molecules (MM). The excitation is slightly smaller for hard sphere molecules because of the more efficient Maxwellization of the high energy tails. Consequently, the laser can inject slightly more energy into a hard sphere gas mixture (right column of Fig. 5.9) as can be seen for a low density of species A.

Semi-continuous

Extended Kinetic

Theory

123

Fig. 5.10 Intensity of the laser pulse induced by the two coherent incident driver-beams for a fringe spacing of X = 5 fim and a pulse-length of Tt = 10 ns.

Furthermore, Fig. 5.9 illustrates how the relaxational behaviour depends on the density of particles A. A lower density of this species leads to less collisional de-excitation events per unit time and therefore the greater relaxation times are observed. For the same reason, the excitation (ratio nc /nc) reaches much higher values for low densities of A than for high ones. In the latter case, internal energy is converted to kinetic energy much more efficiently than in the former. 5.6.1.2

One-dimensional problem

In real LITA experiments, the geometry is at least one-dimensional. In order to perform calculations on a kinetic level, a P3 approximation of the semi-continuous equations for the three-component gas mixture A, C and C* has been implemented. Some results of these calculations are shown in the remainder of this subsection. A typical spatio-temporal evolution of the laser intensity is depicted in Fig. 5.10. According to their intensity, the laser photons excite particles of species C via C + p —> C*. Typical excitation patterns for medium and low densities are shown in Fig. 5.11. We remark that for medium densities the excitation follows the intensity almost linearity. The difference in shape between Fig. 5.10 and the excitation is due to saturation effects. These effects are very pronounced for low densities. Through inelastic collisions C*+N —>• C+N, the excited species release

124

Lecture Notes on the Discretization

medium density —

of the Boltzmann

Equation

low density —

Fig. 5.11 Typical evolution of the gas mixture. Excitation (top), kinetic energy (middle) and density oscillations (bottom) of the gas mixture. The right (left) column shows results for medium (low) gas densities.

their internal energy. The speed of this process depends on the density of the gas mixture. As illustrated in the second line of Fig. 5.11, the kinetic energy increases with time. In the left-hand plot, it even slightly oscillates. Oscillations of the kinetic energy are more pronounced for higher densities. The energy oscillations induce oscillations of the particle density. This is shown at the bottom of Fig. 5.11. We remark that for the medium density the oscillations are strongly damped. Due to free-streaming beyond the boundaries (we use periodic boundary conditions) these oscillations are stronger for the low density.

Semi-continuous

* g 2 5

750 700 650 600 550 500

Extended Kinetic

Y

0.4

r 5

I 450 1

TA

~ 400 J 350 [ 300

T c . --TD • 10

20

125

Theory

30 40 50 time / ps

60

—

70

80

0.2

la, 0

/ / " '

"A

"B —

/V' 10

20

30 40 50 time / ps

60

70

80

Fig. 5.12 First few picoseconds of a fast highly exothermic chemical reaction. The left plot show the evolution of the temperatures and the right plot the evolution of the densities of the products A and B. In the considered time interval, the densities of the reactants C* and D remain virtually constant.

5.6.2

Chemical

reactions

In this section, some results on chemical reactions are presented. They have been obtained by an implementation of the PQ approximation sketched in Sec. 5.3. We discuss fast exothermic chemical reactions taking place in the mixture of hard sphere particles of diameter a = 3.46 A. In order to show results on simple gas-phase reactions, the ground state of C will not be considered. Thus photons and inelastic transitions C* + N —» C + N are excluded. Numerical solutions of the semi-continuous model including photons and C as well as a comparison to a moment-equation approach can be found in [6]. The masses of the four species are chosen as mA = 7, mB = 17, mc = 14 and mP = 10 atomic mass units. The starting point of all calculations is mechanical (but not chemical) equilibrium of all four species at T = 300 K. 5.6.2.1

Simple exothermic reaction

We first study a chemical reaction described by the endothermic reactive cross section a ~ Q(g — gAB) g~/g, where 0(.) denotes the unit step function. For this choice, K can be calculated analytically. The magnitude of the reactive cross section is chosen 60 times smaller than that for elastic interactions. The difference in internal energy is given by AEchem — 125 meV. For the initial densities we choose nA = 10 x nB = 0.0416 mol/m 3 and nD - 10 x nc' = 41.6mol/m 3 . Figure 5.12 shows the evolution of macroscopic quantities during the

126

Lecture Notes on the Discretization

0

0.05

0.1

0.15 0.2 energy / eV

0.25

0.3

of the Boltzmann

0

0.05

0.1

Equation

0.15 0.2 energy / eV

0.25

0.3

Fig. 5.13 First few picoseconds of a fast highly exothermic chemical reaction. The plots show the distribution functions of the products A and B at different instants of time after the ignition of the reaction. The lowest curves show the distribution at t = 0 whereas the highest curves correspond to t = 125 ps.

first 80 ps. Due to the initial rarity of species B, at the beginning, the increase of particle number and temperature is most pronounced for this species. Here, temperature is understood as a measure for the mean kinetic energy per particle, i.e. TN = 2kN/(3kBnN). As can be seen in the right-hand plot of Fig. 5.12, the number of particles B is already 100 times greater at t = 80 ps than at the beginning. During the considered period of time, the generated particles A and B undergo very few elastic collisions. Moreover, due to its smaller mass, A gains more kinetic energy from a reactive collision than B. This implies that the distribution function of A shows extreme deviations from a Maxwellian as illustrated in Figure 5.13. On the other hand, the distribution functions of C* and D stay virtually constant during the first 100 ps. As the evolution continues, elastic collisions drive the gas towards mechanical equilibrium. For species A and B, this is sketched in Fig. 5.14. We see that after about 0.5 ns, all distribution functions have a strong resemblance to a Maxwellian although the generation of particles A and B has by far not finished yet. The distribution functions of species C* and D are illustrated in Fig. 5.15. They only derive little from Maxwellians. The loss of particles C* during the chemical reaction is considerable. On the scale of the left plot of Fig. 5.15, the distribution function almost vanishes in the course of time. On the other hand, we observe only a little loss of species D as well as a shift of the maximum towards higher energies. This corresponds to the increasing temperature. Figure 5.16 shows the evolution of the temperatures and densities of the

Semi-continuous

Extended Kinetic

0.1 0.15 energy / eV

Theory

127

0.25

Fig. 5.14 First five nanoseconds of a fast highly exothermic chemical reaction. Distribution functions of particles A and B at different instants of time after the ignition of the reaction. The highest curves correspond to t = 5 ns, followed by t = 2 ns, t = 1 ns, t = 0.7 ns, t = 0.5 ns, t = 0.4 ns, t = 0.3 ns, t = 0,2 ns, t = 0.1 ns and t = 0 ns, respectively.

