Lecture Notes on the Discretization of the Boltzmann Equation

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

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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 -^ / (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 ) / = 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 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)



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 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 ~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: ,