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 beschrankten Gebieten und verallgemeinerte Geschwindigkeitsmodelle. Dissertation, TU Ilmenau, (1999). [12] D. Gorsch, Generalized discrete velocity models. Preprint M 17/00, Inst. f. Mathematik, TU Ilmenau, (2000), to appear in Math. Models Meth. Appl. Sci. [13] A. Hohmann, Inexact Gauss Newton methods for parameter dependent nonlinear problems. Dissertation, Konrad-Zuse-Zentrum fur Informationstechnik Berlin, (1993).

Steady Kinetic Boundary

Value Problems

157

[14] P. Le Tallec, F. Mallinger, Coupling Boltzmaim and Navier-Stokes Equations by half fluxes. Journ. Comp. Phys., 136 (1997), 51-67. [15] I. Miiller, T. Ruggeri, Rational Extended Thermodynamics. Springer, (1998). [16] N.A. Nurlybaev, Discrete velocity method in the theory of kinetic equations. Transp. Theor. Stat. Phys., 22 (1993), 109-119. [17] L. Pareschi, G. Russo, Numerical solution of the Boltzmann equation I: Spectrally accurate approximation of the collision operator. SIAM J. Numer. Anal, 37 (2000), 1217-1245. [18] F. Rogier, J. Schneider, A direct method for solving the Boltzmann equation. Trans. Theor. Stat. Phys., 23 (1994), 313-338.

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Chapter 7

Computational Methods and Fast Algorithms for Boltzmann Equations L. Pareschi University of Ferrara, Italy

7.1

Introduction

The construction of efficient numerical schemes for Boltzmann equations represents a real challenge for numerical methods and is of paramount importance in many applications, ranging from rarefied gas dynamics (RGD), plasma physics and granular media to semiconductors [2; 14; 18; 35]. The main difficulties are essentially due to the structure of the multidimensional integral that describes the interaction among particles. This integration has to be handled carefully since it is at the basis of the macroscopic properties of the equation and characterize the equilibrium state of the system. Additional difficulties are represented by the large number of variables that characterize the evolution of the equation (in addition to the time we have the three dimensional variables x in space and v in velocity) and the stiffness induced by the presence of small scales, like the case of small mean free path [23; 46]. From the numerical analysis point of view the main questions are related to the computational complexity and the accuracy of the numerical schemes. Typically, if n is the number of parameters (discrete velocities, Fourier or Hermite coefficients, etc.) used to approximate the distribution function in the velocity variable v, the computational cost of a conventional quadrature rule for the evaluation of the collisional integral can be much larger than n 2 . Now, since we have to multiply this cost times the number of 159

160

Lecture Notes on the Discretization

of the Boltzmann

Equation

points in the physical space x the final resulting scheme is prohibitively expensive in many circumstances even with the most advanced computational equipments. As a consequence the realistic numerical computations are based on probabilistic Monte-Carlo techniques at different levels. The most famous examples are the direct simulation Monte Carlo (DSMC) methods by Bird [4] and by Nanbu [40]. Although both methods have been originally developed for RGD their essential features can be easily extended to many other fields. For a detailed description of such methods we refer to [4; 40; 14; 50; 47]. Probabilistic methods present different advantages. First, the computational cost is strongly reduced and approximatively can be considered of the order of the number of points n. Second, these methods do not need any artificial boundary in the velocity space. In fact, particles can have any velocity and thus the discretization points are always well defined independently of the physical problem. At variance, in addition to the computational complexity, a major problem associated with deterministic methods that use a fixed discretization in the velocity domain is that the velocity space is approximated by a finite region. Hence, large regions with a huge number of discretization points are required in some problems (a typical example is the case of very high Mach numbers in RGD), which greatly increase the computing effort. This problem can be partially alleviated by reducing the number of grid points using high order schemes (for example spectrally accurate). On the other hand, probabilistic methods yield low accurate and fluctuating results for nonstationary solutions (where averaging techniques are expensive) with respect to finite differences or finite volume methods and the convergence in general is quite low (0(1/\/nj). In particular, the convergence rate is very slow in the presence of small scales. For continuum flows such methods require so many particles that are not competitive with the corresponding finite volumes schemes for the fluid-dynamic equations [17; 47; 51]. For this reason matching techniques with a fluid-dynamic solver are used for flows with large spatial variations in the mean free path (see [9; 34] for examples in RGD). In these approaches, the domain is subdivided into a Boltzmann domain, and a fluid-dynamic domain. In the first domain the Boltzmann equation is solved, and in the second domain the fluid-dynamic equations are solved. Suitable matching condition are used to couple the

Computational

Methods and Fast Algorithms for Boltzmann

Equations

161

two regions. One difficulty of this approach is the detection of the two regions. It is not trivial, in fact, to establish where the fluid description is sufficient, and where it is necessary to use the full Boltzmann equation. Similar problems are faced when explicit time discretizations are used with deterministic schemes. Some important partial progress in this direction has been recently presented in [22; 23; 31]. The possibility to extend the ideas developed in [23] by using Monte-Carlo methods has been recently investigated in [10; 47]. In order to increase the accuracy and avoid fluctuations, several authors have proposed different deterministic computational approaches especially designed for the numerical solution of problems where particle methods become expensive in order to get sufficiently accurate results (see [1; 6; 7; 16; 17; 26; 28; 31; 33; 38; 41; 42; 43; 46; 53] and the references therein). We also quote methods based on hybrid discretizations that combine finite difference techniques in space with some Monte-Carlo evaluation of the collision operator [15; 56]. In the framework of deterministic approximations, one of the most popular class of methods in RGD is based on the so-called discrete velocity models (DVM) of the Boltzmann equation. All these methods [8; 24; 30; 52] make use of regular discretizations on hypercubes in the velocity field and construct a discrete collision mechanics on the nodes of the hypercube in order to preserve the main physical properties. Unfortunately DVM have the same computational cost of a conventional quadrature rule and due to the particular choice of the integration points imposed by the conservation properties the order of accuracy of the resulting scheme is comparable to that of DSMC. For this reason the interest of DVM in computational RGD is mainly limited to qualitative simulations with a small number of discrete velocities. Thus, in general the requirement of maintaining at a discrete level the main physical properties of the continuous equation makes it extremely difficult to obtain high order accuracy. On the other hand, even if conservation properties are not imposed from the beginning, an accurate scheme would provide an accurate approximation of the conserved quantities. From the above discussion it would clearly be desirable to possess a deterministic scheme with the following characteristics • High accuracy in velocity space. • Possibility to use fast algorithms.

162

Lecture Notes on the Discretization

of the Boltzmann

Equation

• Efficient time discretization even in the presence of small scales.

Here we will focus our attention on the first two points, we refer the reader to [22; 23; 31] for some recent advances on the last aspect. In particular we review some recent results obtained with the use of spectral methods in different field of applied kinetic theory where the Boltzmann equation appears. Spectral methods based on Fourier series have been recently introduced for the Boltzmann [43; 46] and Landau [22; 48] collision operators. It has been shown that these spectral schemes permit to obtain spectrally accurate solutions with a reduction of the computational cost strictly related to the particular structure of the collision operator. A reduction from n2 to n log2 n is readily deducible for the Landau equation, whereas in the Boltzmann case such a reduction does not seem possible. The lack of discrete conservations in the spectral scheme (mass is preserved, whereas momentum and energy are approximated with spectral accuracy) is compensated by its higher accuracy and efficiency. Finally we mention that spectral methods have been successfully applied also to the study of non cut-off Boltzmann equations, like for RGD in the grazing collision limit [49] and for granular flows in the quasi-elastic limit [39]. In particular, during these asymptotic processes it is possible to obtain intermediate approximations that can be evaluated with fast algorithms that bring the overall computational cost to nlog 2 n. We will see how this idea can be used to obtain fast approximated algorithms for the Boltzmann equation. The rest of the chapter is organized as follows. Section 2 is devoted to present the basic ideas of the spectral method using a one-dimensional Boltzmann equation with inelastic collisions. After the derivation of the spectral scheme we present a description of the fast algorithm that reduces the cost from n 2 to nlog 2 n. This derivation is based on a representation of the kernel modes which has been inspired by the so-called quasi-elastic limit of the Boltzmann equation. Numerical results in the space homogeneous case illustrate the accuracy and efficiency of the spectral method and of the fast algorithm. Next in Section 3 the previous ideas are extended to the full Boltzmann equation for RGD. We will show how the direct derivation of the method allows to derive consistency as well as accuracy estimates. The possibility to extend the previous kernel modes decomposition to the multidimensional case is discussed. The section closes with the presentation of some space homogeneous computations.

Computational

7.2 7.2.1

Methods and Fast Algorithms

for Boltzmann

Equations

163

A one-dimensional example The Boltzmann

equation

with inelastic

collisions

Granular materials consist of many small discrete grains, which are inherently inelastic. The lost energy contributes to a variation of the heat of the grains themselves, but, unlike a classical fluid, this energy is not returned. In this sense, there is dissipation of energy. For systems composed of a large number of particles whose size is of a few microns, the natural approach became the study of the evolution dynamics at a mesoscopic level, by means of the classical methods of kinetic theory. Many recent papers (see [2; 20; 37] and the references therein), consider in fact Boltzmann-like equations for partially inelastic rigid spheres. This choice relies on the physical hypothesis that the grains must be cohesionless, which implies the hard-sphere interaction only, and no long-range forces of any kind. The kinetic picture based on the rigid-spheres like collisions does not exhibit formation of clusters and cooling in finite time [24], which on the other hand are considered peculiar behaviors of these materials [3; 36]. For this reason, in recent years the kinetic equations have been modified, to include different collision kernels. Assuming the collision frequency as a function of the kinetic temperature, a dissipative Boltzmann equation for Maxwell molecules [14], where the rate function does not depend on the relative velocity, has been studied in [5]. For a better understanding of the influence of a variable restitution coefficient (which depends of the relative velocity) in the behavior of a gas with inelastic collisions, a one-dimensional Boltzmann-like equation with an almost general rate function, including variable restitution coefficient and dependence of the relative velocity was studied in [55]. We consider this Boltzmann equation in a thermal bath [12]

dtf + vdxf = Q^(f,f)+6dvvf,

(7.1)

where f(x,v,t) is a nonnegative function that represents the density of particles in position x at time t with velocity v, and S is a small diffusion coefficient. In (7.1) Qy is the so-called granular collision operator, which is a quadratic integral operator describing the change in the density function

164

Lecture Notes on the Discretization

of the Boltzmann

Equation

due to creation and annihilation of particles in binary collisions:

QMJ)=

I I JM.JWL+

m{W-^f{vV{w')-\v-w\f{v)f{w)\dwdB. V

J

)

(7.2) In (7.2), the post-collisional velocities v' and w' are related to the precollisional velocities v and w by v' = —(v + w) + -(v — w)h;

w' = -(v + w) — —(v — w)h.

(7.3)

In (7.3) h £ [0,1] is the coefficient of restitution, i.e. v' — w' = h(v — w). Experimental works show that the coefficient of restitution depends on the relative velocity. The grains are close to be elastic for binary collisions with a small relative velocity, while they exhibit a certain degree of inelasticity when the relative velocity in the binary collision is high. For this reason, the coefficient of restitution has the form

^= ^ - H , g ) =

1 1

+ g||>

_B>|7-1,

(7-4)

where the exponent 7 characterizes the asymptotics of the restitution coefficient with respect to the relative velocity. The variable 6 £ R + furnishes a measure of the degree of inelasticity of the collision. Purely elastic collisions are obtained for 9 = 0, while perfectly inelastic collisions correspond to 8 = +00. For any fixed value of the inelasticity parameter 6, 7 > 1 corresponds to grains that are close to be elastic for small relative velocity. Of course, 7 < 1 gives the opposite phenomenon, namely the grains are close to be elastic for large relative velocities. We will refer to this case as the case of "anomalous" granular materials. To any collision the associate kernel is /3(6)\v —w\. This choice takes into account both the rate function of the rigid spheres interactions, and the probability that collisions with a degree of inelasticity 6 occur. Finally, in (7.2) J is the Jacobian of the transformation (v, w) -> (v',w').

j = Jieiv-wp-1) = /i^-H 7 - 1 ) (1 - (7 - i)B\v - w^^hieiv - H7"1)) (7.5) J(6\v — w\J J ) is nonnegative for all 7 < 2. In addition, J(r) is strictly positive on every compact set of E + . For particular choices both of the function /? and of the exponent 7, the collision operator (7.2) reduces to well-known models. If 7 = 1, and 0(6) equals the Dirac delta function S(9 — (1 — q)/q), where q < 1 is a positive

Computational

Methods and Fast Algorithms for Boltzmann

Equations

165

constant, we obtain the Boltzmann equation introduced in [20; 37]. This equation considers the grain-like rigid spheres, and has constant coefficient of restitution q. For collisions of type (7.3) the loss of kinetic energy is v'2 + w'2 - (v2 + w2) = - 1 ( 1 - h2){v - w)2.

(7.6)

Any reasonable kernel has to be such that the total amount of energy transfer is finite. Since l

-(l-h2){v-w)2<e-\v-w\1+\

(7.7)

this corresponds to impose the condition /

0(0)6 d6
(7.8)

JR+

7.2.2

Splitting

algorithms

The main part of our analysis will be devoted to the space homogeneous equation dtf = QM,f)(v,t)

+ 6dvvf.

(7.9)

It is well-known, in fact, that by the standard splitting algorithm we may consider separately the transport dtf + vdxf = 0,

(7.10)

and the relaxation given by (7.9). The overall accuracy of this simple splitting is first order in time. A second order generalization of this method is given by Strang's splitting [54]. Further discretization techniques can be found in [44]. Since the main difficulties are represented by the discretization of the collision operator, it is clear that after the splitting the most difficult part relies in the approximation of the collision step (7.9). For the operator Q 7 , there is conservation both of the mass p(x,t) = / 7K

f(x,v,t)dv,

166

Lecture Notes on the Discretization

of the Boltzmann

Equation

and of the momentum u(x,t) = /

vf(x,v,t)dv,

JUL

while the energy E(x,t)

\v\2f(x,v,t)dv,

= / JR

is dissipated. The steady state when e = S(v), and corresponds to concentration, function that exhibits strong deviations tions (see [12] and the references therein 7.2.3

A spectral

7.2.3.1

0 equals the Dirac delta function whereas when e > 0 is a smooth from classical Gaussian distribufor more details).

method

The choice of the computational domain

In this section we derive the Fourier spectral method for the homogeneous Boltzmann equation (7.2)-(7.9) following [43; 46]. To this aim we define with < •, • > the inner product in L\ (K) and consider the weak form of the equation

=

+5

=

f JR+

m l

\v-wMv')-
(7.11)

JR2

f(v, t)f(w, t) dwdOdv + S / [dvv(p(v)] f(v) dv, JR

for t > 0 and all test functions in the form

= [ W)f JR+

f(v,t)f(v

\q\Mv + q+)-
+ q,t)dqdOdv,

(7.12)

where q — w — v is the relative velocity, and the vectors q+ and q~ that parametrize the post-collisional velocities are given by

Computational Methods and Fast Algorithms for Boltzmann Equations

167

WrwjwsWK**--*™

-3R

-2R I

-R

0

R

2R I

3R

4R

5R

Fig. 7.1 Restriction of the distribution function on the periodic interval

[-2R,2R].

We point out that the possibility to integrate the collision operator over the relative velocity is essential in the derivation of the method. We consider now an initial density function fo(v,t) with compact support Supp(/ 0 (u,i)) C [-R,R]. Then for S = 0 the solution to (7.9) has compact support for all later times. In fact by (7.3), if |u|, \w\ < R then

\v'\<±\v\(l

+ h) +

±\w\{l-h)
and similarly we get \w'\ < R. In addition \q\, \q+\, \q~\ < 2R, thus we have the following L e m m a 7.1 then

/ / the function f(v,t)

is such that Supp(/(i;,£)) C [—R, R]

i) S u p p ( Q 7 ( / , / ) ( M ) ) C [ - # , # ] , ii)

= I ml JM.+

f(v,t)f(v with v + q+,v + q €

I

Mb(« + 9+)-v(f)]

J\v\
+ q,t)dqdffdv,

(7.14)

[-3R,3R].

Remark 7.2.1 The previous result shows that for compactly supported functions / in order to evaluate Q 7 ( / , / ) by a spectral method without aliasing error we can consider the density function / restricted on the interval [—2R, 2R], and extend it by periodicity to a periodic function on [-2R,2R] (see figure 7.1).

168

Lecture Notes on the Discretization

7.2.3.2

of the Boltzmann

Equation

Spectral projection of the equation

To simplify the notation let us take 2R = IT. The approximate function /jv is represented as the truncated Fourier series N

Yl heik\

fN(v)=

(7.15)

k=-N

fk

=

f{v)e-ihvdv.

j h [

(7.16)

A Fourier-Galerkin method [11; 25] is obtained by considering the projection of the homogeneous Boltzmann equation on the space of trigonometric polynomials of degree < N. Hence, taking f = /N and if = e~lkv for k — —N,..., N we have [dtfN - QMN, /JV) - SdvvfN] e~ikv dv = 0.

/

(7.17)

By substituting expression (7.15) into (7.17) we get a set of ordinary differential equations for the Fourier coefficients N

dth=

fifmP(hm)-Sk2fk,

Yl

k = -N,...,N,

(7.18)

l,m = -N l + m = fc

where the Boltzmann kernel modes J3(l,m) are given by 0(1, m) = /

dOP{9) I

\q\ [cos(lq+ - mq~) ~ cos{lq)} .

(7.19)

In fact, by evaluating (7.14) for (p = e~lkv and / = /JV, one obtains J3(l,m)=

[

d8p(0) [

|g|[e-**' + -l]e f l «dg,

(7.20)

and (7.19) follows by using the parities of the trigonometric functions. Note that (7.19) is a real quantity completely independent of the argument v, depending on just the particular kernel structure. This property is strictly related to the use of a Fourier spectral method. Other spectral methods may be developed, however they do not lead to this simplification. In practice all the information characterizing the kinetic equation is now contained in the kernel modes. Clearly, these quantities can be computed

Computational

Methods and Fast Algorithms for Boltzmann

Equations

169

in advance and then stored in a two-dimensional matrix of size 2N. Thanks to symmetry considerations the effective number of kernel modes that need to be computed and stored for the implementation of the method is reduced in practice since $(l,m)=${-l,-m).