/ \ E 10

>

species C* —

if\\

„>

8

ll/\v\

^

6

l/\\w\

SI o

c

~

4 [/

«

2

.Q

-o

\^W$\

^\N§ll^s.

' — ~ ^ ^ ^ * ^ 0.05

0.1 0.15 energy / eV

0.2

0.25

0.25

Fig. 5.15 First five nanoseconds of a fast highly exothermic chemical reaction. Distribution functions of particles C* and D at different instants of time after the ignition of the reaction. The maximum moves from the left (at t = 0) to the right (at t = 5 ns) and decreases in the course of time. The order of the curves is inverse to that given in Fig. 5.14.

various species for the first few nanoseconds of the evolution. We remark that the temperature of the reactants C* and D increase to the common limit of about 387.48 K whereas the products A and B approach the limit from above. After four nanoseconds, a common temperature is reached and the four particle densities have almost arrived at constant levels. In units of mol/m 3 , the final densities are given by nA = 4.19, nB = 4.16, nc = 0.0142 , nD = 37.51. These values are in excellent agreement with the mass action law that predicts in equilibrium nAnB l{nc'nD) = 33.0, which has

128

Lecture Notes on the Discretization

•

650 CJI CJI O) O CJI O

o o o

* perature /

750 ft 700 1

1 450

TA — TB — Tc.--- . TD

\ \ \

\\

'

** 400

-

350 300

. 0.5

1

1.5 2 2.5 time / ns

3.5

of the Boltzmann

_

4.0 3.5

no

;

/^

3.0 2.5 2.0 1.5

1.0 0.5

Equation

nA — nB — . nc.--oi*nD

A /

7

•

'-•-..

•

2 3 time / ns

Fig. 5.16 First few nanoseconds of a fast highly exothermic chemical reaction. The left plot shows the evolution of temperature and the right plot the evolution of the densities of the various gas species. For convenience, the density of species D is divided by ten.

to be compared with the actual value of 32.7 obtained from the numerical results. The discrepancy is less than one per cent. It reflects the error the discretization introduces in the concept of temperature [17]. The temperature as a parameter of the Maxwellian (5.38) differs from the temperature as a measure for the mean kinetic energy per particle. 5.6.2.2

Exothermic threshold energy

Upon introducing a threshold energy r x c5, r 6 N also for the exothermic direction of the chemical reaction, the character of the evolution changes drastically. The exothermic threshold is included by means of the cross section described in [9], whose endothermic variant reads

rtfig)

l

l

9AB

+ 9l

e(p2

(9AB+9D)-

(5.54)

The speed gx is given by g\ =2r S/fj.AB. Due to this complicated structure, the integrals for the Po approximation are evaluated numerically. The dynamics of the reaction depends critically on the threshold energy. According to its magnitude, we observe a more or less rapid ignition of the reaction. Three cases are illustrated in Fig. 5.17, namely the case of a low, a medium, and a high threshold. For the calculations, the distribution function is resolved using 76 energy groups. Measured in units of c5, the energy gap of the chemical reaction is given by q = 50 and the threshold equals r = 25. The three different values for the threshold are obtained by modifying the width S of the energy groups. We choose <5 = 6.25,8.75 and

Semi-continuous

T A _ TB —

1600

TV----

1400

Theory

129

I

TD

temperatur

^

1800

Extended Kinetic

•

•

600 400

^ 300 400 time / ns

J 500

600

700

700

Fig. 5.17 Evolution of temperatures and densities for a low (top), medium, high (bottom) exothermic threshold energy.

11.25 meV, respectively. In units of mol/m 3 , the initial densities are given by nA = 0.0416, nB = 0.00416, nc'= 4.16, nD = 3.33. For a low threshold energy, we first observe that the densities do not decay exponentially to their final values as in the case without exothermic threshold, cf. Fig. 5.16. After an initial phase of about 20 ns where the temperatures of the products A, B go through a local maximum, they raise

130

Lecture Notes on the Discretization

of the Boltzmann

Equation

in concert with the temperatures of the reactants C*,D. This accelerates the reaction. The pronounced peak of TB is a consequence of the low initial density of B. Due to the conversion of internal energy to kinetic energy, the temperatures of A and B are always higher than those of C* and D. When the threshold is increased, the ignition becomes more and more delayed. This can be rationalized as follows. In the beginning of the simulations, the gas mixture is cold. Consequently, few particles have sufficient kinetic energy to undergo a chemical reaction. These few reactions, however, constantly heat the gas mixture. As soon as the temperature of C* and D is high enough, the ignition takes place followed by a rapidly increasing temperature. Obviously, for the highest value of the exothermic threshold energy, the ignition is delayed for the longest time.

5.7

Conclusion

We have introduced a semi-continuous extended kinetic model for a reacting five-component gas mixture interacting with monochromatic photons. The semi-continuous model approximates the continuous Boltzmann equation in a very satisfactory manner. A comparison with the exact BKW mode shows good agreement of the relaxation of the distribution function towards the Maxwell distribution. Even for a moderate number of energy groups the evolution of the entropy is well reflected. The semi-continuous model displays the major properties (.//-theorem and conservation laws) of the full continuous kinetic description. Moreover, Planck's law of radiation is recovered as the self-consistent equilibrium photon intensity. In chemical equilibrium, a mass action law holds. As an application of the semi-continuous model, we have calculated numerically the reaction of the gas mixture to a strong coherent laser pulse. At the level of the distribution functions, we have found different degrees of deviation from Maxwellians due to fast collisional de-excitation processes. The evolution of chemical reactions has also been investigated. In conclusion we can say that the adopted strategy of discretizing the speeds and integrating the directions yields very reliable kinetic models. The treatment of general mass ratios of the involved species is possible along with the inclusion of inelastic and reactive collisions. Thus, semicontinuous models are well suited for the numerical simulation of evolution processes in extended kinetic theory.

Semi-continuous

5.8

Extended Kinetic

Theory

131

References

[1] N. Bellomo, Lecture notes on the mathematical theory of the Boltzmann equation, Series on Advances in Mathematics for Applied Sciences, 33, World Scientific, Singapore, (1995). [2] N. Bellomo, E. Longo, A new discretized model in nonlinear kinetic theory: The semicontinuous Boltzmann equation, M 3 AS: Math. Mod. Meth. in Appl. Sci., 1 (1991), 113-123. [3] C. Cercignani, The Boltzmann equation and its applications, Applied Mathematical Sciences, vol. 67, Springer-Verlag, New York, (1988). [4] E.B. Cummings, Laser-induced thermal acoustics, Ph.D. thesis, California Institute of Technology, Pasadena, CA 91125, (1995). [5] M.H. Ernst, Nonlinear model-Boltzmann equations and exact solutions, Phys. Rep. 78(1) (1981), 1-171. [6] M. Groppi, W. Roller, Kinetic calculations for chemical reactions and inelastic transitions in a gas mixture, in print with ZAMP. [7] C.R. Garibotti, G. Spiga, Boltzmann equation for inelastic scattering, J. Phys. A: Math. Gen. 27 (1994), 2709-2717. [8] M. Groppi, G. Spiga, Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas, J. Math. Chem. 26 (1999), 197219. [9] P. Griehsnig, F. Schiirrer, G. Kiigerl, Kinetic theory for particles with internal degrees of freedom, Rarefied Gas Dynamics: Theory and Simulations (Washington, DC), 159, B. D. Shizgal and D. P. Weaver, Progress in Astronautics and Aeronautics, AIAA, (1994), 581-9. [10] F. Hanser, W. Koller, F. Schiirrer, Treatment of laser-induced thermal acoustics in the framework of discrete kinetic theory, Phys. Rev. E 61 (2000), 2065-73. [11] W. Koller, F. Hanser, F. Schiirrer, A semi-continuous extended kinetic model, J. Phys. A 33 (17) (2000), 3417-30. [12] W. Koller, A semi-continuous kinetic model for bimolecular chemical reactions, J. Phys. A 33 (35) (2000), 6081-94. [13] W. Koller, F. Schiirrer, Pn approximation of the nonlinear semicontinuous Boltzmann equation, Trans. Th. Stat. Phys. 30 (2001), 471-89. [14] J.C. Light, J. Ross, K.E. Shuler, Rate coefficients, reaction cross sections and microscopic reversibility, ch. 8 (1969), 281-320, Hochstim,

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Lecture Notes on the Discretization

of the Boltzmann

Equation

Academic Press, New York. [15] R. Monaco, L. Preziosi, Fluid dynamic application of the discrete Boltzmann equation, Series on Advances in Mathematics for Applied Sciences, vol. 3, World Scientific, Singapore, (1991). [16] J. Oxenius, Kinetic theory of particles and photons, Springer Series on Electrophysics, vol. 20, Springer-Verlag, Berlin, (1986). [17] L. Preziosi, E. Longo, On a conservative polar discretization of the Boltzmann equation, Japan J. Indust. Appl. Math. 14 (1997), 399435. [18] I. Prigogine, E. Xhrouet, On the perturbation of Maxwell distribution function by chemical reactions in a gas, Physica 15 (11-12)(1949), 913932. [19] P. Reiterer, Ch. Reitshammer, F. Schurrer, F. Hanser, Th. Eitzenberger, New discrete model Boltzmann equations for arbitrary partitions of the velocity space, J. Stat. Phys., 98 (2000), 419-440. [20] A. Rossani, G. Spiga, A note on the kinetic theory of chemically reacting gases, Physica A 272 (3-4) (1999), 563-573. [21] A. Rossani, G. Spiga, R. Monaco, Kinetic approach for two-level atoms interacting with monochromatic photons, Mech. Res. Comm. 24 (1997), 237-242. [22] G. Spiga, Rigorous solution to the extended kinetic equations for homogeneous gas mixtures, Lecture N o t e s in Mathematics (G. Toscani, V. Boffi, and S. Rionero, eds.), vol. 1460, Springer-Verlag, New York, 1991.

Chapter 6

Steady Kinetic Boundary Value Problems Hans Babovsky, Daniel Gorsch and Prank Schilder Institute for Mathematics, Ilmenau Technical University Ilmenau, Germany

6.1

Introduction

The numerical calculation of space-dependent solutions of the Boltzmann equation is a demanding task which has attracted much attention during the recent decades. On one hand, there are a couple of important scientific problems relying in the solution of kinetic equations - ranging from problems in space sciences over aerosol dynamics and environment sciences to small scale structures as given e.g. in the semiconductor design. On the other hand, the Boltzmann collision operator acts on a (typically) sixdimensional phase space which makes it very difficult to solve by an efficient numerical scheme. A number of scientific groups has in recent decades been involved in the design of efficiently treatable model systems for the Boltzmann collision operator. Some of the main issues have been (i) find discretized models which are reasonable caricatures of the Boltzmann collision operator; (ii) find domain decomposition strategies to keep the Boltzmann regime as small as possible and to connect it to the dominating fluid dynamic regime (based on the Euler or the Navier-Stokes equations); (Hi) find simplified relaxation models replacing the Boltzmann collision operator. Concerning (ii), there have been many attempts in the 1990's (see, e.g. 133