(7.21)

Moreover it is possible to derive a symmetrized form of the kernel modes

P(l,m) = \f

dep(6) f \q\

* JR+

J\q\
[cos(lq — mq~) + cos(mq+ — lq~) — cos(mq) — cos(lq)] . Note that in the case 7 = 1, an analytic expression for /?(/, m) can be readily computed as 0(Z,m) = 7r2 2 Sinc(prr) - Sine ( ^ ) ~ 2 Sinc(/7r) + Sine

where p = ((I — m) + (I + m)h)/2 and Sinc(x) = sin(x)/x. Finally we can rewrite scheme (7.18) as N

9tfk= Yl

h-mfmfcl,m)-6k2fk,

k = -N,...,N.

(7.22)

ro= —iV

In the previous expression we assume that the Fourier coefficients are extended to zero for |fc| > N. The evaluation of (7.22) requires exactly 0(N2) operations which is smaller than the cost of a standard method based on iV parameters for / in the velocity space since we gain the integration over the variable 6. Thus the straightforward evaluation of (7.22) is slightly less expensive than a usual discrete-velocity algorithm. Remark 7.2.2 By construction the spectral method preserved the mass, whereas momentum and energy are approximated with spectral accuracy if the solution is smooth. These topics will be discussed in more detail in the second part of the chapter when we will develop the multi-dimensional theory.

170

7.2.4 7.2.4.1

Lecture Notes on the Discretization

Fast

of the Boltzmann

Equation

algorithms

Convolution sums

As discussed in the introduction for most applications a computational cost of 0(N2) is too expensive, in particular if compared with the O(N) cost of Monte Carlo methods. In this section we will discuss how we can reduce further the computational cost. First let us remark that we can't speed up the spectral scheme more than 0(ATlog2 N) which is the cost required by the evaluation of the Fast Fourier Transform (FFT). To keep notation simple in the sequel we will assume 6 = 0. To this aim the most natural approach is to search for a kernel decomposition in the form M

P(l,m) = 2 o i ( * ) ^ ( 0 c , - ( m ) ,

(7-23)

where M may be finite or not. Whenever this is possible the scheme results M

N

dth = 5 3 °i(*) £ j=l

h-mbjik

- m) / m c,-(m),

k = -N,...,N.

(7.24)

m=-N

Setting gj(k — m) = fk-mbj(k — m) and hj(m) = fmCj(m) be evaluated through M convolution sums like

the method can

N

J 3 9j{k-m)hj{m),

j = l,...,M.

m=-N

Since by standard transform methods [11] each of these sums requires 0(N log2 N) operations, the final cost of the scheme would be 0(MN log2 iV). Speed up clearly occurs only if M « N/log2 N. In general if (7.23) represents an infinite sum one has to consider a suitable truncation of the expansion which originates a new source of error in the scheme. 7.2.4.2

Kernel decomposition

In order to derive a kernel decomposition like (7.23) we will make use of some of the ideas described in [39; 49] to approximate numerically the quasi-elastic limit of the equation. It is in fact well-known that in this limit

Computational

Methods and Fast Algorithms for Boltzmann

Equations

171

the resulting friction equation can be evaluated by a spectral method with 0{Nlog2N) operations [39]. The quasi elastic limit is obtained by letting the collisions to be elastic, that is 6 = 0 (i.e. h = 1 and q" = 0). Thus we consider the Taylor expansion around q~ = 0 of the integrand in (7.19) cos(/<j+ — mq~)

=

cos(lq) + (I + m)q~~ sm(lq) (l + m)2, _ . , „ , (l + m)3. _ , , . ,, , K ) 3 sm(lq) + ... 03!' ( q 2! -(q f cos(lq) ~ oo

=

^2(-l)j+l

cos(Ig) +

(2j - 1)!

j=i

sin(
q (l + m) ^ - ^ — cos(tg) (2j)

In this way we obtain the expression

^

(2j - 1 ) !

(«")

2J-1

sin(Zg) -

9

J\q\<

JR+

( 2 j)

cos

14? - i) 1 ./R +

~} («")

2J-1

^5) I

\Q\

dq

•'M<*r

sin(Zg) - j — cos(lq) \ dq,

(7.25)

since I + m = k. We can rearrange the previous sum in a decomposition like (7.23) with ^,

(7.26)

\q\(q-)jFi(k)dq,

(7.27)

«;(*) = ( - i ) 6,(/)= [ Ju+

d9(3(6) f

[ [

J\q\<-" Cj(m)

= 1,

(7.28)

where [[.]] denotes the integer part, Fj(lq) = cos(lq) if j even and Fj(lq) = sin(lq) if j odd.

172

Lecture Notes on the Discretization

of the Boltzmann

Equation

Remark 7.2.3 i) At variance with the quasi-elastic limit, here we do not further expand expression (7.25) around q~ = 0 . As a consequence the leading order term does not correspond to the friction equation as in [39]. Moreover it is easy to see that our expansion is equivalent to a Taylor expansion around k = I + m — 0 and thus any truncation to (7.25) preserves the total mass as the original scheme, ii) In the original variables the previous expansion correspond to introduce an expansion of 4. iii) More accurate expansions can be used (for example using Chebychev polynomials). This will be the subject of further studies.

7.2.5

Numerical

examples

As a test case we report the numerical results in the space homogeneous case obtained for 7 = 1 (i.e. the restitution coefficient does not depend on the relative velocity) and 5 = 0 with the choice

m

=

e

-Q-, 0e[o,oo].

The initial data is the sum of two Gaussian distribution f(v,0)

= exp(-(2v + 2) 2 ) + exp(-(2v - 2) 2 ),

with v E [—7r,7r]. The integration time is tj = 8. Here we are interested only in checking the accuracy of our schemes with respect to the variable v and thus the error in t has been neglected. This can be achieved either using very small time steps or a suitably high order time discretization. In all our computations we used a fourth order Runge-Kutta method. We denote by SM the spectral method (7.18)-(7.19) and by FMk the fast spectral methods characterized by (7.18) and (7.25) truncated for M = k. In Figure 7.2 we report the time evolution of the distribution function obtained with scheme SM and FM3 with N = 32 modes. Since the solution

Computational Methods and Fast Algorithms for Boltzmann Equations

Fig.7.2 Time evolution of the distribution function for N 8M (left) and scheme FMa (right).

..

173

= 32 obtained with scheme

j

Fig. 7.3 Time evolution of the energy (left) and distribution function at t for N = 32 and schemes 8M (line), FMa (0).

= 8 (right)

is symmetric in v both spectral methods are conservative in mass and mean velocity. The decay of the energy in time together with the final solution at t I = 8 is reported in Figure 7.3. The good agreement between the two solutions is evident. For larger times in the case 8 = 0 the schemes start to oscillate since the equilibrium state is 8(v). In this situation 8 should be chosen as a numerical viscosity in order to eliminate the spurious oscillations [39]. Finally in Table 7.1 we report the L2 norm of the error at time t = 0.5 obtained with the spectral method for N = 8, N = 16 and N = 32. The 'exact' reference solution has been computed with N = 128 modes. The convergence rate in the L1-norm is about 11.7 when we pass from 8 to

174

Lecture Notes on the Discretization

Table 7.1

Equation

Relative error norms at time t — 0.5 and CPU-time for the spectral method

N=8 N=16 N=32

.2

of the Boltzmann

L1

L<x,

Lt

CPU time

3.988e-003 1.153e-006 8.217e-008

2.559e-003 4.678e-007 8.547e-008

3.017e-003 7.922e-007 7.745e-008

0.012 s 0.027 s 0.084 s

Relative error norms at time t = 0.5 and CPU-time for the fast metho

N=16

Li

Loo

La

CPU time

M=l M=2 M=3

4.291e-001 1.504e-001 7.852e-002

4.984e-001 1.642e-001 8.135e-002

4.506e-001 1.476e-001 7.399e-002

0.0018 s 0.0030 s 0.0040 s

16 modes. The error with TV = 32 modes is of order 10~ 8 which was the tolerance used in the computation of the kernel modes. In the last column the total CPU time in seconds for a single evaluation of the collision integral is reported. Next we test the fast method for different values of the truncation parameter M = 1,2,3 and TV — 16. The results are given in Table 7.2. As expected the spectral accuracy is lost and the method seems to behave as a first order method with respect to the truncation order. Note that since the truncation error dominate the spectral error the same values are obtained also for TV = 8 or TV = 32. For M = 3 and TV = 16 the method is about 7 times faster than the corresponding spectral scheme. The gain increases to a factor 14 for TV = 32. The final solution is reported in Figure 7.4.

7.3 7.3.1

The multi-dimensional case The Boltmann

equation for rarefied

gases

In this section we will first extend the spectral method to the multidimensional case of the Boltzmann equation for RGD and then will carry a detailed theoretical analysis of the scheme. The extension of the method to a multidimensional Boltzmann equation for granular media [5; 12] follows straightforwardly.

Computational Methods and Fast Algorithms for Boltzmann Equations 1DGr*nul>rmeci& energy in time

175

tDGranulafmeofa: solution al 1.0.5

Fig. 7.4 Time evolution of the energy (left) and distribution function at t = 0.5 (right) for N = 16. 'Exact' solution (line), scheme SM (o) and scheme FMz (*).

In the absence of external forces the time evolution of a single component mono atomic gas is governed by the Boltzmann equation (Cf.[13]) j£+v-Vxf=±Q(f,f),

x,vGR3,

(7.29)

where / = f(x, v, t) is a non negative function describing the time evolution of the distribution of particles which move with velocity v in the position x at time t > 0. The number e > 0 is called Knudsen number and is proportional to the mean free path between collisions. On the right hand side, Q(f,f) is the so-called collision operator given by

Q(f,f)(v)=

f

[ B{v-v.,0)[f{v')f
(7.30)

7R3 JS*

where the deflection angle 6 6 [0, n/2], is such that cos6 = (v — v*) • a/\v — v*\. The unit vector a parametrizes the set of all kinematically possible (i.e., those conserving energy and momentum) collisional velocities v' and v't by , v + v* v —— 2

\v-v*\ 2

a, '

, v + v* v„ = * 2

\v-vJ -a. 2

The relative probability of these outgoing velocities depends on the nature of the interaction between the molecules, and this is taken into account in the kernel B. For molecules which interact through an l/rs force law, where r is the distance between interacting particles, physical arguments

176

Lecture Notes on the Discretization

of the Boltzmann

Equation

show that the natural choice for the kernel B is 1) — 1)

B(v - v*,6) = \v - v*\Jb(cos6),

cosO = (n-,c), \v - v*\

where 7 = (s - 5)/(s — 1), and sin 8b(cos 6) is a smooth function except near 8 = 0, where it presents a (nonintegrable) singularity of order (s-t-l)/(s —1). This singularity corresponds to the grazing collisions and when these collisions prevail solutions to the Boltzmann equation converge towards solution of the Landau equation [49; 57]. The case 7 = 0 is referred to as Maxwellian gas whereas the case 7 = 1 yields the Hard Sphere gas. Boltzmann's collision operator has the fundamental properties of conserving mass, momentum and energy

JR )R3

Q(f,f)\

v

\dv = 0,

and satisfies the well-known Boltzmann's

/

(7.31)

H-iheoiem

Q(f,f)log(f)dv<0.

(7.32)

3 JRR

Boltzmann H-theorem implies that any equilibrium distribution function, i.e. any function / for which Q(f,f) = 0, has the form of a locally Maxwellian distribution M(P,u,T)(v)

= ^ — e x p t - l ^ ) ,

(7.33)

where p, u, T are the density, mean velocity and temperature of the gas p=

f(v)dv, JR3

7.3.2

u = - / P JR3

A spectral

vf(y)dv,

T = — / |u - v\2f(v)dv. op yK3

(7.34)

method

After a splitting of the equation we may search for a discretization by spectral methods of the homogeneous Boltzmann equation % = QU,f),

(7-35)

Computational

Methods and Fast Algorithms for Boltzmann

Equations

177

supplemented with the initial condition f(x,v,t

= 0) = fo(x,v).

(7.36)

In (7.35), to keep notation simple, we have fixed e = 1. A simple change of variables permits to write the Boltzmann collision operator Q(f, f) in the form Q(fJ)=

I

dqf

daB(q,6)[f(v

+ q+)f(v + q-)-f(v)f(v

+ q)}, (7.37)

where q = u* - v and the vectors q+ and q~ that parametrize the postcollisional velocities are given by q+ = \(q + kW),

q- = \{q-\q\°).

(7-38)

Again we point out that the possibility to integrate the collision operator over the relative velocity is essential in the derivation of the method. Finally we recall the following identity for the weak form of the collision operator /

Q{f,fMv)dv

=

JR3

[ dv [ JR3

VRS

B(q,9)[ip(v

dq [

da

(7.39)

JS2

+ q+) - f{v)]f(v)f(v

+ q),

for all test functions
The computational

domain

At variance to DSMC methods, where no artificial boundary in velocity space is needed, when constructing a numerical solution for (7.35) by means of a velocity discretization of f(v) a classical problem that one has to deal with is the fact that any numerical approximation to (7.35) requires the given initial density /o in a compact support whereas the steady state solution is a local Maxwellian like (7.33), that clearly does not have this property. As a consequence of this, relations (7.31) are not longer verified in time within a bounded domain. A general way to avoid this problem consists of modifying the collision mechanics or the numerical method in order to satisfy the original conservation laws in a bounded domain. However, it is clear that even with this modification the accuracy of the solution with respect to the original equation is guaranteed only if the size of the velocity domain is large enough.

178

Lecture Notes on the Discretization

of the Boltzmann

Equation

Following the approach we used in the previous sections we observe that if a distribution function / has compact support, Supp(/(u)) C B(0,R), where B(0,R) is the ball of radius R centered in the origin, then by conservation of energy (v')2
Let Supp(/(v)) C B(0,R)

then

i) Supp(Q(f,f)(v)) C B(0,V2R), ii) !

Q(f,fMv)dv

JR3

=

[ JB(0,V2R)

[ JB(0,2R)

[ B(q,6)

(7.40)

JS2

[tp(v + q+) - (2 + V2)R, assuming f(v) = 0 on [-T, T]3 \ B(0, R), and extend it by periodicity to a periodic function on [—T,T]3. Note that by virtue of the periodicity property of the function it is enough to take T > (3 + ^/2)R/2 to prevent intersections of the regions where / is different from zero (seefigure7.5). Remark 7.3.1 i) The previous proposition when Supp(/) € B(0,R) guarantees the absence of intersection between periods of the distribution function and thus permits to develop spectral methods on the cube without aliasing errors. However, since the support of / increases with time in practice we should choose T large enough in order to minimize the errors due to aliasing. ii) The choice of the integration domain #(0,2R) for g guarantees the equality ii). However (see [14]) it is important to note that for any M > 0 the collision operator with cut-off QM (/,/)> where g is integrated on the ball B(0, M), satisfies the conservation laws

Computational Methods and Fast Algorithms for Boltzmann Equations

,*'

/

(2+72)"^---, ,*'

& ^

;

!

HP

; -T

179

W;;;

T

\

J9^ /

"x

..--''

'v,

y'

Fig. 7.5 Restriction of the distribution function on the periodic box [—T, T] x [—T, T] in two dimensions. and the Boltzmann H-theorem we saw in Section 2. Clearly, the choice of M < 2R will not satisfy ii), and gives rise to different approximations of Q(f,f)- Aliasing errors in this general case are avoided if T > (M + (1 + y/2)R)/2.

7.3.2.2

Spectral projection of the equation

To simplify the notation let us take T = n and hence R = XTV with A = 2/(3+\/2). Hereafter, we use just one index to denote the three-dimensional sums with respect to the vector k = ( f c i , ^ , ^ ) € Z 3 , hence we set N

N

E -

£

•

fc=_JV fci,fc2,*3=-JV

The approximate function /M is represented as the truncated Fourier series AT

fN(v) = J2 heik\ k=-N

(7.41)

180

Lecture Notes on the Discretization

fk

=

of the Boltzmann

Equation

f(v)e~ikvdv.

7Z&[

(7.42)

As in the ID case, we obtain a set of Ode's for the fk coefficients by requiring that the residual of (7.35) be orthogonal to all trigonometric polynomials of degree < N [11; 25]. Hence, we have / •/[-TT.Tr]

3

[dtfN-Q(fN,fN)]e-ikvdv

= 0,

k = -N,...,N.

(7.43)

By substituting expression (7.41) into (7.43) and using the identity (7.39) for (p = e~lk'v we get a set of ordinary differential equations for the Fourier coefficients N

dtfk=

Yl

fifmP(l,m),

k = -N,...,N,

(7.44)

!,tr. = - i V

where the Boltzmann kernel modes J3(l, m) are now given by P{l,m) = I JB(0,2\n)

dq f

doB(q,B)

JS2

[ e - i ( ,+m >-« + _ l]

L

J

iql e

.

(7.45)

Again it is remarkable that (7.45) is a scalar quantity with the same form of (7.19) completely independent of the function fjy and on the argument v, depending just on the particular kernel structure. In practice these quantities can be computed in advance and then stored in a multidimensional matrix. It is possible to show that the kernel modes are functions only of \l — m\, \l + m\ and of the angle between r) — (l+m) and (i= (l — m) (see [43]). In particular we have 0(1,m) = F(\l + m\,\l-m\,ri-n)

= F(\l-m\,\l

+ m\,T) • n).

(7.46)

Since from (7.45) it follows easily J3(l,m) = $(—l, —m), from (7.46) we get also $(l, m) = j3(—l, —m) and hence the kernel modes are real functions. Obviously, these properties are useful to reduce the storage requirements of the method. We refer to [46] for a more detailed discussion on this topic.

Computational

Methods and Fast Algorithms for Boltzmann

Equations

181

Finally we can rewrite scheme (7.44) in the form, k = —N,..., N Bf -£=

N

Y,

fk-mfm$(k-m,m).