134

Lecture Notes on the Discretization

of the Boltzmann

Equation

[14] and the literature cited therein). However, the initial anticipation put into the concept of coupling kinetic and fluid mechanic equations has much cooled down since it lacks mathematical foundations as well as convincing numerical results. The aspect (Hi) has attracted attention e.g. in form of generalized BGK-models (e.g. the ES-BGK model studied in [l]). However, these models lack of a substantial part of gas kinetics, the two-particle interaction mechanics. The approach (i) has been (and still is) a vivid section of numerics for the Boltzmann equation. The proposed schemes include (i) Discrete velocity models which are (numerical) approximations of the Boltzmann collision dynamics (e.g. [5; 9; 16; 18]); (ii) fast solvers for the Boltzmann collision integral [6; 7]; (Hi) spectral methods for the Boltzmann collision integral, as represented in [17]. The approach presented in this paper concerns the design of appropriate discrete models for the Boltzmann collision operator as well as the use of efficient spatial resolution schemes. Concerning the approximation of the Boltzmann collision operator, special attention is paid to small discrete models which give rise to numerical systems of reasonable dimension. The scope of the paper is as follows. At first, we sketch ways for the design of (small) discrete velocity systems which are reasonable weak approximations of the Boltzmann collision operator - or which at least conserve its main properties (Section 2). Then we propose a scheme for the spatial discretization (Section 3). Finally we present some results concerning steady kinetic problems. In all cases we restrict to 2D velocity spaces.

6.2

Discrete kinetic models

When trying to find discrete approximations for the continuous Boltzmann collision operator for numerical purposes, there is a conflict between the two requirements (i) the model should be as close as possible to the original operator (including if possible some convergence proof); (ii) the model should be accessible to an efficient numerical scheme. We first investigate models for which the first requirement is relaxed (Section 2.1). After that we introduce a convergent approximation of the

Steady Kinetic Boundary

Value Problems

135

continuous operator (Section 2.2). 6.2.1

General

classes

of collision

models

6.2.1.1

Moment preserving collision kernels

For sufficiently regular functions / , the Boltzmann collision operator is given as J[f, f](v) = 1 1 JRd

K\v - w\, v)(f(v')f(w')

-

f{v)f{w))dw{rj)dw

JSd~1

= :J+[fJ](v)-J-[f,f](v).

(2.1)

Relevant for the construction of an efficient numerical scheme is the knowledge that v' and w' are points on the sphere between v and w (this is intimately connected to the conservation of momentum and energy): v' = v — (v — w, rj),

w' = w + (v — w, rj)

(rj € S1),

where (.,.) describes the usual Euclidean scalar product in R 2 . For two functions f,g, define the scalar product (.,.) by := /

f(v)g(v)dv.

JRd

With this notation, the weak formulation of J[.,.] is given as follows. Let $ be a test function. Then ($,J[f,f})= [

f

JRd JRd

17

k(\v-w\,r,)mv')-$(v))Mv)

L^Sd-1

f(v)f(w)dwdv.

(2.2)

(This result is well-known in kinetic theory and may be found in the relevant textbooks, e.g. [10].) If {3>n,n = 0,1,...} is a complete system of test functions, then J[f, f] can be reconstructed in principle from the knowledge of all ( * „ , . / [ / , / ] ) . Since the numerical solution of J[f, f] is quite expensive, one way out is to weaken the assumption of an exact pointwise evaluation. We may require that only a (small) finite number of products ($„, J[f, / ] ) , n = 0 , . . . , N, is reproduced correctly. Suppose / is defined on some grid; then it is a simple task to evaluate the loss term J~ [/, / ] numerically. More involved is the

136

Lecture Notes on the Discretization

of the Boltzmann

Equation

integration of the gain term since it includes the integration over a manifold which is not well approximated by the grid points. Thus we introduce the following principle. 2.1 Basic principle: In (2.2), replace the integral

L

k(\v — w\,r}) ($(v') - $(v)) dw(rj)

with an integral of the form /

(*(z) - *(«)) p(z\v,

w)dn(z),

where the density p(.\.,.) and the measure dfi are chosen such that /

$n(z)p(z\v,w)d[i(z)

=

$n(v')(k(\v-w\,r))dtj(r]),

(2.3)

n = 0,...,N. This ensures that ($„,
Suppose that fg "lives" on some regular grid Q and that the measure n(A) :— #{z 6 £/nA} counts the number of grid points in A. If the support of p(., v, w) lies in Q, then we obtain a discretized version of the Boltzmann collision operator. 2.2 Discretized collision operator on the grid Q:

Js(fg,fg)(z) = Yl P(z\v,w)fG{v)h{w) -fs • Yl K(\z-w\)fg(w) (2.4) with K(\V — W\)= /

k(\v - w\,r})du)(ri) —

p(z\v,w)dfi(z).

Of course, the minimal requirement is that among the test functions $ „ , n = l,...,N, are the functions 1, vi} \v\2, which ensure the physical moment

Steady Kinetic Boundary

Value Problems

137

conservation laws. This general ansatz gives much place for special choices. For example, one may try to find a good pointwise approximation (with some convergence theorem; this is contained in section 2.2); or one may try to find an ansatz which can be efficiently numerically evaluated. Such models may be based on microreversible collision laws or on transition probabilities not satisfying microreversibility. 6.2.1.2

Axiparallel models

Consider the case of two velocity dimensions, d = 2. (The case d = 3 is technically slightly more complicated.) By changing the integration order, the collision operator (2.1) can be written as

/

/

Js1 L/R2

k(\v~w\,T])(f(vr,,w±)f(wtl,v±)-f(vn,v±)f{wn,w±))d' (2.5)

du>(v), if we write V = Vv • 77 + V± • T]-1 = : (vv, V±) ,

and w = wn -r) + w± • T)1- —:

(wn,w±).

(77-*- is a unit vector orthogonal to the unit vector 77.) If we consider (for notational simplicity) 77 = (1,0) T and write v = (vx,vy)T and w = (wx,wy)T, then the inner integral of (2.5) reads

^ 7 [ / . / ] ( « ) = / k(\v-w\,6)(f(vx,wy)f(wx,vy)-f(v)f(w))dw,

(2.6)

where 6 is the angle between v — w and (vx, wx) — (wx,vy). In this section we consider only kinetic models described by collision operators of the form (2.6). Since in this case a transition of a velocity pair (v,w) into a pair (v', w') is possible if (and only if) v and w are the end points of the diagonal of an axiparallel rectangle, and v' and w' are the end points of the other diagonal, we call them axiparallel modeb.

138

Lecture Notes on the Discretization

of the Boltzmann

Equation

Of course, such models are not physical in the sense that they have too many collisional invariants. In fact, one can easily prove, that all quantities of the forms

1>x(vx) := Y^ f(v).

V'yK) == Yl f(v)

Vy

Vx

are collisional invariants. Nevertheless they are of interest since they may be the basis for efficient numerical simulations. According to the artificial invariants, there are ways to suppress them (see [3; 4]). Suppose Q = Q° x Q° with Q° = { £ i , . . . , £R} is a finite grid of the velocity space 1R . For notational simplicity we write Q° = { 1 , . . . , R}. Following the recipe of 2.1.1, the operator (2.6) is discretized as J [f,f]ik= 22 Ailjkfilfjk — 2 J Aikjlfikfjl,

(2.7)

j.ieQ0

j,ieg°

for i,ke Q°. Suppose V° = { P i , . . . , Pr} with r < R is a partition of Q°. Then V = V° x V° is a partition of Q. The operator (2.7) gets a special structure if the coefficients Aikji depend only on the partition. Precisely: 2.3 Assumption: Define the mapping [.] : { 1 , . . . , R} —> { 1 , . . . , r} by [i] = a

T h e n Aiktji

<=>def

i S Pa •

= Ajjpj^p.

Defining now

fa/3 •=

fik

Y

aIld

^^' f}aP : =

i€Pa,kePf)

X] W' A* ' iZPcktPp

we find that J[f, f]ap is quadratic and depends only on / : J[f,f]a/3 = 2__, A.apitaipfapifaip a',/3'

— 2_^ Aaptaij3ifapfa'j}i

.

(2.8)

a',/3'

Thus these components represent a closed system of the dimension r x r. As is shown in [2], the full operator J[f, / ] can be transformed into an operator with a small number of nonlinear components given by (2.8) and a

Steady Kinetic Boundary

Value Problems

139

large number of linear components. This has a huge effect when trying to establish an efficient numerical scheme. Next to their numerical efficiency, the main advantage of these models is that they allow for the establishment of a hierarchy of models giving rise to the construction of a "bridge" between coarse discrete and continuous kinetic systems. Moereover, modifying collision frequencies on different levels of the hierarchy, we arrive at specific closure relations (reducing the infinite hierarchy to a finite system) which are based on kinetic principles (rather than on e.g. heuristic maximum entropy principles as used in Extended Thermodynamics [15]). We will demonstrate this with a simple two-level Broadwell system. (For details, consider [3].) A n Example Consider the following configuration. A

B

C

D

If A, B, C and D represent points in velocity space and if we introduce the correct collision relations, we end up with the well-known Broadwell system [8] given (in the space homogeneous case) by the equations

dtfoo = dtfu

= -dtfio

= -dtfoi

=

T(/IO/OI

- /oo/ii) •

If the blocks A, B, C and D themselves represent Broadwell models and if in addition there is a block interaction between A, B, C and D according to the Broadwell kinetics, we arrive at a 4 x 4-velocity model called the twolevel Broadwell system. By recursion, we may construct an infinite-level Broadwell hierarchy which resembles in its structure somehow an infinite Haar wavelet system. (For details, see [3].) Here, we restrict for simplicity to the two-level system and study a homogeneous relaxation problem. The velocities are numbered by four boolean indices, (iojo, HJi), where (io,ii) represents the block, and (jo,Ji) gives the

140

Lecture Notes on the Discretization

of the Boltzmann

Equation

local position. Thus the numbering is as follows. 00,00 00,01 00,10 00,11

01,00 01,01 01,10 01,11

10.00 10.01 10.10 10.11

11,00 11,01 11,10 11,11

We are looking for solutions of the initial value problem which are symmetric with respect to the two diagonals, i.e. /oo,oo(*) = / i i , n ( t ) = /oi,oi(*) = fw,w(t)

=

foi,oo(t) = foo,oi(t) = / n , i o ( 0 = /io,n(*) = /n,oo(0 = foo,n{t) = /io,oi(£) = /oi,io(*) = /io,oo(£) = /n,oi(*) = foo,io(t) = foi,n(t)

= 4>6{t).

Choosing 71 for the block collision rates and 72 for the local rates, and using the transformation introduced in [3], it is a straightforward matter to show that the solution according to the initial values 4>i(0) = a,

0 2 (O) = / 3 ,

h = 5 =

with 2 (a + /?) = 1 is given as

Mt) = \ ;(s+^)M*) + r + w; h(t) = \ '(\-a-^)e+(t)+r+(t)

03(f)=

K^

e+w_r+(t)

)'

Mt) = \

•(i+^y_{t)+r_{t)

Mt) = \

[(i_£^) Mt) _ r . w ]

Mt) =

\(\e-(t)+r-(t)

(f>6=0,

Steady Kinetic Boundary

141

Value Problems

Here we have written for brevity e±(t) = l ± e x p ( - 7 i t / 2 ) ; r± are solutions of the initial value problems dtr+ = 0.0625(a - f3)2 ( 7 l e l + -y2e2+) - 0.125( 7l + 72)e+r+ r+(0) = 0.5(a + /J), 2

9 t r_ = -0.0625(a - /3) (lxe\

2

+ l2e -) - 0.125(7i + 72)e_r_ r_(0)=0.

F2

Fig. 1: (FO) (Fl) (F2)

Space homogeneous solutions Initial distribution, Equilibrium state, Projected initial distribution

142

Lecture Notes on the Discretization

of the Boltzmann

Equation

The initial distribution and the corresponding equilibrium state for a = 1/3 and /? = 2/3 are demonstrated in Fig.l (FO), (Fl). Replacing the local collision rate 72 with 1/e and passing to the limit e —> 0, the initial distribution instantaneously passes over to the projected initial distribution given by Fig.l (F2). (The equilibrium configuration is not affected by this procedure.) Investigating the relaxation of moments according to this solution we find that most of the lower moments do not depend on r± (which are not given analytically). However, the shear stress is given through

pvj% = r - + 3 ( a - / 3 ) J ( e + - f i _ ) + 2 ( r + + r _ ) .