(7.47)

m=-N

In the previous expression we assume that the Fourier coefficients are extended to zero for \kj\ > N, j = 1,2,3. Remark 7.3.2 i) The evaluation of (7.44) requires exactly 0(N6) operations. We emphasize that the usual cost for a method based on N3 parameters for / in the velocity space is 0(naN6) where na is the numbers of angles discretizations. Thus as in the ID case the straightforward evaluation of (7.44) it is slightly less expensive than a standard discrete-velocity algorithm. ii) In the VHS case, B(q, 6) = C 7 l^l7, the dependence on the scattering angle disappears and (7.45) reduces to a one-dimensional integral [46] ${l,m)

= Cx,7 I r 2+7 [Sinc(|Z + m|A7rr)Sinc(|Z-m|A7rr) JO

-

Sinc2(2|m|A7rr)]dr

=

C A , 7 [F7{\1 + m\, \l - m\) - F 7 (|m|, |m|)],

where C\n = (47r)2(2A7r)3+7C7. In addition for integer values of 7 < — 3 the previous integral can be computed explicitly (see [46]) given rise to very simple and easy implementable approximation formulas. We report the expressions for the the case of Maxwell molecules 7 = 0 and hard spheres 7 = 1 F0(\l +

m\,\l-m\)

Fi(|l + m | , | J - m | )

p sin(q) — q sin(p) p2(qsm(q) + cos(g)) 2£r]p2q2 q2(psin(p) + cos(p)) — 4£r)

2|^V

'

where £ = \l + m\\Tr, rj — \l- m\XK, p = (£ + 77), q = (£ - 77).

182

Lecture Notes on the Discretization

7.3.3

Properties

of the spectral

of the Boltzmann

Equation

method

In this section we will analyze in detail the main properties of the spectral approach, with a particular attention to the physical properties and to the accuracy and stability properties of the spectral method. Let us first set up the mathematical framework of our analysis. For any t > 0, fiv(v,t) is a trigonometric polynomial of degree N in v, i.e. fN{t) € FN where P ^ = span {eikv

\ - N < kj < N, j = 1,2,3} .

Moreover, let T>N • L2([—7r,7r]3) -*• FN be the orthogonal projection upon P ^ in the inner product of L 2 ([-7r,7r] 3 ) (see (7.43)):

v^ePw.

>=0, We denote the L 2 -norm by

ll/l|2 = « / , / » 1 / 2 . With this definition V^f = /N, where /AT is the truncated Fourier series of / (7.41). Since the operator VN is self-adjoint the following property hold V / ,

V for the truncated Fourier series which is defined by a multiplication of each Fourier coefficient by a factor o^ <

VNI,
>=< f,Ps

=< VNf,VN(f

>

N

SfN(v)=fN=

E ^heik\

(7.49)

where a^ = a^ (Tk2 &k3 are required to be real non-negative numbers such that
Approximation by truncated Fourier series

First we prove some approximation properties of the projection operator VN, m particular those concerning positivity of the density function and approximation of the macroscopic quantities. Let us remark that, in general, when we approximate a non-negative function by a partial sum of its

Computational

Methods and Fast Algorithms for Boltzmann

Equations

183

Fourier series, that partial sum may be negative. The results are summarized in the following proposition Let f € L 2 ([-7r,7r] 3 ), / > 0, V v, and let us define

Proposition 7.2

I pu ) := I / I v I dv. \pe J •/[-".']' V M2 /

(7.50)

Then we have: i) If the factors o~k are such that N

1 + 2 ^2 oh cos(k -v)>0,

(7.51)

then

ii) The moments of /JV can be defined equivalently as

(PN \ puN

] •=

V PeN J

J 3

/N

7r]3

• -^

V n2 /

^

(

/

v

/JV

r /[

I M

r l

f

\ dv = I

/

J{

VN

Vfi

1 du

2 )N

-^]3 V (v :

\

f sk0 \

N
VJV

l

I

2

/fc

tifc

* = - " V (»2)* /

V (V )N J

where VN — VNV, (V2)M = VNV2, Sko is the Kronecker delta, and Vk and {v2)k are the Fourier coefficients of v and v2. Hi) The following relations hold P = PN,

\pu-puN\

\pe-

< ^_||/||2,

peN\ <

AT3/2

184

Lecture Notes on the Discretization

of the Boltzmann

Equation

Proof. Note that the smoothed series (7.49) can be represented in integral form, as [11] /AT(W) = 7^-T3 / (2TT)3

KN(v

- w)f(w)

dw,

J{_„^]3

where the kernel KN is given by N

Kff(v) = 1 + 2 Y^ crfc cos(A; • v). fc=i

Prom the positivity of KM follows the positivity of fN and hence property i). Property ii) is a simple consequence of the fact that the projection operator is self-adjoint and follows choosing ip(v) = l,v, \v\2 in (7.48). The last identity can be obtained by direct substitution of the truncated Fourier series /JV and using the properties that fk = /_ f c , where f_k denotes the complex conjugate of fk, and the following relations:

eikvdv,

Sko = 7 ^ [

(v2)k , k

ik

v2e~ikv

=— ^ /

dv,

(2TT)3 ; [ _ , , , ] ,

ve ikV

= TT^I /

dv

~

-

(2TT)3 y[_w>w]s

The first equality in Hi) is a consequence of PN = (27r)3/o = /

J[-ir,Tr]3

f(v)dv

= p.

The estimates for puM and pejy are derived observing that for each ip € L2([—7r,7r]3), using Schwartz inequality, we have |

- | <
><||/||2||V-¥»JV||2-

Computational

Methods and Fast Algorithms

for Boltzmann

Equations

185

Now, by direct computation we obtain (yj)Q = 0, j = 1,2,3 and {v2)0 = ir2 whereas for k ^ 0 we have 3

u

(vi)k = —. nk--14~\

n

r

3

6k

v sin k v

'° /

( j j)

2 cos

d

i

dv = i

3~

/

Sk

n

-\t.

'°~i—>

(?-52)

j = 1,2,3, and

2

1

3

3

,

.

3

/•

ip )k = T^ZS 53 I I **«° / fa) \^>

,- = i ,=i

3

2

/JX*,-

( —1)

(M;) «> = 53II **.onfr

•/[-*,*]

, = 1 ,=1

K

j

(7.53) Using Parseval's identity and (7.52)-(7.53) we obtain the following estimates \V-VN\h

< ATl/2'

the conclusion follows taking
O

Remark 7.3.3 • The positivity requirement i) can be satisfied using the factors [11] ^ = (

1

- ^ ) '

i = 1.2,3.

(7.54)

In fact, these smoothing factors correspond to a non-negative kernel, the so-called Fejer's kernel, KN given by

This is also equivalent to replace the truncated Fourier series by the arithmetic means, or Cesaro sums, of the truncated series. However, Fejer's kernel produce a heavy smearing of the function near a singularity point. In most applications it is desirable to have a sharper representation of the function by using different smoothing at the expense of retaining some oscillations or small negative values.

186

Lecture Notes on the Discretization of the Boltzmann Equation

• Since the smoothed projection SVN = Vff satisfies < P'Nf,

=< f,VsNv

>=< VNf,VsNV

V / , ip € L 2 ([-7r,7r] 3 ), (7.55) by using the properties of o>, the analogue of points ii) and Hi) can be proved for f^. • From ii) and (7.52) the j-component of the momentum of /AT depends only on the imaginary part of fk and only on the N Fourier coefficients on the axis ki = 0,1 ^ j . Similarly from ii) and (7.53) the energy of /jv depends only on the real part of fk and only on the 3iV Fourier coefficients on the three orthogonal axes. • The estimates given in Hi) can be strongly improved if / is smooth. If / € Hp([-Tt,n]3), where r > 0 is an integer and Hp([-ir,ir]3) is the subspace of the Sobolev space Hr([—7r,7r]3), which consists of periodic functions [11], for each ip € L2([—7r,7r]3) we have >

l < /,¥> > - < f, I < 1Mb 11/ - IN\\2 < ^ r l M b l l / H ^ , where || • \\nr denotes the norm in Hp([—7r,7r]3). This inequality shows that the projection error on the moments decay faster than algebraically when the solution is infinitely smooth. 7.3.3.2

Consistency and spectral accuracy

Now we can discuss the main properties of the semi-discrete scheme defined by (7.44). For simplicity we will restrict our discussion to the VHS model introduced previously. We will denote by Qx(f, f) the Boltzmann operator with cut-off on the relative velocity on B(0,2Xn) and, as usual, we denote by Q^(/N) = 'PNQx(fN,fN)In order to prove a consistency result for the method we need the following L e m m a 7.2

Let f,g£

£ 2 ([-7r,7r] 3 ), then

i) QX{fN,9N)£V2N, ii)

IK2 A (/,5)l|2
Computational

Proof. formula

Methods and Fast Algorithms

for Boltzmann

Equations

187

The first statement follows immediately from the representation N

N

J2 fi9mP{l,m)ei«+m>v

Q\fN,9N)= £ l=-N

m=-N 2N

= £ Q%eik\ k=-2N

where N

Qk=

£

tempera).

(,77l=-JV

The point it) for the gain part of the collision operator is a consequence of the estimates in [27; 32] IIQ A ' + (/,3)l|2 < C1II/H2IMI1 < (27r) 3 / 2 C 1 ||/|| 2 || 9 || 2 . The corresponding result for the loss part can be computed directly observing that Lx(f) = f *q Bx where *q denotes the convolution operation with respect to q and Bx is the VHS kernel with cut-off over the relative velocity q on the ball B(0,2\n). Hence 11/(9 *, 5*)|| 2 < ll/lbllff *, BA||oo < ||/||2||B A ||oo||5||i < C 2 | | / | | 2 | | 5 | | 2 , with C2 =

(2TT) 3 / 2 ||5 A ||OO

= (27r)3/2C7(2A7r)T.

D

Remark 7.3.4 In the previous proposition the constant C depends on the size of the domain of integration in v. Thus the estimate becomes meaningless when the integration domain is the whole E 3 space as in the original equation. This drawback does not affect us, since here we are interested in proving the spectral accuracy of the method when the integration domain is fixed. Then we observe that the method defined by equation (7.44) can be written in equivalent form as

^ f = QXN(M,

(7.56)

188

Lecture Notes on the Discretization

of the Boltzmann

Equation

with the initial condition fff(v,t

= 0) = fo,ff(v),

(7-57)

x

x

where, to simplify the notation, we use Q (f) instead of Q (f, / ) . We point out that because of the periodicity assumption on / , and hence on Qx(f), the collision operator Qx(f) preserves in time the mass contained in the period. On the contrary, momentum and energy are not preserved in time. From Proposition 7.2 it is also clear that the projected collision operator QN(/N) will preserve the mass in time. This can also be derived directly from the properties of the kernel modes, in fact, from p = (27r)3/o we have TV

dt

=~ ^2 f-mfmP(-m,m) = 0, m= — N

since J3(—m,m) = 0. Next we state the consistency in the L 2 -norm for the approximation of the collision operator Qx{f) with <9JV(/W), Theorem 7.1

Let f €

L2([-TT,

TT]3),

then

( \Q\f) - QUMh < c [||/ - fsh + " Q~N fr : ) l k ) , vr > o,

(7.58) where C depends on ||/||2Proof. \\Q\f)

First, we can split the error into two parts -

QNUN)\\2

< \\Q\f)

-

QXUN)\\2

+ WQHfw) -

QxN(fN)\\2.

Now from Lemma 7.2, <5 A (/JV) 6 ^2N and hence <3A(//v) is periodic and infinitely smooth together with all its derivatives thus [11] \\QX(fN)-QXN(fN)\\2<-^\\QX(fN)\\H;,

Vr>0.

(7.59)

By application of Lemma 7.2 and from the identity [14] Q\f)-Qx{g)

= QxU +

g,f-9),

(which is a direct consequence of the bilinearity of Qx), we have IIQ A (/) - QX(fN)\\2 = \\QHf + fN,f~

fN)\\2

< C l | | / + fNfoWf - fM\2

Computational Methods and Fast Algorithms for Boltzmann Equations

189

<2CI||/||2||/-/JV||2.

This concludes the proof.

•

Remark 7.3.5 From (7.59) it follows that, except for the projection errors on the initial data, the variations of momentum and energy introduced by the semi-discrete scheme are spectrally small and hence the observed variations with respect to the projected moments are mainly due to the aliasing of periods. In fact, using (7.59), from Schwarz inequality we have I < QXUN),
> - < QXN(fN),

I < ^rlM|2||QA(/iv)||^.

(7.60)

The estimates on the conservation laws can be derived by considering ip = v,v2. The previous theorem states that the rate of convergence in the L 2 -norm of QJV(/JV) to Qx{f) depends only on the speed of convergence of /jy to / . Hence if JN is spectrally accurate so it is Q ^ ( / J V ) - The following corollary states the spectral accuracy of the approximation of the collision operator Corollary 7.1 \\Q\f) Proof.

Let f € Hrp{[—7r,7r]3), r > 0 then ~ QXNUN)\\2

< §;

(\\f\\Hr + \\QX(fN)\\Hi)

,

(7.61)

It is enough to observe that l|/-/,v||2<^||/|k;.

Remark 7.3.6

Q

From the previous corollary it follows

I < Qx(f),v >-< QXNUN),V> I < ^ I M h (11/IIH; + \\Qx(fN)\\H;) , (7.62) and hence, by taking tp = v,v2, the spectral accuracy of the moments. 7.3.3.3

Stability of a smoothed positive scheme

Let us rewrite the homogeneous Boltzmann equation for VHS kernels with cut-off over the relative velocity in the equivalent form ^+»f with J * ( / , / ) = Q * ( / , / ) + / z / .

= JX(f,f),

(7.63)

190

Lecture Notes on the Discretization

of the Boltzmann

Equation

It is easy to check that for the loss part of the collision operator we have the inequality Lx{f)

< C747r(2A7r)'1' f

•/[-TT.Tr]3

fLx(f)

f(v) dv = C747r(2A7r)7/o.

Thus for H > C747r(2A7r)7/9,

(7.64)

JX(f> f) is a positive monotone operator in the sense that JX(f,f)>JX(9,9)>0

if

/>0.

Then we have the following result regarding the stability of a smoothed spectral scheme [45] Theorem 7.2 There exist a unique positive solution /AT(<,I') 6 C 1 ([0, T], IP^), with ||/JV||I — 9, for arbitrary time T > 0 to the initial value problem - ^ - + M/JV = SJN(/N),

v G E 3 , t > 0,

(7.65)

fN(v,t = 0) = flN(v), provided that /o € L2([—n,ir}3) is non-negative, \\fo\\i = p, (J, satisfies (7.64) and the smoothing operator S satisfies (7.51). The proof is similar to the proof of existence and uniqueness of a solution for the homogeneous Boltzmann equation for hard sphere gases with cutoff on the relative velocity. It makes use of the conservation of mass of SQxv(fpf) and on the positivity preserving properties of Vfj [45]. Prom Theorem 7.2 it follows that the L1 norm of the spectral solution is constant in time and hence the smoothed positive scheme is stable in the Z^-norm. Remark 7.3.7 For practical purposes the positive scheme (7.65) introduces too much smoothing, and spectral accuracy is lost. Note that, without smoothing, scheme (7.65) and scheme (7.56) are the same. However, it is interesting to remark that, as pointed out in [45], the main reason in the lack of accuracy of (7.65) is represented by the smoothed projection of the initial data.

Computational

7.3.4

Methods and Fast Algorithms for Boltzmann

Accelerated

Equations

191

algorithms

Similarly to the one-dimensional case in this section we will discuss how we can reduce further the computational cost. To derive a kernel decomposition like (7.23) we will use the same ideas described in [39; 49] for the grazing collision limit. In this limit the resulting Landau equation can be evaluated by a spectral method with 0(N3 log2 N) operations [22; 48]. Thus we consider the multivariate Taylor expansion around q~ = 0 of the integrand in (7.45) exp(—il • q+im • q~)

=

exp(—il-q) + (l +

+ =

2{

;

m)-q~exp(—il-q)

exp(-»Z • g) + . . .

/ -, N ^((i + m)-q-)j exp(-il • q) + 2 ^ — ; '

, ., , exp(-il-q).

3

j=i

'

In this way we obtain the expression

~[J-JR+

J\q\
((I + m) • q~Y exp(—il • q) dq

= E77/ W ) / ~[

J- JR+

M

7 |g|<7r

(k • q-y exp(-«7 • q) dq,

(7.66)

since I + m = k. Every term that appears in (7.66) can be computed by means of convolution sums. Note that the number of convolutions we have is of order 0(j2) for every j in the sum, and thus if the sum (7.66) is truncated to the first M terms the total number of convolutions is 0(M3). Since by standard transform methods [11] each of these sums requires 0(N3 log2 N) operations the final cost of the scheme would be 0(M3N3 log2 N). Speed up now occurs only if M 3 < < JV 3 /log 2 N. Remark 7.3.8 Since we do not further expand expression (7.66) around q~~ = 0 as in the grazing limit, the leading order term does not correspond to the Landau equation as in [49]. Again it is easy to see that our expansion is equivalent to a multivariate Taylor expansion around k = I + m = 0

192

Lecture Notes on the Discretization

of the Boltzmann

Equation

and thus any truncation to (7.66) preserves the total mass as the original scheme. In the original variables the previous expansion corresponds to introduce an expansion of Q 7 ( / , / ) that involves multidimensional derivatives of increasing order in the v variable. Thus truncating expression (7.66) may originate unstable PDEs approximations of the Boltzmann equation. 7.3.5

Numerical

tests

The time integration of the Boltzmann equation represents in itself a challenging problem, since the nonlinear collision operator becomes highly stiff near the fluid regime. However, here we are just interested in testing the accuracy of the velocity discretization we have seen in the previous sections and therefore we will always assume a fixed e = 1. Clearly, for a low-order time discretization, one expects that the error of the fully discretized scheme will be dominated by the temporal errors. Thus if we want to check the spectral accuracy of the method either very small time-steps or else high-order time-differencing methods are required. In our tests we use the spectral method coupled with an explicit fourth-order Runge-Kutta method in time. This Runge-Kutta scheme provides the high temporal accuracy needed to demonstrate spectral accuracy in velocity. In this section we study only the performance of the spectral method defined by (7.44)-(7.45). Numerical results based on the fast algorithm (7.44)-(7.66) are actually under study. We apply the spectral scheme to the following test problems: 2D Maxellian molecules We consider the 2D Maxwellian molecules case (i.e. 7 = 0), with Co = 1/(2*). The initial condition is f(v,0)

=

v2

^exp(-v2/a2),

and the integration time: tmax = 40. This problem has an exact solution given by [49]

Computational Methods and Fast Algorithms for Boltzmann Equations

Fig. 7.6 2D Maxellian case. Left: Relative Loo (dots), L\ (line), L2 (dash), norms of the error in the density function vs time, for N = 16 modes per direction. Right: Positivity error vs time with N — 16 (continuous line) and an N = 32 (dashed line). where 5 = 1 — exp(—
(zl-Nm)-f(vi,t)\pY/p

with i = (11,12), P = 1 and p = 2. A similar expression is used for the L^ error. Note that the relative error increases initially, and then it decreases almost monotonically in time. After a long time the error starts increasing again. This effect is due to aliasing. When the number of Fourier modes

193

194

i

Lecture Notes on the Discretization of the Boltzmann Equation

u

is

n

s

»

s

m

S

10

15

Fig. 7.7 2D Maxellian case. L\ relative norm of the error vs time for N = 8,16, and 32 modes per direction for a = TT/6 (left) and with a more compact initial condition a = 7r/7 (center) and a = TT/8 (right).