To find simplified relaxation models for certain important moments, many authors restrict to heuristic assumptions on relaxation rates. Here, we derive a specific time behaviour from kinetic principles by speeding up the local collision rate. To this end we replace 72 with 1/e and pass to the limit e —> 0. From standard asymptotic analysis follows then that r± are given as

r±(t) =

±0.5(a-/3)2e±(t)

thus reducing the above second moment to the form

pvZvj = 2 ((a - p) + 1.5)2 • e x p ( - 7 i t / 2 ) .

Similar simplifications are achieved for certain higher moments. Taking the above limit means that after an infinitesimally small initial time the initial distribution changes due to local collisions to a new initial distribution (compare Fig.l (F2)). The relaxation rates for the original model (Fig.2, solid line) and the simplified model due to increase local relaxation rates (Fig.2, dotted line) are demonstrated in Fig.2.

Steady Kinetic Boundary 10

Value Problems

T

I

r

j

i

L 15

143

l

pxy_mod 0.01

0

5

10

20

t

Fig. 2: Shear stress relaxation rates Such arguments may become interesting for higher-level systems for the truncation (resp. relaxation) of higher moments which lack of physical significance.

6.2.2

Discretizations

of the Boltzmann

collision

operator

In this section we discuss how discrete kinetic models can be constructed which are close to the original continuous Boltzmann collision operator. For their derivation we have to cope with two problems. First, the Boltzmann collision operator has to be restricted to a bounded domain Q. Second, a collision model on a finite grid covering Cl has to be constructed. As in the previous section, we consider only a 2D velocity space. 6.2.2.1

Restriction of velocity space

In order to discretize the velocity space 1R2 we have to restrict to bounded domains f2. That means that we have to define a new collision operator JQ acting on this domain. The approach used here was presented in [11; 12], so we only intend to give the main idea. At first, the collision operator J[f, f] is split into the gain term G[f] and the loss term L[f]. Second, let / be a function with support in fl. Multiplying a test function $ G C 1 (]R 2 ) with the gain term G[f] and integrating over

144

Lecture Notes on the Discretization

of the Boltzmann

Equation

the velocity space IR yields (compare (2.2)) /

JR2

*(«) G[f](v) dv = f f f Ja Jn JS1

$(v')K(v-w,r))f(v)f(w)dr)dvdw.

The introduction of an appropriate measure nn(dz, v, w) leads to /

$(v)

G[f](v)dv

« / f f$(z)K(\v-w\)f{v)f(w)nn(dz,v,w)dwdv Jn Jn Jn = f *(z) / f K(\v - w\) f(v)f(w) Jn Ja Jn

(2.9)

fin(z, v, dw) dv dz,

(2.10)

where the "w"-sign is meant as follows. For the collision invariants $(v) = l,vx,vy, \v\2, both sides of (2.9) are equal; otherwise, the equation / $(v')K(v Jsi

— w,r)) dr] = $(z) K(\v — w\) Jn

no.(dz,v,w)

holds for all v, w with v' € Q, for all r). The exchange of the measures Hn(dz,v,w)dwdv and /j.n(z,v,dw)dvdz in (2.9) and (2.10) has been carefully justified in [ll]. (2.10) gives rise for a weak definition (in the sense of distributions) of a gain operator GQ, restricted to ft. 2.4 Definition: The restrictions of the gain operator G[f] and the loss operator L[f] on the domain Q, are defined by Gn[f]{z) := f f K(\v-w\)f(v)f(w) Jn Jn

vn(z,v,dw)

dv

and Ln\f](v) = f K(\v - w\) f(v)f(w) Jn

dw.

The operator Jn[f, f] = G^[f] — La[f] denotes the Boltzmann collision operator on Q. By construction, Jn have the same properties as the original collision operator J regarding the conservation of mass, momentum, and energy. Furthermore, in [ll] it was proven that they approximate J in a weak sense if the domain Q becomes larger and larger.

Steady Kinetic Boundary

6.2.2.2

145

Value Problems

Discrete collision operators

To discretize the operators JQ we introduce a regular equidistant grid ilh with \Q,h\ = n points over the rectangular domain D, in velocity space. Further we replace the distribution function nsi{z,v,w) by a tensor M^ in such a manner that vh, wh) » Yl *("*) Mio >

/ n $(z) m(dz,

fc=i

Jn h

l

h

J

(211)

l

h

where v = v , w = i> , and v ,yi £ Q . Now we consider again the weak formulation of the gain operator GQ :

= / / / $(z) f(v)f(w) Jn Jn Jn = [ [ f(v)f(w)

K(\v-w\)

K(\v-w\)

(in(z,v,dw)

dv dz

[ $(z) /xn(dz,i;,i«) du; dv.

(2.12)

Replacing the functions / and K by step functions on the right-hand side of the identity above and using the ansatz (2.11) give

/ *(z) Gn[/](*) dz in Jn

~ E /( u ')/( wi ) ^(i yi - ^i) E $ ( yfc ) M « L2 ij

k

= E $(wfe) E M Rv M"J L2 . A:

(2-13)

ij l

with the abbreviations f(v ) = fi, f[yi) = fj, and K(\v* — vi\) = K^ as well as the constant L = \il^\/n. Here and in the following 1
£(/) := E Mj R*i M"i L

9k

and

#M

:=

/* E /*R* L •

146

Lecture Notes on the Discretization

of the Boltzmann

Equation

The expression j £ (/) :=
Fig. 3: Weights Af§, (a)

We have developed a software package for the numerical calculation of the weights on grids of different sizes for a constant Boltzmann collision kernel, for Maxwell molecules and for VHS (variable diameter hard spheres) models. Figs. 3 and 4 show the weights Mt* for particular pairs (u*,i^) of velocities on a 10 x 10 grid. The marked fields on the left demonstrate the support of M^. In Fig. 4, the circle of all momentum and energy preserving collision results passes the domain of calculation; this induces boundary effects for the weights.

Steady Kinetic Boundary Value Problems

147

Fig. 4: Weights M%, (b)

As was pointed out in [12], this approach outperforms alternative schemes as presented e.g. in [5; 9; 16; 18] at least in one point. It overcomes the problem of approximating surfaces of spheres (as they appear in the interior integral of the Boltzmann collision integral) by points of a regular grid. As a consequence, meaningful numerical results are achieved even for quite coarse grids £lh (see section 4). For this, one of the crucial features in the derivation was the decoupling of gain and loss terms.

6.3

F E M for the semi discrete steady Boltzmann equation

Denoting & := vk, the discretization procedure demonstrated above turns the steady Boltzmann equation into the semidiscretized system #V*/ f c (:r) = J f e h (/),

* = l,...,n.

(3.1)

For its spatial discretization we will use finite elements. Further we are going to solve this nonlinear equation by quasi linearization and inexact Newton methods. The decision for a finite element discretization was made to be able to model complex spatial geometries. Inexact Newton-Methods are the methods of choice if large nonlinear systems have to be solved.

148

Lecture Notes on the Discretization

6.3.1

Inexact

Newton

methods

of the Boltzmann

and quasi

Equation

linearization

The solution of large nonlinear systems in the frame of a Newton method requires the solution of (large) linear systems. Having in mind to ensure quadratic convergence of the Newton process it is not necessary to solve these linear subproblems exactly since the Newton iterates themselves are only approximations of the solution. We would expect that for iterates far form the solution it is sufficient to solve the corresponding linear system inaccurately. Approaching the solution we have to spend more and more effort on the linear problems. On one hand we can decrease drastically the computional effort needed for the solution of the whole problem. On the other hand this formulation of the Newton method represents the numerical fact that the solution of even linear subproblems can not be an exact one. We consider the general nonlinear problem F(x)=0,

(3.2)

with the differentiable mapping F : D c X —* Y of an open subset D c X of a Banach space X in a Banach space Y. The Newton method for the problem (3.2) is defined by xi+1=xi+Axi,

0 = F'(xi)Axi+F{xi),

1 = 0,1,2,....

Using numerical approximation methods it is not possible to solve the linear problem determining Ax1 exactly. Its solution has to be approximated. This implies the transition to inexact Newton methods xi+1=xi

+ si,

ri=F'(xi)si+F(xi),

t = 0,l,2,....

If the inner residuals rl are controlled by the outer residuals F{xl)

|H|< £ i 11^)11, with the adaptively chosen £j then the quadratic convergence of the inexact Newton process can be ensured under weak conditions on the mapping F. A more detailed description of inexact Newton methods including convergence theorems and control strategies for the parameters £j can be found in [13]. In the context of inexact Newton methods we can approximate the solution f(x, £) of a nonlinear boundary value problem for the steady semi discrete

Steady Kinetic Boundary

149

Value Problems

Boltzmann equation (3.1) by the solution of linear boundary value problems. The approach is know as quasi linearization. Its starting point is the following formulation of the boundary value problem

/tfV x A-J?(/)\ ThU)

= 0,

:=

£V x /„-J*(/) where bh(f) denotes the semi discrete boundary conditions. The linear system for the determination of the inexact Newton correction sl (T")'(/*) sz + Tn(f)

= rl

(3.3)

needs the explicit expression of the derivative of the operator Jh. Since the operator Jh is bilinear its derivative can be calculated easily. This results in the following form for the system (3.3) / gVxs[

- 2 J f (/', *') + £f V x / j - Jf (/*) \

(3.4) ^V

x

4 - 2J%(f, s*) + £ V X / * - J*(/«) B i s i + 6' , (/ i )

with Bl = b'(f%). If we want to solve this system numerically we need to be able to calculate the finite element discretization of • the differential operator ^ V / f e . • the linear collision operator (Lh(f0) f)k = 2J£(/o, / ) • the nonlinear collision operator J%(f) for k = 1 , . . . ,n. 6.3.2

FEM discretization

of the linear Boltzmann

equation

We presume that a regular triangulation is given for the spatial domain ttx. Further we suppose that a set of linear, continuous functions {^>j}^Li

150

Lecture Notes on the Discretization

of the Boltzmann

Equation

with compact support is given on this triangulation. The finite dimensional subspace of C(f2 x ) used for an approximation of the solution of the discrete Boltzmann equation consist of all functions / which are linear combinations of the given
f(x) = J2uJ
(3-5)

j=i

We start from the linear semi-discrete Boltzmann equation n

#V/*(a:)-X;&(aO/j(aO=0,

k = l,...,n.

(3.6)

i=i

Here the scalar functions Z£; (x) are the components of the linear Boltzmann collision operator Lh(fo) = (JH)'(fo) = 2Jh(fo,-). Inserting the ansatz (3.5) in equation (3.6) we get m

m

n

X X ^ ^ K - E E t o ^ W =0, k = l,...,n. (3.7) j=i

j=ii=i

Calculating the ^ ( I R ) scalar product (•, •) of ifi and equation (3.7) leads to m

m

n

X><< ZlVxvM - £ £<W. lhkm)u\ = 0, j=i

j=i

(3.8)

i=i

k = 1 , . . . ,n, i = 1 , . . . ,m. By introducing the abbreviations a*- = {
m

a

n

E « * - Z ! Z X u l =°. k = l,...,n,i = l,...,m. 3=1

u

(3.9)

3 = 1 1=1

In order to produce a flat index structure in the involved quantities we define the following global indices p=(i-l)n

+ k,

q = (j-l)n

+l

and

q = (j-l)n

Due to this renumbering, equation (3.9) reads as m

mn a

22 pq q — 2li bpq ui 3=1

u

9=1

=

0,

p=l,...,mn.

+ k.

(3.10)

Steady Kinetic Boundary

Value Problems

151

Thus we got the first component for setting up a finite element discretization for the equation (3.4) in order to solve the semi-discrete Boltzmann equation by quasi linearization.