Error in tha fourth moment n=8,16, 32

j

1—

,— • - ' •

'

•

'

r -i

•

Fig. 7.8 2D Maxellian case. Error in energy (left) and in the fourth moment (right) vs time for n = 8,16, and 32 modes per direction. increases, t h e effect of aliasing becomes dominant over the error due t o the spectral approximation. Clearly, because of t h e loss of positivity, t h e L\ n o r m of / is not conserved. T h e positivity error can be measured by t h e quantity Cn •

El/jv(«m)|

-HfNJVm)

:

Computational

Methods and Fast Algorithms for Boltzmann

Equations

195

As it is expected, the positivity error is spectrally small (see Fig. 7.6 right), and for a grid as coarse as 162 is less than 10~ 3 . We refer to [45] for a detailed discussion about the connections between positivity and spectral accuracy. This effect is more evident in Figure 7.7 where we show the relative L\ norm of the error in the density function for N = 8,16, and 32 modes per direction. A time step At = 1 has been used. # modes n 8x8 16 x 16 32x32

Error at time t == 5 TT/8 TT/6 7r/7 1.51 x H T 1 2.09 x l O " 1 2.71 x l O - 1 3.01 x H T 3 9.78 x l 0 ~ 3 1.78 x l 0 ~ 3 3.79 x l O - 6 0.81 x l O - 5 0.61 x l O " 5 Table 7.3

Convergence rate TT/6

TT/7

TT/8

5.64 9.64

4.41 10.23

3.91 8.19

Convergence test for Boltzmann integral in 2D.

If we fix the attention to Figure 7.7 at time t = 5 we observe the results for the relative L 1 norms of the error reported in table 7.3 for different choices of the scaling parameter a = n/6, n/7, 7r/8. In the last three columns the order of accuracy is reported. The decay of the error with the increase of the number of modes is an indication of spectral accuracy. For longer times the aliasing effect dominates in the case N = 32. This suggests that a compromise should be obtained between aliasing and spectral error, and that the period should be increased when increasing the number of Fourier modes. Roughly speaking, the period should be chosen in such a way that the two contributions of the error are of the same order of magnitude. To show that the effect of aliasing becomes dominant when the number of Fourier modes is increased, we repeat the previous calculation using a more compact initial condition, which is equivalent to using a larger period. The results of the computation are shown in Fig. 7.7 (right). With 32 modes per direction, when increasing the period with respect to the numerical support of the function, the error for short times increases, because of the loss of resolution (since the number of modes is the same), but the error for long time decreases, since the effect of aliasing is less pronounced. In Figure 7.8 the error in the conservation of energy and in the evolution of the fourth order moment is shown. Energy is not exactly conserved, but if aliasing error is small, energy is conserved within spectral accuracy.

196

Lecture Notes on the Discretization

of the Boltzmann

Equation

The exact value of the fourth moment is §R-1vif{v,t)d2v — 8S(t)(2 — S(t)). The cusps in the figures are due to the fact that the exact and the numerical values cross at a certain time, and therefore at that time the error is zero (which is shown by a cusp in a semi-log plot). Mass (i.e. J f(v) dv) is conserved exactly by the scheme, while conservation of momentum is maintained within round-off error, because of the particular symmetry of the problem. Because of spectral accuracy, a semi-log plot has been used for the norms in most figures. 3D Maxwell molecules Next we consider the 3D Maxwellian molecules case, with Co = l/(47r). The initial condition is the sum of two Gaussians

f(v,0) =

2 3

6XP /

(

2(27T(T ) /2

| w - 2 < r e i | 2 \ +, eXP /

2*^)

{

\v+

2oe^\\

2^J. '

with a = XTT/6 and the integration time is tmax = 10. Here e\ = (1,0,0) denotes the unit vector in the direction vx. This test is used to check the evolution laws for the moments of the distribution function (temperature and stress tensor). The derivation of the evolution law for the stress tensor is presented in [6]. Let Pij, i,j = 1, • • • ,3 denote the components of the stress tensor defined as Pij = /

f(v)(Vi - Ui)(Vj - Uj) dv,

JR3

where u, are the components of the mean velocity. Because of the axial symmetry of the initial condition, the stress tensor is diagonal, with P22 = P33. The expression of the components is given by 2 pu=p+-w(t),

1 P22=p-^w(t),

and w(t) = w0 exp(—2-KpCot). U>Q = 2a2, p = pT, T = (7/3)
Computational Methods and Fast Algorithms for Boltzmann Equations

197

Fig. 7.9 3D Maxellian case. Stress tensor component P n and P22 vs time for N = 16. Exact (line) and numerical (dash).

Fig. 7.10 Maxellian case. Relative error in the computation of the stress tensor: P l l (line) and P22 (dash) in the 3D case for N = 16 (left) and 2D case for JV = 32 (right).

2D VHS

molecules

Finally we consider t h e 2D VHS molecules case, with Co = l/(27r). T h e initial condition is the sum of two Gaussians

f(v,0) =

\v — 2ae\\ 2(2TT<7 2 )

exp

2o^~

I -j +expf-^ + 2a2trei 2

198

Lecture Notes on the Discretization

of the Boltzmann

Equation

with a = \ir/6 and the integration time is £max = 10- This test is used to check the evolution of moments (temperature and stress tensor), and to compare the relaxation to equilibrium of Maxwellian molecules, with the relaxation of other VHS molecules. For Maxwellian molecules, an expression similar to the 3D case can be derived for the components of the stress tensor:

Pu{t)=P+^w(t), where w(t) = w0exp(—irpCot). therefore irpCo = 1/2.

P22(t)=p-^w(t), In our case p = 1, and Co = l/(27r),

Fig. 7.11 2D VHS case. Relaxation of the stress for different VHS molecules. 7 = 0 (line), 7 = 0.5 (dot-dash), 7 = 1 (dots), and 7 = 2 (dash). The relative error in the computation of the stress is below 10~ 4 , and this allows an accurate computation of the relaxation of the moments for general VHS molecules. Note that, at variance with the case shown in the previous figure, in this case the error increases monotonically. The reason is that with this choice of the number of modes and size of the computational domain, the aliasing error is dominant with respect to the resolution error. As an example, in Fig. 7.11 we show the relaxation of F n for molecules with 7 = 0,0.5,1, and 2. A semi-log plot of the relaxation shows that the decay is essentially exponential (see Fig. 7.11 right). We refer to [46] for a comparison of the results obtained with our spectral method with those obtained by a Monte Carlo scheme.

Computational

7.4

Methods and Fast Algorithms for Boltzmann

Equations

199

References

[1] K. Aoki, S. Kosuge, S. Takata, Shock-wave structure for a binary gas mixture: finite difference analysis of the Boltzmann equation for hardspheres molecules, Eur. J. Mech. B-Fluids., 20 (2001), 87-126. [2] D. Benedetto, E. Caglioti, M. Pulvirenti, A kinetic equation for granular media, Math. Mod. Numer. Anal., 31 (1997), 615-641. [3] B. Bernu, R. Mazighi, One-dimensional bounce of inelastically colliding marbles on a wall, J. Phys A, Math. Gen., 23 (1990), 5745-5754. [4] G.A. Bird, Molecular gas dynamics and direct simulation of gas flows Clarendon Press, Oxford, UK, (1994). [5] A.V. Bobylev, J.A. Carrillo, I. Gamba, On some properties of kinetic and hydrodynamics equations for inelastic interactions, J. Statist.Phys., 98 (2000), 743-773. [6] A.V. Bobylev, S. Rjasanow, Difference scheme for the Boltzmann equation based on the Fast Fourier Transform, Eur. J. Mech., B/Fluids., 16, 2 (1997), 293-306. [7] A.V. Bobylev, S. Rjasanow, Fast deterministic method of solving the Boltzmann equation for hard spheres, Eur. J. Mech. B Fluids., 18 (5) (1999), 869-887. [8] C. Buet, A discrete velocity scheme for the Boltzmann operator of rarefied gas dynamics, Trans. Theo. Stat. Phys., 25 (1996), 33-60. [9] J.F. Bourgat, P. Le Tallec, B. Perthame, Y. Qiu, Coupling Boltzmann and Euler Equations without overlapping, Domain decomposition methods in science and engineering, Como 1992, Contemp. Math., 157 (1994), 377-398. [10] R.E. Caflisch, L. Pareschi, Implicit Monte-Carlo methods for rarefied gas dynamics I: the space homogeneous case, J. Comput. Phys., 154 (1)(1999), 90-116. [11] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral methods in fluid dynamics, Springer Verlag, New York, (1988). [12] J.A. Carrillo, C. Cercignani, I.M. Gamba, Steady states of a Boltzmann equation for driven granular media, Phys. Rev. E (3) 62 (2000), no. 6, part A, 7700-7707. [13] C. Cercignani, The Boltzmann equation and its applications, Springer-Verlag, Berlin, (1988). [14] C. Cercignani, R. Illner, M. Pulvirenti, The mathematical theory of dilute gases, Springer-Verlag, New York, (1995).

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Lecture Notes on the Discretization of the Boltzmann Equation

15] F.G. Ceremisin, Fast solution of the Boltzmann Equation, Proceedings of the 17th International Symposium on Rarefied G a s Dynamics, edited by A. Beylich, Weinheim, (1991), 273-284. 16] A.J. Chorin Numerical solution of Boltzmann's equation, Comm. Pure and App. Math., 25(1972), 171-186. [17] F. Coron, B. Perthame, Numerical passage from kinetic to fluid equations, SIAM J. Numer. Anal., 28 (1991), 26-42. [18] P. Degond, B. Lucquin-Desreux, The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case, M3AS, 2 (1992), 167-182. [19] L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing, Trans. Theo. Stat. Phys., 21 (1992), 259276. [20] Y. Du, H. Li, L.P. Kadanoff, Breakdown of hydrodynamics in a onedimensional system of inelastic particles, Phys. Rev. Lett., 74 (1995), 1268-1271. [21] M. H. Ernst, Exact solutions of the nonlinear Boltzmann equation and related kinetic models, Nonequilibrium Phenomena I. The Boltzmann equation, North-Holland, (1983), 52-119. [22] F. Filbet, L. Pareschi, A numerical method for the accurate solution of the Fokker-Planck-Landau equation in the non homogeneous case, J.Comp. Phys. to appear. [23] E. Gabetta, L. Pareschi, G. Toscani, Relaxation schemes for nonlinear kinetic equations, SIAM J. Numer. Anal., 34 (6) (1997), 1967-2011. [24] D. Goldstein, B. Sturtevant, J.E. Broadwell, Investigation of the motion of discrete velocity gases, Rar. Gas. Dynam., Progress in Astronautics e Aeronautics, 118, AIAA, Washington, (1989). [25] D. Gottlieb, S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM CBMS-NSF Series, (1977). [26] Y.N. Grigoriev, A.N. Mikhalitsyn, A spectral method of solving Boltzmann's kinetic equation numerically, U.S.S.R. Comput. Math. Phys., 23 (6) (1983), 105-111. [27] T. Gustaffson, LP-estimates for the nonlinear spatially homogeneous Boltzmann equation, Arch. Rat. Mech. Anal., 92, (1986), 23-57. [28] J.C. Hang, J.Y. Yang, Rarefied flow using nonlinear model Boltzmann equations, J. Comp. Phys., 120, (1995), 323-339. [29] R. Illner, H. Neunzert, On simulation methods for the Boltzmann equa-

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Methods and Fast Algorithms

for Boltzmann

Equations

201

tion, Trans, Theo. Stat. Phys., 16, (1987), 141-154. [30] T. Inamuro, B. Sturtevant, Numerical study of discrete velocity gases, Phys. Fluids A., 2, 12 (1990), 2196-2203. [31] S. Jin, L. Pareschi, Discretization of the multiscale semiconductor Boltzmann equation by diffusive relaxation schemes, J. Comp. Phys., 161 (2000), 312-330. [32] P.L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications. I, J. Math. Kyoto Univ., 34-2 (1994), 391427. [33] E. Longo, L. Preziosi, On a conservative polar discretization of the Boltzmann equation, Japan J. Indust. App. Math., 14 (1997), 399435. [34] P. Le Tallec, F. Mallinger, Coupling Boltzmann and Navier-Stokes Equations by Half Fluxes, J. Comp. Phys., 136 (1997), 51-67. [35] P. Markowich, C. Ringhofer, C. Schmeiser, Semiconductor equations, Springer-Verlag, Wien-New York, (1989). [36] S. McNamara, W.R. Young, Inelastic collapse and clumping in a onedimensional granular medium, Phys. Fluids A., 4 (1992), 496-504. [37] S. McNamara, W.R. Young, Kinetics of a one-dimensional granular medium in the quasi-elastic limit, Phys. Fluids A., 5 (1993), 34-45. [38] Y. Morchoisne, Une methode de differences finies pour la resolution del Vequation de Boltzmann: Traitement du terme de collision C. R. Acad. Sci. Paris, t.313. Serie II, (1991), 1513-1518. [39] G. Naldi, L. Pareschi, G. Toscani, Spectral methods for a singular Boltzmann equation for granular flows and numerical quasi elastic limit, preprint 2001. [40] K. Nanbu, Direct simulation scheme derived from the Boltzmann equation. I. Monocomponent Gases, J. Phys. Soc. Japan., 52 (1983), 20422049. [41] T. Ohwada, Structure of normal shock waves: Direct numerical analysis of the Boltzmann equation for hard sphere molecules, Phys. Fluids A., 5, 1 (1993), 217-234. [42] L. Pareschi, On the fast evaluation of kinetic equations for driven granular flows, Proceedings Enumath 2001, to appear. [43] L. Pareschi, B. Perthame, A Fourier spectral method for homogeneous Boltzmann equations, Trans. Theo. and Stat. Phys., 25, 5(1996), 369383.

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Lecture Notes on the Discretization

of the Boltzmann

Equation

[44] L. Pareschi, G. Russo, Implicit-Explicit (IMEX) Runge-Kutta schemes for stiff systems of differential equations, Recent Trends in Numerical Analysis, Edited by L.Brugnano and D.Trigiante, 3 (2000), 269-289. [45] L. Pareschi, G. Russo, On the stability of spectral methods for the Boltzmann equation, Transp. Theo. Stat. Phys., 3-5 (2000). [46] L. Pareschi, G. Russo, Numerical solution of the Boltzmann equation I: spectral approximation of the collision operator, SIAM J. Num. Anal. (2000). [47] L. Pareschi, G. Russo, An introduction to Monte Carlo methods for the Boltzmann equation, ESAIM Proceedings, 10 (2001), 35-75. [48] L. Pareschi, G. Russo, G. Toscani, Fast spectral methods for the Landau-Fokker-Planck collision operator, J. Comp. Phys., 165 (2000), 216-236. [49] L. Pareschi, G. Toscani, C. Villani, Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit, Numerische Math, to appear. [50] B. Perthame, Introduction to the theory of random particle methods for Boltzmann equation, in Advances in Kinetic Theory and Computing, B. Perthame Editor, World Scientific, (1994). [51] D.I. Pullin, Direct simulation methods for compressible inviscid ideal gas-flow, J. Comp. Phys., 34 (1980), 231-244. [52] F. Rogier, J. Schneider, A direct method for solving the Boltzmann equation, Trans. Theo. Stat. Phys., 23 (1994), 313-338. [53] Y. Sone, T. Ohwada, K. Aoki, Temperature jump and Knudsen layer in a rarefied gas over a plane wall: Numerical analysis of the linearized Boltzmann equation for hard-sphere molecules, Phys. Fluids A., 1-2 (1989), 363-370. [54] G. Strang, On the construction and the comparison of difference schemes. SIAM J. Numer. Anal., 5 (1968), 506-517. [55] G. Toscani, One-dimensional kinetic models of granular flows, M2AN Math. Model. Numer. Anal., 34 (2000) 1277-1291. [56] Z. Tan, P. Varghese, The A-e method for the Boltzmann equation, J. Comp. Phys., 110 (1994), 327-340. [57] C. Villani, On a new class of weak solutions for the spatially homogeneous Boltzmann and Landau equations., Arch. Rat. Mech. Anal., 143 (1998) 273-307.

Chapter 8

Discrete Velocity Models and Dynamical Systems A.V. Bobylev and N. Bernhoff Department of Mathematics Karlstad University 651 88 Karlstad SWEDEN

8.1

Introduction

The discrete kinetic theory [1; 2] was at the early stage of its development quite separated from the "continuous" kinetic theory. Then it became clear that the Boltzmann equation can be approximated by DVMs with high accuracy and that such approximations can be used for numerical methods. The first method of this kind was proposed in [3], the proof of its convergence was done in [4; 5]. This stimulated some new works on DVMs with arbitrarily large number of velocities, though certain results for general DVMs were obtained before ( see, in particular, [6; 7; 8] and references therein ). In the present paper we continue to study DVMs in directions formulated in [9; 10]. It is almost a trivial observation that, for a class of solutions depending on one scalar variable, the general DVM becomes a system of ODEs ( dynamical system ). This system is not very interesting in the spatially homogeneous case, however it is much less trivial in the case of plane stationary ( or shock wave structure ) problem. A relatively new point of view is perhaps to consider the Boltzmann equation ( for such problems ) as a limiting case of the dynamical system when the number of discrete velocities tends to infinity. The DVM is considered as its finite dimensional approximation which in principle can explain many "qualitative" features of the solution of the Boltzmann equation itself. In such terms one can 203

204

Lecture Notes on the Discretization

of the Boltzmann

Equation

speak about "phase space", "saddle points", "phase portrait", etc. In particular, the shock wave is understood as the phase trajectory connecting two singular ( saddle ) points of the vector field. In this paper we make only a few first steps in the study of such dynamical systems. The first problem is always to clarify the nature of singular ( stationary ) points which are Maxwellians in the case of DVMs. We show that main characteristics of such points ( dimensions of corresponding stable, unstable and neutral manifolds ) can be evaluated exactly in the most general case. These results can be applied not only to DVMs, but also to any reasonable finite dimensional approximation of kinetic equations ( e.g., moment equations ). This makes the well-known behavior of solutions of the evaporation/condensation problem for the Boltzmann equation quite transparent and explains why the number of free parameters at infinity is different for sub/supersonic evaporation and condensation. This result at the level of the linearized Boltzmann equation was actually obtained long ago [11], we just made it almost obvious for any reasonable finite dimensional approximation. We also describe in detail a behavior of solutions in the neighborhood of any stationary ( Maxwellian ) point. The paper is organized as follows. We formulate the main problems in Section 8-2 and discuss in more detail the connection between DVMs and dynamical systems in Section 8.3. The behavior of solutions near the stationary point is studied ( in the most general case ) in Sections 8.4 - 8.5. Theorem 8.1 ( Section 8.5 ) gives the exact formulas for numbers of positive and negative eigenvalues of the linearized operator. Applications to the half-space problem are discussed in Section 8.7 ( Theorem 8.2 ). Generalizations to the weakly nonlinear case (by using the Grobman-Hartman theorem and the theory of bifurcations for ODEs) are briefly discussed in Section 8.7. Some results of this paper were earlier presented without proofs in [9,10].

8.2

Statement of the Problem

The general discrete velocity model ( DVM ) reads

(Jt + Vi§i) fi {X'i} = Qi (/) = Qi (/l' -'fn),i = *' •"'"'

(8-1)

Discrete Velocity Models and Dynamical

205

Systems

where x € Rrf and t € R+ denote the position and time respectively, {v\,...,vn} C R d denote a set of discrete velocities of the model. The functions /j (x, t) are understood as spatial densities of particles having velocities Vi G R d , usually d = 2,3 in applications. The operators Qt (/) ( collision integrals ) are given by equalities n

QiU)= E

T

i(fifi-hfk),

(8.2)

j,k,l=l

such that Tfk = Yjk\ = Vft > 0. It is also assumed that Tfk = 0 unless the conservation laws Vj + vi, \vi\2 + \vk\2 = | ^ | 2 + \vi\2

Vi+vk=

(8.3)

are satisfied. DVM (8.1), (8.2) is called "normal" if any solution of equations * (Vi) + * (vk) = * (Vj) + * (vi), where indices (i,k;j,l)

(8.4)

take all possible values satisfying (8.3), is given by V(v) = a + b-v + c\vf,

(8.5)

with some constant a, c € R and b € R d . We consider below only normal DVMs ( methods of their construction are described in [12] ). In such a case the equality n

X>(OQi(A,...,/n) = 0

(8.6)

has ( on arbitrary f\, ...,fn > 0 ) the general solution (8.5), moreover n

D(f)

= E^)

l o

S-^

0

'

(8-7)

i=l

D(f)

=

0 if and only if / 4 = M{ =

e^Vi\

where * (v) is given by Eq. (8.5). We are interested in this paper in special classes of solutions of DVM (8.1) when / , (x,t) can be represented as functions of one scalar variable. Then DVM (8.1) becomes a system of nonlinear ODEs and one can use well-developed methods of the theory of dynamical systems. There are at least three interesting classes of such problems:

206

Lecture Notes on the Discretization

of the Boltzmann

Equation

(A) Spatially homogeneous problem: fi = /*(*), 7j£ = Qi ( / ) . *" = L •••»»» t

€ R

(8-8)

= l,...,n,

(8.9)

(B) Plane stationary problem: we denote x = (x\...,xd),Vi

= (v},...,vf),i

and assume that /* = /»(a: 1 ), then DVM reads

v] J^- = Q4 (/), i = 1,..., n, x1 € R.

(8.10)

(C) Shock wave problem: we assume that fi (x, t) = fi(x-

ct) = Ft (x1 - cH) ,

(8.11)

where c = (c^O, ...,0) is the velocity of the shock wave, then we denote y = x1 — c : i and obtain (vj - c1) -£

= Qi (F) ,i = l, ...,n, y e R.

(8.12)

Remark 1. If we consider the Boltzmann equation ( the "continuous limit" of DVM (8.1) as n —>• oo [5; 6] ) then the case (C) reduces to the case (B) because of Galilei invariance. The cases (B) and (C) are, however, always different for DVM with finite number of velocities. Remark 2. We assume in the following that the set of velocities V = {vi,...,vn} is chosen in such a way that vj ^ 0, i = 1, ...,n. The simplest example is the 12-velocity plane ( d = 2 ) DVM with velocities (±1,±1), (±3, ±1), ( ± 1 , ±3) with all possible combinations of signs. One can easily prove that such DVM is normal by using method of the paper [12]. We also assume that c 1 ^ vj, i = l,...,n, in the case (C). These assumptions are not necessary, they just simplify some proofs. One more simplifying assumption is that we consider only such sets of velocities V that are symmetric with respect to a^-axis: if Vi = (v},vl

...,vf) 6 V then (v},±vl....

±vf) £ V,

(8.13)

for any combinations of signs. The equations (8.8), (8.10), (8.12) admit in such case a class of solutions satisfying fi = fv if vj = v], and |t>i|2 - K I2 ,

(8-14)

Discrete Velocity Models and Dynamical

Systems

207

and only this class of "flows" symmetric with respect to a^-axis is considered below. This reduces the number n of equations (8.8), (8.10), (8.12) to the number n < n of different combinations (t>j,K| ) in the velocity set, however the structure of the collision term (8.2) ( in slightly different notations ) remains unchanged. We assume below ( without changing the notation n, etc. ) that such a reduction is already made. The symmetry assumption is not very important ( our main results do not depend on it ), but it simplifies some considerations and makes the results more transparent. The above cases (A), (B) and (C) can now be considered in the unified form if we denote independent variables ( t, x1 and y in cases (A), (B) and (C) respectively ) by t: D^-=Q{f),feRl,QeKn,teIi,D at

= diag{dud2,...,dn),

(8.15)

where (A) dk = 1; (B) di = v\; (C) dt = v] - c\; i = 1, ...,n.

(8.16)

We have now three ( and only three ) conservation laws defined by identities ( for arbitrary / € R™ ) (Q,l) = (Q,u) = (Q,w) = 0,

(8.17)

where brackets denote the usual scalar product in R n , u = (ui,...,un), w = (wi,...,wn) are chosen in such a way that p = (f,u) and E = (/,«;) are given in the initial notation (8.1) by equalities n

n

p=^2fiv},E = Y,fi\vi\2, i=l

(8-18)

i=l

( note that the number n in Eqs. (8.1) and (8.15) is not the same if d > 2, the difference depends on concrete DVM ). Some general properties of equations (8.15) are studied in the next section. 8.3

D V M as Dynamical System

All solutions of the system (8.15) having physical meaning can be represented by phase trajectories / = f(t) in the physical domain R " ( /* > 0,

208

Lecture Notes on the Discretization of the Boltzmann Equation

i = l,...,n) of the whole phase space R n . The conservation laws (8.17) define the invariant subspace parametrized by three numbers {31,32,33) such that h = {Df, 1), n = (Df, u), j 3 = {Df, w),

(8.19)

to which each phase trajectory / (£) belongs for all t 6 R . The next step is to find singular ( stationary ) points of the system given by the equality g(/) = 0,/eR£.

(8.20)

Its general solution is given by f = M = (Mi, ...,M„), Mi = AeaUi~^Wi,

(8.21)

where the three parameters (A,a,(3) depend of parameters (ji,J2,J3) of the corresponding invariant subspace such that Eqs. (8.19) hold. For brevity we do not consider in detail the transcendental equations (8.19) with / = M for (A,a,(3) as functions of {31,32,33)- We just note that DVMs with large number of velocities can approximate the Boltzmann equation ( on regular lattices Zd ). In such cases the dependence of (A, a, /?) on {31,32,33) is practically the same as for the Boltzmann equation. The most important qualitative feature of this dependence is the following: (A) If D = I then the equations (8.19) have a unique positive solution M (ji,J2,J3) or no positive solutions at all. (B — C) If D is defined as in cases (B) and (C), then, for some values of {ji,32,jz), t n e equations (8.19) have two positive solution: sub- and supersonic Maxwellians M\t2(ji,J2,J3)We assume that all DVMs considered below also have such properties. Even this simple observation yields a clear image of the shock wave solution as a phase trajectory connecting two singular points in the case (C). The famous Mott-Smith approximation [13] is simply its approximation by the straight line. We note that the similar solution in the stationary case (B) can also be considered as a shock profile, however for the model with "shifted" velocities v\ = Vi+c. Thus it is easy to find (under above assumptions) all stationary points in the physical domain of the phase space of the dynamical system (8.15). It is more difficult, however, to define all topological characteristics of the points (dimensions of corresponding neutral, stable and unstable manifolds in the neighborhood of each point). Surprisingly, this problem can also be

Discrete Velocity Models and Dynamical

209

Systems

solved exactly in the most general case. The solution is given in Sections 8.4 - 8.5. In order to study a neighborhood of a given Maxwellian M we denote / = M + VMF,

(8.22)

where \[M is defined in an obvious way. Then the equations (8.15) are transformed to dF D^- = LF + S(F), at

(8.23)

where the linearized operator L ( n x n matrix ) is given by equalities n

j,k,l=l

i

= l,...,n.

(8.24)

The nonlinear term reads: n

Si(F)= Y, T*y/Jh(FjF,-FiFk),i = l,...,n.

(8.25)

j,k,l=l

By using the symmetry relations Y\k = T^ki = T^k > 0 one can easily verify that (1) (Lf, g) = (/, Lg); (2) (Lf, f) < 0.

(8.26)

Hence the matrix L is symmetric and semi-negative. The diagonal matrix D (8.15) ( under our assumptions ) has no zero elements on the diagonal. If we assume that F\t=0 = Fo then the formal solution of Eq. (8.23) reads t tD lL

F(t) = e ~ F0 + f e^-T^D~lL o

[S (F)} (r) dr.

(8.27)

Moreover the nonlinear term is not very important for the topological classification of the stationary point F = 0 of the system (8.23). Therefore we consider in the next section the general linear problem related to this equation.

210

8.4

Lecture Notes on the Discretization

of the Boltzmann

Equation

Linearized Equation

Let A and B be real symmetric n x n matrices, such that (x,Ax) > 0 , d e t B ^ O ,

(8.28)

where x 6 R " is an arbitrary vector. The linearized kinetic equation and its formal solution read Bxt + Ax = 0 =• x (t) = e~tB~lAx

(0), t € R.

(8.29)

We study below the general case of any symmetric A and B satisfying (8.28). In the particular case of DVM, the matrices B = D, A = —L are defined in Eqs. (8.15), (8.16), (8.24). First we study the eigenvalue problem Ax = XBx or equivalently B~1Ax = Xx,

(8.30)

and denote N(A) Im(A)

= =

{xeRn\Ax = 0}, n

n

{xeB. \3y€R :x

(8.31) =

Ay}.

We assume that dim N (A) = p > 1 and introduce an orthogonal basis in N (A) such that N{A) = Span(y1,...,yp),

{yhyj)

= % i, j = 1, ...,p.

(8.32)

The following construction will be used several times below. Construction: We denote v = (vi,...,vk),

u = (ui,...,u fc ),

(8.33)

where Vi,Uj € R n are arbitrary vectors, k y ) M ) ^ = {xt, Myj),

(8.34)

and consider the change of variables x' = t/ix, y' = U2x,

(8.35)

Discrete Velocity Models and Dynamical

Systems

211

where x, x ' , y , y ' are formally understood as fc-dimensional vectors (8.33). Then it is easy to verify the following formulas: (x'®y)M

=

C/i(x®y)M, (x®y')M = (x®y)Mt/2T,

( x ' ® y ' )M M

=

^ itfi<x®y>M' (x®y)Mf/2T-

(8.36)

The first application of this construction is that we can always choose the basis (8.33) in such a way that (yi,Vj) = kj, (Vi,Byj) = uSij, i,j = 1, ...,p,

(8.37)

by using the transformation (8.35) with k = p, x = y, U\ = Vi = U, where U is an orthogonal ( UT = U~l ) matrix which diagonalizes the matrix ( y ® y ) e - We assume below that the conditions (8.37) are fulfilled and denote ( for any vectors x, y 6 R " ) (x, y)B = (z, By),

(x, y)A = (x, Ay).

(8.38)

We can always choose the numeration such that H = (yi,yi)B

7^0 Hi = l,...,k; 7i =0tfi

and denote z\ = yk+\, —,zi=yp,l=p the following form:

= k+l,...,p,

(8.39)

— k. Finally the basis in N(A) has

N(A) = Span(yi,...,yk,z1,...,zi),

k + I = p,

(8.40)

where (yi,Vj)

=

Sij, (zr,zs)=Srs,

(yi,Zj)=0;

(8.41)

{yuVi)B = 7i%, 7. ¥" 0; (yi,Zj)B = (zr,zs)B = o, i,j = l,..k,r,s

= l,...l.

We distinguish below (a) the non-degenerate case I = 0 and (b) the degenerate case / > 1. In the second case we shall also need generalized eigenvectors for A = 0 satisfying the equation {B^A)2

x = 0, or equivalents A S - 1 Ax = 0.

(8.42)

We denote therefore V{B~1A)

= {z€Rn

(B~1A)2x

= o},

(8.43)

212

Lecture Notes on the Discretization

of the Boltzmann

Equation

assuming that A and B satisfy conditions (8.28) and that N{A) is given by Eqs. (8.40), (8.41) with fixed p > 1 and 0 < I < p. Lemma 8.1

If 1 = 0 then V {B~lA)

V(B~1A)

= N(A).

Ifl>l

then

= Span(y1,...,yk,zi,...,zi,vi,...,vi), x

Vi = A~ Bzi,

(8.44)

i = l,...,l,

where the operator A"1 : Im (A) —>• Im (A) is uniquely defined in the usual way. Proof. It is clear that N(A) C V (B^A). If the equation (8.42) has a solution x £ N(A) then z = B~lAx e N(A) => Bz £ Im (A), z^Q. On the other hand Im (A) = {x G R n | (x, y)=0\/y£

N{A)} ,

(8.45)

and therefore (z, y)B = 0 for any y £ N(A). Such z ^ 0 obviously exists if and only if / > 1 in Eq. (8.40). Then the vector v^A^BzelmiA)

(8.46)

is uniquely defined and satisfies Eq. (8.42). The general solution of a; € Im (A) of Eq. (8.42) is clearly a linear combination of vi,...,vi defined in Eq. (8.44). It remains to verify that (vi,...,vi) are linearly independent ( their independence with vectors from N(A) is obvious ). The I x / matrix with elements (vi,vj) =

(A^BZ^A^BZJ)

= (A-2Bzi,Bzj)

(8.47)

is non-singular since {A~2u, u) > 0 if u elm (A),

(8.48)

and this completes the proof of Lemma 8.1.

•

a

a

Remark. If we replace A by A , a > 0, and note that N (A) = N (A ), then it is clear that dim (V [B-lA))

= dim (V (B^A01))

=p + l

(8.49)

in notations (8.40) - (8.43). This property will be used below ( with a = 1/2 ) in order to evaluate a number of non-zero eigenvalues of the problem (8.30).

Discrete

Velocity Models and Dynamical

213

Systems

Lemma 8.2 The problem (8.30) with A and B satisfying conditions (8.28), (8.40), (8.41) has d — n—p — l non-zero eigenvalues ( with account of multiplicity ) {Ai,...,A<j}. The corresponding eigenvectors {ui,...,Ud} can be chosen in such a way that (ui,Uj)B Proof.

= XiSij, Xt ^ 0, i,j - l,...,d.

The substitution y = All2x Axl2B~xAxl2y

(8.50)

transforms the equation (8.30) to = Xy,y£

Rn,

(8.51)

with symmetric matrix AXI2B~XAXI2. The problem (8.51) has therefore n real eigenvectors {y\ ,...,yn} with corresponding real eigenvalues {Ai,..., An} which are relabeled in such a way that Ai ^ 0,..., A<j ^ 0 and \d+i = ••• = Xn = 0 ( 1 < d < n is still unknown ). The eigenvectors {yi,---,yn} form an orthogonal basis in R n : (yi,Vj) = $ij, i,3 = l , - , n .

(8.52)

The number of zero eigenvalues is equal to dim (V(B~1A1^2)) in the notation (8.43). Hence accordingly to Eq. (8.49), the problem (8.51) has p + I eigenvalues A = 0 (we mean, of course, the dimension of N{A1/2B~XA1/2)). Therefore d = n — p — I. If y is a solution of Eq. (8.51) then u = B~1Axl2y is a solution of Eq. (8.30) with the same A since Bu = A^2y, Au = A^iA^B^A^y

= XAl'2y.

(8.53)

The vectors m = B-xAx>2yu

i = 1,..., d

(8.54)

are therefore eigenvectors of B~XA that correspond to eigenvalues X\,..., Ad ^ 0. The orthogonality conditions (8.52) lead to (ui,Uj)B Xi

=

{Axl2yi,B-xAxl2yj)

^

0,i,j = l,...,d.