6.3.3

FEM discretization equation

for the nonlinear

Boltzmann

For the discretization of the nonlinear discrete Botzmann equation #V*A(:r)-•/£(/)=<),

k = l,...,n

(3.11)

we need the same tools used for the discretization of the linear one. Starting from this we introduce the ansatz (3.5) in equation (3.11) resulting in

$3 eJVxW «£-•#(«), k = l,...,n. Taking the L2OR ) scalar product of tfi and this equation leads to m

J2&i>ZkVx
l,...,n.

3= 1

Using the abbreviations a^- = (i,£jVx¥>j) and c^(u) = (
] C aij uk ~ ci(u)'

k = l,...,n,

i=

l,...,m,

i=i

or with global indices (3.10) m

^2aP9u9 ~ CP(U) ~ ° '

P=l,---,mn.

i=i

This is the second expression we need for setting up a finite element discretization of equation (3.4). Before we can write down the whole system for the Newton correction s% in the case of the semi-discrete Boltzmann equation we have to pay some attention on the semi-discrete boundary conditions bh(f) in the next section.

152

Lecture Notes on the Discretization

6.3.4

Semi-discrete

boundary

of the Boltzmann

Equation

conditions

The common boundary conditions for the Boltzmann equation are linear conditions. In general we apply boundary conditions only to the components fk(x) of f(x) corresponding to instream velocities at the boundary point x, that is, n(x)T£k > 0 where n(x) is the surface normal of Clx at x. Furthermore the boundary conditions only act on the velocity dependence of / at the point x, that means there is no coupling between the values of / in two different boundary points x. Therefore we can concentrate on the boundary condition at one special point x». If we collect all instream components at a;* of /(#*) in the vector f>o(%*) we can write the boundary conditions at

B*1f>0(x*) + B*2f<0(x*) = b*. Two kinds of boundary conditions are of special interest: (1) Instream boundary conditions. The boundary matrix B* = (Bt B%) has the form B* = (I

0).

In this case the vector b* contains the density function which is assumed to stream in. (2) Reflexion boundary conditions. The boundary matrix B* has the form B* = (I

B*2).

The vector b* is equal to zero for this condition.

6.4

Numerical results

We have developed a programming package kinLib which contains the main features relevant for the simulation of steady kinetic problems. At present we are restricted to problems which are derived from 2D channel flow geometries. (However, the approach presented above covers more complicated geometries.) The following numerical examples have been obtained with a 6 x 6-velocity model of the system presented in Section 2.2.

Steady Kinetic Boundary

Value Problems

153

Fig. 5: FE discretization of channel

As a first test case we have chosen a backward facing step followed by a ramp. The flow is entering the region from the left. The FE-grid for the spatial discretization is demonstrated in Fig. 5. Figures 6 and 7 show the velocity field and the density distribution obtained for the steady solution. In a second test case, the flow around a wall is calculated. The geometry and the grid points are shown in Fig. 8. In Fig. 9, the velocity field around the wall is presented. Though based on a very coarse grid, the results are in reasonable agreement with results known from other simulations.

154

Lecture Notes on the Discretization

of the Boltzmann

Equation

Velocity-Raid I

I

0.6

0.7

0.350.30.250.2 —

- = * » - -

0.15 — 0.1

l

>—TTC

0.05 0-0.05-

-0.1 -0.15_ l _

L_

0.2

0.3

0.4

0.5

Fig. 6: Velocity field of channel flow

Density

i

.

02

03

04

05

i

i

06

07

Fig. 7: Density distribution of channel flow

Steady Kinetic Boundary

0

0.2

0.4

0.6

0.8

Value Problems

1

1.2

1.4

155

1.6

1.8

2

Fig. 8: Grid for flow around wall Velocity-Field 2 1.8

U

.

I

1

.

1

_

-

-

—»

_,

—*

""*

"**

>* ~* ^ -»—»-•

**"'

-

"""*'

-

s

_ ' ^/;p----t.,t" \

-

- 'V-

•

^

\

•»

N

\

"*

-

— _

-

* ^

0.2 0

„

-».

0.8

0.4

-

-

1.2

0.6

1

.

.»»

1.6 1.4

.

" ^

" - .. •

-

;

•

•

Fig. 9: Velocity field around wall

-

156

6.5

Lecture Notes on the Discretization

of the Boltzmann

Equation

References

[I] 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). [2] H. Babovsky, Reducible kinetic equations, Preprint M 20/00, Inst. f. Mathematik, TU Ilmenau, (2000) (zur Veroffentlichung eingereicht). [3] H. Babovsky, A kinetic multiscale model, Preprint M 24/00, Inst. f. Mathematik, TU Ilmenau, (2000) (to appear in Math. Models Meth. Appl. Sci.). [4] H. Babovsky, Hierarchies of reducible kinetic models, in: Discrete Modelling and Discrete Algorithms in Continuum Mechanics, Th. Sonar and I. Thomas (Eds.), Logos Verlag, Berlin (2001). [5] A.V. Bobylev, A. Palczewski, J. Schneider, On the approximation of the Boltzmann equation by discrete velocity models. C. R. Acad. Sci. Paris I 320(5)(1995)639-644. [6] A.V. Bobylev, S. Rjasanow, Fast deterministic method of solving the Boltzmann equation for hard spheres. European J. Mech. B Fluids, 18 (1999), 869-887. [7] A.V. Bobylev, S. Rjasanow, Numerical solution of the Boltzmann Equation using a fully conservative difference scheme based on the Fast Fourier Transform. Trans. Theor. Stat. Phys., 29 (2000), 289-310. [8] J.E. Broadwell, Shock structure in a simple discrete velocity gas. Phys. Fluids, 7 (1964), 1243-1247. [9] C. Buet, A discrete velocity scheme for the Boltzmann operator of rarefied gas dynamics. Trans. Theor. Stat. Phys., 25 (1996), 33-60. [10] C. Cercignani, The Boltzmann equation and its applications. Springer, New York, (1988). [II] D. Gorsch, Boltzmann operatoren auf be