= {yi,Axl2B-xAxl2yj)

= XiSii, (8.55)

Hence, the vectors (8.54) are linearly independent ( t h e equation aiui + ... + ctdUd = 0 has only the trivial solution ct\ = ... = eta = 0, as it follows from Eqs. (8.55) ). It is easy to see that no other solution (A ^ 0, u) of Eq. (8.30) is possible since it would lead to a new solution y — Axl2u ^ 0 of Eq. (8.51). The proof is complete. •

214

Lecture Notes on the Discretization

of the Boltzmann

Equation

We have constructed in Lemmas 8.1, 8.2 a set of n vectors ( see Eqs. (8.44), (8.54) ) {yx, -,yk,zi,-,zi,vi,

...,vi,ui,...,ud}

, k + 2/ + d = n,

(8.56)

and intend to prove The set (8.56) forms a basis in R n .

Lemma 8.3

Proof. We need just to verify that vectors (8.56) are linearly independent. Let numbers ai,...,ak, bi,...,bi, ai,...,cu, /?i, ...,Pd be such that k

l

d

x = ] P a,iyi + ^2 (bjZj + ctjVj) + ^2 0rur = 0. i=l

,7=1

We note that Bur € Im (A) and therefore {ur,y)B r = l,...,d. On the other hand,

(VJ,UT)B

= 0 for any y £ N(A),

= — (vj,Aur)

= — (Avj,ur)

A/*

— (Bzj,ur)

(8.57)

r=l

=

Af

= 0, j = l,...,l, r = l,...,d, since Zj £ N(A).

Hence

Ay

/3r = (x, ur)B =0,r = l,...,d. The linear independence of the remaining system ( the basis (8.44) in V(B^1A) ) was already proved in Lemma 8.1. • One can easily see that the matrix has in the basis (8.56) the standard Jordan form since B~xAyi X

B~ AZJ

— 0, i = l,...,fc; B~lAur -

1

0, B~ AVJ

= Xrur,r

= 1, ...,d;

(8.58)

- Zj, j = 1,...,/.

These formulas allow to construct the matrix exponential (8.29) in the form k

l

Q

d

b

x(t) = J2 i W Vi + ^2 ( 3 (*) 3 + i (*) i) + Yl Pr (*) Ur' i=l

Z

a

v

j=l

(8-59)

r=l

where = l,...,k;pr(t)=/3T(0)e-x"t,

Oi(t)

=

ai(0),i

aj(t)

=

aj(0),bj(t)=bj(0)-taj(0),j

r = l,...,d; = l,...,l.

(8.60)

The representation (8.59), (8.60) is however not sufficient in cases when t € R plays a role of spatial variable ( cases (B) and (C) from Section 8.2 ),

Discrete

Velocity Models and Dynamical

Systems

215

we need to know the exact number of positive and negative eigenvalues. This problem is solved in the next section.

8.5

Indices of Stationary Points

It is convenient to transform the basis (8.56) to the form orthogonal with respect to the product ( , ) B . It was already proved in Lemma 8.3 that {ur, x)B = 0 for any x £ V(B~1A) and any r = 1,..., d. Therefore we need to consider just the basis (8.44) of V(B~1A). Changing vi,..., vi to k w

i

=V

J -"^2 — (vi>yi)B ' 7t = (yi,Vi)B » J = !> •••)'.

( 8 - 61 )

we exclude also {j/i, •••,yk} since (Vi,Zj)B

=

(yi'wj)B

= 0, i = l,...,k,j

= 1,...,/.

Hence, it remains to orthogonalize ( with respect to ( , ) 21 vectors {z1,...,zhw1,...,wi}

(8.62)

B

) the system of

, AWJ = BZJ, j = l,...,l.

(8.63)

In order to do this we denote ( see Construction in the beginning of Section 8.4) z = (zi,...,zi),

w = (wi,...,wi),

(8.64)

= 0 , (zi,Wj)B

= (wi,Wj)A,

(8.65)

and note that (zi,Zj)B

i,j = 1,...,/.

We consider two symmetric I x / matrices ( in the notation (8.34) ) A = (z® w)B = (w w)A , B = (w (8) w)B ,

(8.66)

and note that the matrix A is positive since (Bzi,Wj) = {Bzi,Vj) =

{BZUA~XBZJ)

, Bzt e Im(A),

i,j = l,...,l. (8.67) Therefore there exist (1) the inverse matrix A~x and (2) the orthogonal matrix U such that the matrix S — UBUT is diagonal. The transformation

216

Lecture Notes on the Discretization

of the Boltzmann

Equation

of the basis (8.65) z' = A-1Uz,

w' = t/w, UT = U~1,

(8.68)

leads to equalities ( see Eqs. (8.35), (8.36) ) (z' ® w %

=

(w'®w')B

=

UA.-1 (z ® w ) B UT = T, (z' ® z % = 0, T

U{w®w)BU

= S = diag{si,.

(8.69)

..,s(},

where I is the unit I x Z matrix. Hence we obtain the basis {z[,..., z[, w[,..., w\} such that (z'i'z'j)B

=

°'

= Si

(Z^W'J)B

i>

(W'^W'J)B

= Si5i

*>•? = i . - ^ -

i>

( 8 - 70 )

All above transformations were obviously invertible, therefore vectors {z[, ...jzljw'i, ...,w[} remain linearly independent. The final step is to denote w

t = WJ ~

S

-^Tz'v

W

J = w'i ~ ^

4

3 = 1, - , I,

(8-71)

then (wt,wf)B

= sij> (wliwj)B

= -&ij> (wt>wj)B

= °> iJ = !> •••»'• (8.72)

Thus, the following statement is proved. Lemma 8.4

The basis (8.56) in R™ can be transformed to the form {vi, -,yk,wt>

—>w*>wi ,...,wf,ui,...,ud)

,

(8.73)

k + I = p, p + I + d = n, such that all vectors (8.73) are pairwise orthogonal with respect to the product ( , ) B and {Vi,yi)B

w w

( t> t)B

=

H^O,

(ur,ur)B=

l

W

W

Xr^0,

(8.74)

1

= > ( 7> 7)B = - >

where i = l, ...,k, j = 1,...,/, r — 1, ...,d. In applied problems the number p is relatively small compared to n ( note that formally n = oo for the Boltzmann equation whereas p < 5 ). Therefore numbers k, I, 71, ...,7*, ( k + I = p ) and the subspace N(A) of invariants are usually known in ( almost for DVMs ) explicit form. On the

Discrete Velocity Models and Dynamical

217

Systems

other hand, the matrix B is usually diagonal ( see Eqs. (8.15), (8.16) ) and therefore all its eigenvalues are also known. This explains an importance of the following statement. Theorem 8.1 / / (1) the nxn matrix B has n+ positive and n~~ negative ( n+ + n" = n ) eigenvalues and (2) the p x p matrix B, with entries bij = (xi,Bxj) such that N (A) = Span (xi, ...,xp), has k+ positive, k~ negative and I zero eigenvalues ( k+ + k" + I = p ) , then the numbers m± of positive and negative eigenvalues of the nxn matrix B~l A are given by equalities m+ =n+ -k+ - I, m~ =n~ -k~

- I.

(8.75)

Proof. The proof is based on the well-known fact ( Sylvester's theorem ) that the number of positive and negative coefficients of the non - degenerated quadratic form (x,x)B does not depend on basis. If are eigenvalues of B and {ei,...,e„} corresponding eigenvectors such that (e,,ej) = Sij, i,j = l,...,n, then n"*"

{x,x)B

are

n~

= ] T & + c ? + X!6n++Acn++fc' ci i=l

=

{x,ej),j

= l,...,n.

(8.76)

k=\

On the other hand, we can represent x by using the basis (8.73) k

X

l

= 5Z aiVi i=l

+

Yl ^Wt

d

+ VJWj) + Yl PrUr,

j=l

(8.77)

r=l

then

(X,X)B = £><»? + E ((W")2 - (^7)2) + E X^ri=l

j=l

(8-78)

r=l

We note that 71, ...,7* are non-zero eigenvalues of the matrix B defined in Theorem 8.1, k+ of them are positive and k~ = k — k+ are negative. Then the comparison of Eqs. (8.76) and (8.78) leads to equalities (8.75). The theorem is proved. • We introduce Definition 8.1

The number A = (m + -m~)

- (n+ -n~)

,

(8.79)

218

Lecture Notes on the Discretization

of the Boltzmann

Equation

where rri* and n ± are numbers of positive and negative eigenvalues of matrices B~1A and B respectively, is said to be an index of the saddle point x = 0 of the equation (8.29). The numbers n+, n~ ( n+ + n~ = n) are usually known in advance, the numbers k+, k~, I, (k++k~~+l=p) can be easily found by considering eigenvalue problem for symmetric p x p matrix B. Then the knowledge of A yields values of m+ and m" since m + + m~ = n—p — l. In terms of the index A Theorem 8.1 simply states that A = AT - k+.

(8.80)

It is remarkable that A remains finite ( — p < A < p ) if n —>oo. In applications to DVMs the matrix B is fixed whereas the matrix A = — L (8.24) depends on the concrete Maxwellian M. The analysis of the behavior of A in the "continuous limit" ( Boltzmann equation ) was actually done ( in different terms ) in the paper [11] and therefore we do not repeat it here ( see also [9] ).

8.6

Applications to Half-Space Problems

If t € R is a spatial variable ( cases (B) and (C) from Section 8.2 ) then we are interested in solutions which are bounded ( or at least have no exponential growth ) if t —> oo. The case t -> — oo is not considered here since Eq. (8.29) is invariant under the transformation t' — —t, B' = —B. We can clarify the structure of such solutions provided numbers n*, A^ and I are known. Two possible cases are described below on the basis of the general solution (8.59), (8.60) and formulas (8.75). (A) Bounded at infinity solutions k

x(t)

=

^Oiyi

I

+ ^bjZj

i~l

j=l

m+

+ ^Prute-***,

(8.81)

r=l

k

=

k+ + k~, m+ =n+ -k+

r

=

l,...,m+,

-I, A+ = ( u + , u + ) B > 0,

form a (n + + fc~)-parameter subspace of R n .

Discrete Velocity Models and Dynamical

Systems

219

(B) Slowly increasing at infinity solutions k

i

i

x f

() = EaiVi + E h z i + E i=l

j—l

m+ a

i (UJ ~ tzo) + E P r u t e ~ K \

j=l

r=l

(8.82) form a (n + + k~ + I) - parameter subspace of R™. The half-space problem on [0, oo) usually means that we prescribe n + boundary conditions at t = 0 and certain number rioo of conditions at infinity. The formal analysis ( just on the number of free parameters ) shows that in the above cases we obtain: (A) noo = AT; (B) n^ = k~ +1.

(8.83)

It is clear that the case (B) is typical for boundary layer problems when the Mach number M ^ ( for the Boltzmann equation ) equals zero. The case (A) is typical for the evaporation-condensation problem, moreover I = 0 if Mx> 7^ 0,1 (for the Boltzmann equation ). For brevity we consider in more detail just the non-degenerate (1 = 0) case (A) and show the existence of solutions of the linearized half-space problem. If {ei,...,e n + ,wi,...,wn-} is the basis of normalized eigenvectors of B such that (ei,ej)

=

(wi,Wj) =6ij,

(ei,ej)B

=

bfSij,

bf,...,b++

>

(ei,Wj) =0,

(wi,wj)B=b^Sij,

(8.84) (ei,wj)B

= 0,

0,bi,...,b~_<0,

then the boundary conditions at t = 0 are given by equalities xf = (x(0),ei),i

= l,...,n+.

(8.85)

The solution x (t) (8.81) at t = 0 can be written in the case I = 0 as Jfc+

k~

a

m+

( 8 - 86 )

x(0) = J2 4vt + E 7v7 + E # V • i=l

i=l

r=l

We note that zoo = lim x(t) = J2 afvt '^°°

i=\

+E i=i

a

7v7

( 8 - 87 )

220

Lecture Notes on the Discretization

and prescribe k

of the Boltzmann

Equation

parameters a~ =

(x(t),y-)n . \ %

,Vj )

B

= canst.,j

= l,...,k~,

(8.88)

B

which are uniquely denned by Zoo. Then we obtain a linear system k~

X

a

k+

e

m+

a

t-Y. 7(y7> °)

= T, t &,<>•)+Y,ft{<>&.), (8.89)

j=l

i—1

s

=

r—1

l , . . . , n + = m + + k+,

for unknown af, ...,a~£+,Pf,...,0^+. This system has a unique solution. Otherwise there exists a non-trivial solution a^~, ...,a£+, /3+,...,/3++ of the homogeneous system and therefore there exists a vector x ^ 0 such that k+

m+

n~

* = E «tyi + E ft< = E w • i=l

r=l

(8-9°)

r=l

This leads, however, to contradiction since the equality k+

m+

2

{x,x)B = Y,(yi>yt)B&) i=l

„

n~

A

+ E +(#") = E 6 7 s 2 < ° r=l

(8-91)

r=l

is possible only for x = 0. This proves the following Theorem 8.2 The boundary value problem (8.85), (8.88) for the equation (8.29) has in the non-degenerated case I = 0 a unique solution. This result is similar to the results of the paper [11] obtained for the linearized Boltzmann equation. We note that no assumptions except two general conditions (8.28) and non-degeneracy (1=0) were made. The degenerate case can be considered in similar way.

8.7

Concluding Remarks

The results of Sections 8.4-8.6 can be applied to any dynamical system of the form (IT

B— = -Ax + ...,x£lln, at

(8.92)

Discrete Velocity Models and Dynamical

Systems

221

where A and B are symmetric matrices satisfying conditions (8.28), dots standing for nonlinear terms ( quadratic in the case of DVMs of the Boltzmann equation ). It was shown that all important characteristics of the saddle ( in the general case ) point x = 0 can be studied in detail and that some of them do not depend of the dimension n of the system and give right answers ( for example, the number of conditions at infinity in the half-space problem ) even in the case of the Boltzmann equation which formally corresponds to n = oo. In this paper we actually restrict our considerations to the linearized equation. There are, however, some well-known facts from the general theory of dynamical systems [14] which in principle allow us to generalize the results to the weakly nonlinear case. On one hand, it is the Grobman - Hartman theorem on the structural stability of the saddle point which states that the nonlinear system (8.92) is, in the neighborhood of x = 0, topologically equivalent to its linear part. It is probably possible to construct the solution of the weakly nonlinear half-space problem on the basis of this theorem. On the other hand, the problem of the weak shock wave ( M = 1 + e, M denotes the Mach number ) structure is the well-known ( in the theory of dynamical systems ) problem of the bifurcation of the saddle point. In such a case, a pair of stationary points is born from one initial point and there is one ( and only one ) phase trajectory connecting these two points [14, p.243]. These remarks cannot replace proofs, but they explain in some sense why certain successful results were obtained before ( see [7; 8; 15] ) for weakly nonlinear problems, though methods of these papers were not based on the general theory of dynamical systems. Acknowledgement. The research was supported by NFR ( Sweden ) grant 20005092.

8.8

References

[1] R. Gatignol, Theorie cinetique des gaz a repartition discrete de vitesses, Lecture Notes in Physics 36, Springer (1975). [2] H. Cabannes, The discrete Boltzmann equation, Lecture Notes, Berkeley (1980). [3] D. Goldstein, B. Sturtevant, J.E. Broadwell, Investigations of the mo-

222

[4]

[5]

[6]

[7]

[8] [9]

[10] [11] [12] [13] [14] [15]

Lecture Notes on the Discretization

of the Boltzmann

Equation

tion of discrete - velocity gases, Rarefied Gas Dynamics: Theoretical and Computational Techniques, AIAA (1989), 110-117. A.V. Bobylev, A. Palczewski, J. Schneider, On approximation of the Boltzmann equation by discrete velocity models, C.R. Acad.Sci. Paris, 320 (1995), 639-644. A. Palczewski, J. Schneider, A.V. Bobylev, A consistency for a discrete velocity model of the Boltzmann equation, SIAM J. Numer. Anal., 34 (1997), 1865-1883. C. Cercignani, R. Illner, M. Pulvirenti, M. Shinbrot, On nonlinear stationary half- space problem in discrete kinetic theory, J. Stat. Phys., 52 (1988), 885-896. S. Ukai, On the half - space problem for the discrete velocity model of the Boltzmann equation, Advances in Nonlinear P D E s and Stochastics, Ser. on Advances in Math. Appl. Sci. 48, World Scientific (1998), 160-174. C. Bose, R. Illner, S. Ukai, On shock - wave solutions for DVMs of the Boltzmann equation, Transp. Theory Stat. Phys., 27(1998), 35-66. A.V. Bobylev, Relationships between Discrete and Continuous Kinetic theories, Rarefied Gas Dynamics Vol 1, Cepadues - Editions (1999), 1-19. A.V. Bobylev, Half - space problem for finite - dimensional approximation of kinetic equations, Oberwolfach Conference (April 2001). F. Coron, F. Golse, C. Sulem, A Classification of Well - Posed Kinetic Layer Problems, Comm. Pure and Appl. Math., XLI (1988), 409-435. A.V. Bobylev, C. Cercignani, Discrete velocity kinetic models without non - physical invariants, J. Stat. Phys., 97 (1999), 677-686. C. Cercignani, Theory and Application of the Boltzmann equation, Scottish Academic Press (1975). V.I. Arnold, Additional topics in the theory of ODEs, Nauka (1978), (in Russian). R.E. Caflisch, B. Nicolenko, Shock profile solutions of the Boltzmann equation, Comm. Math. Phys., 86 (1982), 161-194.

Chapter 9

Numerical Method for the Compton Scattering Operator C. Buet 1 , S. Cordier2 1

9.1

Commissariat a I'Energie Atomique, 91680 Bruyeres-le-Chdtel 2 Laboratoire MAPMO, UMR 6628, Universite Orleans, 45067 Orleans, France

Introduction

We are interested in the dynamics of a low energy, homogeneous and isotropic photon gas that interacts via Compton scattering with a low energy electron gas in thermodynamical equilibrium. In this paper, we first review the main properties of the Quantum Boltzmann equation (QBE) and its "grazing" collision limit that is called the Kompaneets equation (which is a Fokker-Planck like equation). We will recall recent result of Escobedo and Mischler [EM, EM2] about the equilibrium state of such equation that behaves under some conditions as a "concentration near the origin" (section 9.2). In section 9.3.1, we present a numerical scheme for the QBE that preserves the properties of the equation described above. Then, we present a numerical scheme for the Kompaneets equations that has the same properties i.e. are compatible with all the properties of the continuous equation. This allow us to observe the "concentration" phenomena. We present two ways for deriving this scheme. The first way is based on the asymptotic that transform the QBE into the Kompaneets equation at the discrete level. The second one is based on the same ideas as the Chang-Cooper paper [7]. These two methods lead surprisingly to the same scheme up to a multiplicative constant (that goes to 1 when the 223

224

Lecture Notes on the Discretization

of the Boltzmann

Equation

mesh is refined) and this fact allows us to use either method usually devoted to Boltzmann like equation (e.g. entropy decay) or to Fokker-Planck type equation (Maximum principle). We shall also deal with the difficulty due to the mass concentration phenomena. For this, we shall defined generalized Bose-Einstein distribution function (with negative value of v) that converge (when the mesh is refined) toward the Planck distribution plus a Dirac measure at origin. We then illustrate the schemes on the following result by Escobedo et al. [13] : even if the initial density is smaller than the Planck one, the solution of the Kompaneets equation (with the boundary conditions) is not always global in time (on the contrary, the solutions of the QBE are well defined for any time) provided that the initial density (arbitrary small) is close enough to the origin. This is due to a balance between the "Burgers" term (with a negative velocity that push the particle toward the origin and the diffusion term that spreads the solution. We shall observe this "blow-up" in finite time. This paper is related to previous works of the authors on the Boltzmann or Fokker-Planck-Landau operator [4; 5] . In these papers, numerical discretizations are designed in order to be compatible with relevant physical properties such as conservation laws and entropy decay using Discrete Velocities Method. These methods allow to insure the correct large time behaviour i.e. the trend to thermodynamical equilibrium. The numerical efficiency of the proposed algorithms have been tested also for the non homogeneous case (where the distribution function depend on the space variable too) in a paper by the authors and F. Filbet [3] .

9.2

Quantum Boltzmann and Kompaneets Equation

In this section, we briefly present the equation we are interested in, namely the Quantum Boltzmann equation and its "grazing" limit the so-called Kompaneets equation. We refer to [14] for the original paper on these equations and to [11; 12] for a more recent and mathematical presentation.

9.2.1

The Quantum

Boltzmann

equation

We consider an isotropic and homogeneous photon gas scattered by cold electrons at thermodynamical equilibrium. The distribution function of the

Numerical

Method for the Compton Scattering

225

Operator

photons / depends on time and on the energy variable k : f(k, t) represents the number of photons that have energy k > 0 at time t > 0. This function / obeys the following scaled quantum Boltzmann equation (QBE)

e = 6(fc fc)(/,(i+/}

exp( fc) /(i+/,}exp( fc),dfc (92}i

% r ''

~~

~ ' --

where we omit the variable k and t for simplicity and / ' as usual denotes f(k',t). The so-called cross section b - positive and symmetric- is related to the probability for a given particle at energy level k to be scattered to the energy level k'. The exponential terms represents the distribution function of electrons. Defining g = k2f, equation (9.2.1) can be rewritten as ^

= J°° &(*', k){g'(k2 + g) exp(-fc) - g(k'2 + g') exp(-k'))dk'.

(9.2.2)

Let us define the function h of two variables as H9,k)d=f

T^—(9-2.3) kl +g In the reminder, we shall also note h = h(g,k) — h(g(k),k). Using this notation, the QBE equation (9.2.2) reads in the more symmetric form ^

= JCOb(k',k)ekek'(k2+g)(k'2+g')(h(g',k')ek'-h(g,k)ek)dk'.

(9.2.4)

In the reminder of the paper, we write QB for the Quantum Boltzmann operator defined above as the right hand side in (9.2.4). Multiplying (9.2.4) by a test function *(fc), integrating over k and replacing k by k', we have the weak symmetrized formulation of the QB operator

^{k)dk

= -J

J

(*(k)-*(k'))b(k',k)e-ke-k'

(fc2 + g)(k'2 + g')(h(g', k')ek' - h(g,

(9.2.5)

k)ek)dk'dk.

Using this weak formulation, one obtains easily the unique law of conservation for the QBE, that is the conservation of the total density Nd-f

gdk,

JtN(g)

= 0.

(9.2.6)

226

Lecture Notes on the Discretization

of the Boltzmann

Equation

The second main property of the QBE is the decay of entropy. We first define the function s(x, k) of two variables as s(x, k) d= zlog(a;) + k2 \og(k2) + kx - (k2 + x) log(fc2 + x), and the functional entropy, H as H(g)d=f

[°° s(g(k),k)dk. Jo

(9.2.7)

Then, H decays in time

±H<0. dt ~ This property, so-called H-theorem can be easily checked, at least formally, on the weak formulation of the QBE. Indeed, we have

Hence, the result is obtained by choosing

*(*) = log(h(g,k)eMk)) = log (j^f)

>

in the weak form (9.3.5) using that b > 0.

9.2.2

The Kompaneets

equation

When the cross section b concentrates on small modification of the energy (this asymptotic can be related to the grazing collision limit of the classical Boltzmann equation, see [9]), one obtains a Fokker-Planck type equation that has first been derived by Kompaneets in [14] . This equation shares the same properties (total density, N, conservation and entropy decay) with the QBE. Let us refer to [12] for recent mathematical work on this equation. It is proved that solution of the QBE tends to the solution of the Kompaneets equation which we shall now describe. In this "grazing" collision limit, which is detailed in the above reference and presented at a discrete level in section 3.2.1, the QBE becomes dtf(k,t)

= k-2dk(k4(f

+ f2 + dkf)),

(9.2.8)

Numerical Method for the Compton Scattering

227

Operator

i.e. the sum of a convective term, a Burgers like nonlinear term and a diffusive term with weights k2 and k4. This so-called Kompaneets equation (denoted by K equation in this article) can be equivalently written as dtf(k,t)

= k-2dk (fc*(i+ / ) * ( _ £ _ + 0

f t r

^)) .

(9.2.9)

A third expression of the Kompaneets equation reads dtg = dk ( e x p - ( * + n)(g + k2fdk^k)k^j

,

(9.2.10)

where we use g = k2f, h defined by (9.2.3) and the so-called Bose-Einstein state (see section 9.2.3)

for fi > 0. This is easily checked using the identity h(Bli(k),k) exp— (k + fi).

=

The properties of this equation can be verified on its weak formulation. Multiplying (9.2.10) by a test function \P(fc) and integrating over k, we have the weak formulation of the Kompaneets equation

J™ ^(k)dk

= y°° &g(fc)exp-(k+»)(g+k2)2dk ^k)k)dk.

(9.2.12)

The H functional entropy (resp. N total density) denned by (9.2.7) (resp. by (9.2.6)) is decaying (resp. constant) as it can be checked using $ = log(h(g, k) exp(fc)) (resp. \P = 1) in the above weak form. As usual (see [3; 4; 5]), the formulations are (formally) equivalent at the continuous level. This is no more the case once discretized : the properties of the discretized equation will depend on the discretized form.

9.2.3

Entropy

and equilibrium

states

Let us now turn to the equilibrium states or stationary solutions of QB or K equations. Such functions should minimize the entropy H for a fixed total density N. The function s being concave with respect to its first variable, it is easy to find the minimum as zero of its derivative (with respect to a;)

228

Lecture Notes on the Discretization

of the Boltzmann

Equation

for any fixed k. This gives the so-called Bose-Einstein distribution Bll(k), denned by (9.2.11), with /i. The coefficient fj, has to be positive in order that the density is finite i.e. the function B^ is integrable. The limit case fi = 0 is related to the Planck distribution. This parameter fi is a function of the density - we denote by N^ = N(Bli)- and this function is decreasing since the Bose-Einstein states are ordered : if fj, < v then £?M > Bv. The application fi i-> iVM maps [0, oo) into ]0, N(Bo)]Thus, there is a critical or maximal density No = N(Bo) that corresponds to the Planck distribution v = 0. For any initial density with a density greater than No, there is no classical equilibrium state. Caflisch and Levermore prove in [6] that, in this case, the minimum of entropy is realized by the Planck distribution Bo plus a Dirac measure located at k = 0, 6Q, that does not change the value of the entropy H i.e. H(BQ + a8o) = H(Bo). We summarize these results in the following H-theorem for the QB or the K equation: Theorem 9.1 For any positive weak solution of QB or K equation with initial data g°, one has -^H(g) < 0 with equality if and only if g — M, the equilibrium state M being defined by • ifN(g°) < NQ then M = Bil with /j, > 0 such that N{g°) = iV(BM) • if N(g°) > No then M = B0 + (N(g°) - N0)S0 For the details and the proof of this result we refer to [6; 11; 12] . Escobedo and Mischler study the evolution problem and prove the convergence in large time of the (weak) solution of QBE toward these equilibrium states starting from smooth initial data. More precisely, they prove that, in the second case i.e. for an initial density larger than the critical one, the regular part of the solution converges to the Planck distribution (in L 1 (e, oo) for any e > 0) and the reminding part of the density condensates near the origin. Let us precise the result for the Cauchy problem associated with the QB or the K equation with an initial data g° 6 S defined by

S = {ge L\0M n L$>OO[, kg e L\oMglog(g)

e L[ 0iOo[ }.

Numerical

Method for the Compton Scattering

229

Operator

The existence and uniqueness for the QBE in this space is proved in [11; 12]. However, this hypothesis for g° is not sufficient to guarantee the existence of a global positive solution for the K equation. Blow up in finite time can occur even if N(g°) < N(P). A sufficient condition to ensure existence of global in time, positive solution is that g° < P. We also use another form of the entropy, so-called relative entropy. We note H(g\M) = H{g) — H(M) with M the equilibrium state associated with g° 6 S, defined in the theorem 9.1. A simple calculation gives H{g\M) = J {glog(g/M) - (k2 + g) log((fc2 + g)/(k2 + M))) dk. (9.2.13)

9.2.4

A Maximum equality

principle

and a Czizar-Kullback

like in-

We first prove a maximum principle for positive and global solutions g 6 C 1 ([0, oo[, S) for the QB or the K equation. Our result can be stated as Lemma 9.1 Let g € C'1([0, oo[, S) such that the H-theorem holds i.e. the functional defined by (9.2.7) decays • if there exists fj, such that a = sup h(g X < +oo then sup hM' L < a. • if there exists \i such that ^g'X > 1 then hM' L > 1. Using that for any \x > 0, QBE (9.3.5) as

/I(JB M ,

k) = exp— (k + fi), we can write the

y°° ^*(k)dk=i y°° y°° <*(*) - *(*')) ne, *) exP - < * + ^ exp - ( , ' + M )(*»

+ g)iS

+

MM£L

-

^L)dk>dk, (9.2.14)

and the K equation as (9.2.12). These weak formulations are useful to prove the decay of the relative entropy. The proof of this Lemma is postponed in Appendix A.

230

Lecture Notes on the Discretization

of the Boltzmann

Equation

Note that in the first case ( sup h(g X < +00 ) the solution behaves as g ~ Ck2 exp(—k) for sufficiently large k. Remark 9.2.1 This result gives an upper bound of the solution only for k > max(0, log(a)). For some special cases we deduce the following corollary Corollary 9.1 If g £ C 1 ([0, oo[, 5) and if there exist fj,i and ^2 such that B»2 <9° < B M1 then BM2 < g < B„ 1 . Proof. We just have to prove the second inequality since the first is exactly the second part of Lemma 9.1. Since the function h(g, k) is monotone in g for fixed k, we have

h{B^,k) Lemma 9.1 gives h(g,k) < /i(i?M1,fc) and using again the fact that is increasing in the variable g the result follows.

h(g,k) •

Concerning the trend to equilibrium, we prove the following CzizarKullback like inequality Lemma 9.2 If g e C 1 ([0, oo[, S) and sup ^(g'kl < +00 then for any ko > 0 there exists a constant C depending only on ko, N(g°) and sup jJfl M such that ||S-M||„

[fc 0 ,oo[

Proof. The proof is divided into two parts, one part for "small" k and the second one for "large" k. The entropy H(g\M) is defined by (9.2.13). Using the identities 1 - h(g,k) = -p^ as H(g\M)

=

and ^hU'k'j

=

t*+M ' w e

can rewrite

H

Numerical Method for the Compton Scattering

Operator

231

Then, using the inequality, see [15], Pl

+ | ( p ! - p 2 ) 4 > 2(pi - p 2 ) 2 ,

l o g ( ^ ) + (1 - P l ) l o g ( J — £ - ) > 2 ( P l -p2f

(9.2.15) which holds for any 0 < p\ < 1 and 0 < p2 < 1, for p\ — h(g,k) and we obtain P 2 = h(M,k), « (

9

| i W ) > 2 / ( ^

+ 9

) ( ^ - ^ )

!

^

Note that M ^ k2 + MJ

(*» + „)' fc2+# »

-

fc4|ff M|2 ' (k2+g){k2 + M)2'

Then, for any fco > 0 and R > ko, we obtain using Cauchy-Schwarz inequality

/

\g-M\dk<

It is easy to check that there exists C\ > 0 such that R

2 (k2 + g)(k + M)2 J7 . yA -i-ctt fc4

i Jko

1/2

^ , , ,, D Ar.
with Ci(fc0,-R) which goes to +oo as ko —»• 0 or i£ —> oo. Thus, we have /

|5-M|dfc7V(flt,))H(s|m).

•/fco

We now consider the case of "large" fc. Define y(t) as y(t)=H(Mt),

Mtd=f

M +

t(g-M).

Thus

!/'(*) = [ Jk>0

(g- M)dgs(Mt)dk,

(9.2.16)

232

Lecture Notes on the Discretization

of the Boltzmann

Equation

and using M t = 0 = M, which is the minimum of s, we have y'(t = 0) = 0,

Using now the Taylor's formula with integral reminder gives us

Using Ml = M + t(g - M) < M + g, we obtain

that is, by taking t = 1 i.e. M t = 1 = g and y(£ = 1) = H(g)

,*, +k2{g)(M f,7,M)2+ g + ,o^k Vy| Jfk>0(M k*) <" 2H(g\M). Using again Cauchy-Schwarz inequality, for all R > 0

f

\g- M\dk <

Jk>R

\Jk>R

&

J

\Jk>R(M + g)(g + k* + M) J

and

Jk>R

k

Jk>R

Jk>R

k

the first term of the r.h.s. is bounded by 2N(g°). For the second term, since we have assumed that a = sup A9B X < +oo, and using lemma 9.1, we h a v e SU

P lj§3) < a a n d ' f o r k > max(log(a),0), we have g < " . ? l " K o • Choosing R > log(a) there exists a constant C2 = C2(N(g°),R) such that

/Jt>R \g-M\dkR (M ,M + k9)(s }f-M+ J& + M ) M^'2 2

Numerical Method for the Compton Scattering Operator

233

which gives us f \g-M\dkR

(9.2.17)

Using R - a in (9.2.16) and in (9.2.17), we have \\g-M\\Lj

with C = C(fc0, N(g°),sup ffi^ ffi) = max(Ci, C 2 ) as we claimed.

D

Remark 9.2.2 It can be checked that the constant C — C(ko,N(g°),swp Ap' fc /) blows up as fco tends to 0. However if there exists n such that g° < B M and /x > 0, then by means of Corollary 9.1, g < £?M. This implies that g ^ s is bounded by exp(fcl„\_! which is L 1 and this is still true with fco = 0 and thus the constant only depends on the initial data C = C(g°). In some cases, the entropy permits to control the trend of g toward M . Lemma 9.3 For initial data such that g° < Bo, the solution of the K equation verifies limt->+oo H{g\M) = 0. Proof. We know that for such positive solution, [11; 12] . in t, monotone decreasing that limt_,.+0o H(g\M) = a H'{gn\M)(tn) -4 0, with gn

initial data that there exists a global in time Thus, H(g\M) is well denned at any time, C 1 and positive. Thus, there exists a > 0 such and an increasing sequence tn -> oo such that = g(tn), as n-> oo.

Since for all n, g„ < B0, using Dunford-Pettis theorem, up to an extraction gn —> g, in the sense of measures and g is such that H'(g\M) = 0, that is g = M. We can also prove that, up to an extraction, {gn} is bounded in W,£ •, for anyfco> 0. The K equation can be written as

dg^ _ d£ dt~dk' with F defined by F = (fc2 - 2k)g + g2 + fc2§§. Thus

234

Lecture Notes on the Discretization

of the Boltzmann

Equation

and then r+°° f)a

J

l

r+°° i

ldk

r+°° i

lFldk

dk ~J

¥

+

-j^\(k2-2k)g + g2\dk. (9.2.18)

J

Since g < BQ, the second term of the r.h.s. of the above inequality (9.2.18) is clearly bounded by a constant which does not depend on ko. Using Cauchy-Schwarz inequality, we have

f+°° i

lrlJ,

( f+°° i g(k2+g) „ V / 2 ( f+ao i

F2

„V/2

Let us now consider the production of entropy dH

r . (

h(g) \ dg „

Then, using — = —— and integrating by parts, one get at ok

or, using the definition (9.2.3) of h, f+0°

dH dt

JJo 0

1

F2 dk, k g(k2+g) 2

and then

ill"2' Using one more time g < BQ, one can also verify that ' + » 1 g(k' + g) -dk,

/ k0 Jk,

&

k*

is bounded by a constant which depends only of ko. The result can be summarized as, for each ko there exist two constants C\ and C^Afo) such that

/

+

feo Jk,

~§|^
d t

Numerical

Method for the Compton Scattering

Operator

235

By construction, the sequence gn is such that H'(g) tends to zero. Thus, the sequence H'(gn) is bounded and thus for anyfco> 0, the sequence gn is bounded in W,/ *. [«o>oo[

Now by means of this estimate, Helly theorem and diagonal extraction, up to an extraction we have gn —»• M a.e. One can also easily check that if 0 < g < B0 then \s(g,k)\ < \s(B0,k)\ and s{B0,k) is indeed in ^([O.ooQ. Thus, using Lebesgue dominated convergence theorem lim

H(gn) = H(M),

t n ->+oo

that is a = 0.

9.3

•

Semidiscretization

Let us now turn to the discretization in the energy variable of the QB and of the K equation. We consider a uniform grid in k denoted by

for i = 0 • • • n, with k0 = nh. 9.3.1

Semidiscretization

for the QBE

We shall detail the scheme for two asymptotic cross sections, the uniform one and the grazing or concentrated one. Let us recall the QBE, once restricted to a bounded domain dt

= f ° b(k, k')((k2 + g)e-kg' - (k'2 + g')e-k'g)dk'

(9.3.1)

Jo and the associated symmetrized weak formulation reads k

°

/

da

1 rk°

*ft = 2j0

fk°

J0 K(k2+9)e-kgl-(k'2

+

g')e-k'g)(*-V')dk'dk,

(9.3.2) for any test function * . The discretization of (9.3.1) is based on a standard quadrature formula for the above integral d

N

Hki, kj) (9j{k2 + gi)e~ki - 9i(k] + 9j)e-^) .

-j^^hY, 3=0

(9.3.3)

236

Lecture Notes on the Discretization

of the Boltzmann

Equation

This gives a numerical method that is conservative and entropy decaying. Indeed, from (9.3.3) we have the discrete analogue of (9.3.2) N

A

7,2

i=0

N

N

i=0 j=0

V

9i

9j

)

Then, using the above weak discretized formulation, we obtain the discrete version of the H-theorem for the QBE with the discrete entropy defined by H{g) — h^2i-os{9i->^i)-> that is ^H. < o. By construction Bose-Einstein functions are equilibrium points of the discrete QBE. The relations defining the discrete equilibrium states, ^ = 0, imply that (with Pi — (Bo)i) •

h(gi,ki) _ h(9i,kj) h{Pi,ki) ~~ HPJJT)

r

,, • • ,J

• 9o (9j - exp(-kj)(k]

n >

+ gj)) = 0, for all j .

If g0(t = 0) = 0, one can check on (9.3.3) that go(t) — 0 for all t and, in this case, the equilibrium states M associated to g are Bose-Einstein function JBM eventually with \i negative. This fact happens, as we will see also for the K equation, in the case of N(g°) > N(Bo). If go(0) 7^ 0, the situation is quite different: if N(g°) < N(B0) then necessarily M = B^ with a positive fi, which implies that limt-^+oo g(k = 0,i) = 0, and, if N{g°) > iV(J50), then M = B0 + aSQ with a = N(g°) N(B0) (this is not proved because Mo = 0 and M = £?M with negative fj, is also a possible equilibrium state for the differential system). These nonlinear ordinary differential systems can be very simplified in the two particular cases of uniform and concentrated cross sections. 9.3.1.1

Uniform cross section

When b(k,k') = b(k)b(k'), the evaluation of the double integral (9.3.2) reduces to two (simple) moments :

^

= (jkJb{k')g'dAb{k){k2+g)e-kk0 _

(i

b(k')(k'2 + g')e-k'dk'

\ _

b(k)g.

(9.3.5)

Numerical Method for the Compton Scattering Operator

237

Note that this expression is compatible with the measure values solution of the form g = greg + ado considered in [12] (which satisfied a system of equation for both greg{k,t) and a(t)1 described in [11] ). One replaces the integral by a discrete sum. Assume b(k) = 1 for example (as in [12] ). Let us note Mo the discrete density and define the first moment as n

M! = h^iti

+ 9i)e~ki •

i=l

We consider the following explicit scheme g

'"

~ g* = M0k2e~ki

+ (Moe~ki - M?)g?.

(9.3.6)

The conservation of density can be verified by summing the r.h.s. in the above equations. Positivity is preserved at any iteration, provided that

Indeed, multiplying (9.3.6) by eki and summing over i gives

<

MoMj" - M?(M? - ] T

<

CM? ~

with C = M 0 + E k2ekih.

k\ekih)

(M?)2,

Then,

M?+1 < Mf + CAtnM?

- Ain(Mr)2,

and using At„ = j ^ r , we have by induction that M " < C and the time steps are bounded from below.

9.3.2

Semidiscretization

for the Kompaneets

equation

For the discretization of the K equation we proceed in two ways. The first one is based on the scheme for the QBE and the fact that for concentrated cross section around k = k', the K equation is the asymptotic limit of the QBE, see [11] . The second one is adapted from the method proposed by Chang and Cooper, see [7] , for linear equation of Fokker-Planck type.

238

9.3.2.1

Lecture Notes on the Discretization

of the Boltzmann

Equation

Concentrated cross section

One consider a cross section sequences on the form

where & is a positive and even function. This gives the K equation (when b(k,k')=ek'2ek''2). If we assume, for simplicity, that b is compactly supported with supp(b) =] — 3/2,3/2[ and choose e = h. The cut-off/smoothing parameter £ is equal to the mesh size. Then, in the double sum (9.3.4), only the terms such that (i — j) — ± 1 do not vanish. We obtain the following scheme

dgt

at

^

9i+i(A? + 9i)eh/2 - 9i(k2+1 + gi+1)e-V>

£

+ 9i)e-hl2

+9i-i(kf

- 9i{kU

+

(9.3.7)

9i-i)eh'2.

Note that this system (with 6(1) = 1) can also be written as

dg%

at

±{gi+1kleh'2

+

2

ft_xA?e-*/

( ^ + 1 e - " / 2 + kUehl2)9i)

-

+ +^{eh'2

(9.3.8) - e-hl2){gi9i+l

- g ^ ) .

The first part corresponds to a tridiagonal linear system and the second part to the non linear Burgers term f2. One can write this system in the form

£ = s('»t-'-»)•

(^

Numerical

with -F_i/2 = Fi+i

Method for the Compton Scattering

239

Operator

— 0 and for i = 0, ...,n - 1

FN+I/2

= - (exp{-)k?gi+1

-exp{--)k?+igij

+

e x p ( - ) - e x p ( - - ) J gigi+1

= Iexp(-^±i)(pi + ( h(9i+i,kj+i)

_

fc82)(ffi+1+^+1)

h(gi,kj)

\h(Bi+1,ki+i) h{Bi,ki) 1 1 . (r> M + ki+1 h + ki+i\ f h(gi+i,ki+i) h B o )> hT. \V^ 2 " 2 J \h(B ,k )

=

i+1

i+1

h(gi,ki)

h(Bi,ki)

for any Bose-Einstein state B^, since /i(B^,fc) = exp(fc + fi). This is the consistency of this scheme for the Kompaneets equation. Indeed, the flux of the K equation (9.2.10) is

Using a Taylor expansion around ki+\ = '

i+1 2

we obtain

F ^ ^ G ^ A ^ + OC/*2).

9.3.2.2

Chang and Cooper method

We now follow the method proposed by Chang and Cooper [7] . We use from (9.2.9) which is close to the linear Fokker-Planck expression and discretize the fluxes at the interface Fi+i between ki and fc;+i in a standard centered finite difference for the diffusion part /

\

1 + /A+1

=

1 (

fi+i

h\l + fi+1

fi \

1 + fiJ

=

fi+i - fi

hil + fi+^il + fi)'

a linear combination for the convective part

(lT7) i+1 = (1 -^ ) rT^T-^rf7:'

240

Lecture Notes on the Discretization

of the Boltzmann

Equation

where 0 < 6i+i < 1 has to be defined further, and, using a harmonic average for the 'diffusion coefficients"

(k4(i+f)2)i+i

=

k}ki+l{\+mi+fi+1).

This last choice permits to simplify the denominator in the diffusion discretization which is now linear in / . Then, the flux can be written after some simple calculus as (with gi = fc?/j) Fi+x = Si+igi(gi+1

+ kf+1) + (1 - S i + i ) g i + 1 ( g i + kf) + -^{kjgi+i - fc?+iff»)-

(9.3.10) The coefficients 54,i are now to be determined. We shall choose them in order that the Bose-Einstein equilibrium states are preserved by this scheme. We impose that for gt = Mi = ( .. + '„_ 1 with any real value of a, the flux vanishes. This gives 1_ 1_ *** = r - T ^ - r -

(9-3-11)

Note that these coefficients are independent of i for such a uniform grid and have the following Taylor expansion for small h:

This means that when h -> 0, the scheme becomes a symmetrized approximation of the convective part. Using this choice (9.3.11), the formulae (9.3.10) can be simplified into Fi+h = gi+1(gi + kf) + -^—^{k2i9i+i

~ kf+l9i).

(9.3.12)

The same expression can be found for the flux F^i F f _i = gi(gi-i + fci_x) +

h

_ 1 (fci-iffi -

fcift-i).

Then, we obtain the same semidiscretization (using the notation F _ i / 2 = Fn+i/2 = 0) as in (9.3.9) up to a multiplicative factor that goes to 1 as /i->0

ph/2 _ p-ft/2

1

7

Numerical Method for the Compton Scattering Operator

241

Remark 9.3.1 Note that, for a non uniform grid, the same method applies, replacing/i by its local value Afc i+ i = (ki+\— hi). However, the two schemes are no more proportional although the ratio converges to 1 (when the mesh is refined i.e. max; Afc i+ i -4 0). 9.3.2.3

Properties of the two schemes

A discrete integration by parts gives the weak form for the two above schemes: n-l

t=0

i=0

Thus, if we define the discrete density N(g) as ^27=0 ^9* an^ t n e discrete entropy as H(g) = 5^=0 ^ ( S t ; ^ ) o n e verifies the conservation of density (•jfiN(g) = 0) and the decay of entropy. Indeed we have

thus using (9.3.10) d -H{g)

N

= ~E

h C

^ + 9i)(kli

+ 9i+i) (log(A i+1 ) - log(AO) (A i+1 - A.),

i=0

(9.3.14) with Aj+i = C = for the flux (9.3.10) or C = e *liplh)-i^ (9-3.12), and C is strictly positive. Using the classical inequality (x — y) (log(x) — log(«/)) > 0, we have +

£{pt +\'kk\++\], f o r t h e flux

exp(- fei+ 2 fci + 1 )

|H(9)
Existence of global positive solution

We write system (9.3.8) by factorizing the gain term G(g) and the loss term L(g) as usual for Boltzmann equation ^

= ~ (G(g)i - L{g)i9i),

(9.3.15)

242

Lecture Notes on the Discretization

of the Boltzmann

Equation

with G(g)i > 0 and L(g)i > 0. We have the existence local in time using the Cauchy-Lipschitz theorem starting from g® > 0 for all i > 0. Then, the loss term being bounded (for a given initial density N), we have that the solution remains positive. Finally, using the conservation of density, we have an upper bound for the semidiscretized solution
Discrete equilibrium state

We shall prove that the discrete equilibrium states are the generalized BoseEinstein distribution functions. We restrict ourselves to the case /°(0) < oo that is <7°(0) = 0. It is easy to verify that if g0(t = 0) = 0 then g0(t) = 0. Thus, the distribution function is discretized on [h, k0] i.e. for i = 1 • • • n. From (9.3.14), it is easy to verify that -^H(g) = 0 if and only if g = B^ with n eventually negative. Let us consider, for any fixed density N, the discrete Bose-Einstein distribution B k

^)

k2 = e t + a_iX*>fc»

where a is such that n x=i Note that Nh is decreasing with a and is one-to-one from ] — h, oo[ to [0, oo[. Thus, any arbitrary density N can be associated with a generalized discrete Bose-Einstein state provided that the mesh is refined enough such that N > N° or Nh < k0. When n ->• oo or h -»• 0, one has for N < N° Nh - • iV(£?M), and for N > N°, Nn -> BQ + (N - N°)60- Indeed, in the second case, one has —h < a < 0 and thus a —> 0 as h —> 0. One can check easily that if Nh(g°) > Nh(Bo), the equilibrium state associated to g° is a Bose-Einstein state B% with a € [—h, 0[. Let us now precise the relation between such equilibrium state and B0 + (N(g°) — N(B0))S0. For simplicity, we consider the problem in continuous variable k, thus Nh is now

Jk>h

Numerical Method for the Compton Scattering

Operator

243

We shall prove the following result Proposition 9.3.2 For any N > 0, B% converge when h -» 0 in the sense of distribution toward the continuous equilibrium functions that is, when N < No, there exists ft > 0 such that B^ —> Bp i.e. a —»• /3 as h —> 0 and when N > N° B^B0

+ (N-

N°)S0,

where Bo is the Planck distribution and N° its density. Moreover, the associated entropy converges i.e. H{Mh) -> H{Bo) when N > N° and H{B^) ->• H(Bp) when N
Maximum principle for the discrete equation Kompaneets

Let us prove a stronger result that the existence of a positive solution when the initial data is in between two Bose-Einstein functions. Indeed, as for the linear Fokker-Planck operator, the Kompaneets equation satisfies a Maximum principle that is verified on its discretized version. Proposition 9.3.3 Let g a solution of (9.3.8). If there exists p such that g(t = 0,k) < BM(fc) (resp. g(t = 0,k) > Bli(k)), then, for all t > 0, one has g(t,k) < B^k) (resp. g(t,k) > B„(k)) . Proof. First, one checks easily that the scheme (9.3.8) can be written in the form d

r,fh(9,k)\

9i_r

^(

h(g,k)\

,„„.„,

where we note h(g, k) = k$^ |t> and D is a finite difference operator defined as Di-± = 4>i+i

-i,

and Ci+i are non-negative coefficients (with the notation C_i/ 2 = Cjv+1/2 = 0) that depend on the function g and of A;. Note that for any Bose-Einstein function B^, h(B0,k)

=

fc2

°

= exp(-fc) =

h(B^,k)exp(-fj,).

244

Lecture Notes on the Discretization

of the Boltzmann

Equation

Therefore, one can change B0 into any Bose-Einstein B^ in formula (9.3.16) with coefficient multiplied by exp(-/x). Moreover, h(g,k) is an increasing function of g since dh{g,k) dg

_ k2 "(PH-V)"

We denote by _ l

~

h(gi,kj)

HB^ki^ki)'

then the (Aj) satisfy a system of the same form i.e.

^

=

Ei+hD{\)i+h-Ei_iD{\)i_h,

with non-negative Ei+1. Define As = maxj=i...jv Aj. The function As is a piecewise C 1 function of t. By definition of As, we have, Vt > 0, there exists a subset Is(t) of {1, • • • , N}, such that Aj(i) = Xs(t) for all i £ Is(t). Thus, Vi 6 Is(t),Wj € {1, • • • , iV}, A, > Xj and using the positivity of the coefficients E, we obtain that ^f- < 0. The same idea proves that the minimum of Aj increases with time. • Remark 9.3.4 Assume that for some fi, g(t = 0) < B^, then h(g(t = 0),fc) < /i(SM,fc) or equivalently As(0) < 1 then for all t, \s(t) < 1 i.e. g(t,k) < Bfj,(k). In the same way, if for some fi, g(t = 0) > B^, then

git^yB^k). Thus, using the Kullback like inequality (9.2), which is also valid for discrete measures, we can prove the following lemma Lemma 9.4 When g(t = 0) < Bo, the distribution function converges toward its equilibrium. Proof. Using H(g\M) = H(g) - H(M), where M is the discrete equilibrium associated to g, is decreasing in time and is positive. Thus, there exists an increasing and diverging sequence £& such that H'(tk) -> 0. The zero of the derivative of H are the equilibrium M. Hence, g(tk) goes to M and H(g(tk)\M) -> H(M\M) = 0. Since H(g\M) is decreasing, we have necessary lim H(g\M) = 0, t—>oo

Numerical Method for the Compton Scattering

245

Operator

and, by lemma 9.2, we obtain, the convergence of g toward its equilibrium

M.

•

Remark 9.3.5 The results of this section are valid for the discrete QBE, the proofs being analogous.

9.4

Time Discretization for the Kompaneets Equation

We eliminate explicit time discretization since, when Bose condensation occurs, time step to ensure positivity would be in 0(/i 3 ) to compare with At = 0(h2) for classical parabolic problems. Therefore, we shall only consider implicit scheme. We shall see that a fully implicit scheme has good properties but seems hard to implement at a reasonable cost. We propose an alternative implicit scheme with a low cost but for which we cannot prove all the features of the fully implicit scheme. As illustrated by numerical examples, this scheme works well.

9.4.1

Fully implicit

scheme

Let us consider an implicit scheme of the form (g represents gn at iteration n and g denotes gn+1) g = g + tQ(g).

(9.4.1)

Assume that the scheme is positive i.e. gn > 0 => gn+1 > 0, we shall prove that it is automatically entropy decaying i.e. Hn+1 = H(gn+1) < H{gn) = Hn. Indeed, H(g) = f s(g,k) denned by (9.2.13) and using a second order Taylor expansion, at point g with integral reminder, we have H(g)-H(g)

= dgH(g)(tQ(g))+t2

f

(l-z)Q(g)Td2ggH(g+z(g-g))Q(g)dz,

Jo

where dgH denotes the functional derivative

dgH = Idg(s(g,k))dk = jlog {^^~)

dk,

246

Lecture Notes on the Discretization

and we have dgH(g)(tQ(g)) respect to g is negative

of the Boltzmann

Equation

< 0. Moreover, the second derivative with

1 d2yyggs(g,k) ,k) = = g - - g—+ ^kz< 0 , and this concludes the proof. The existence of a positive solution for the implicit scheme is ensured by the Brouwer fixed point theorem. We choose C > 0 such that CN(g)f + Q(f) is a positive operator for all positive / such that the density of / less or equal to N(g). Then (9.4.1) can be rewritten as (/ denotes gn+1 and g denotes gn): / ( l + N(g)Ct) = g + N(g)Ct ( / + ^ ^ j

.

(9.4.2)

The mapping / h+ T(f) Ttn

1

N{g)Ct

(

Q(f)

\

is continuous from the convex compact set E = {/ > 0 such that density of / is less than or equal to N(g)} into itself thus the Brouwer fixed point theorem insure the existence of an element / * of E such that / * = T(/*) and necessarily / * has the same density and energy that g. Despite its good properties, since the implicit scheme is non-linear an iterative procedure is needed and have to be stopped before exact convergence.

9.4.2

Semi-implicit

scheme

The method we suggest is linear part semi-implicitly semidiscretized system are tem (9.3.8) can be written

to treat the linear part implicitly and the nonbut in such a way that the properties of the preserved. As we have seen the differential sysas

^9i = l(F{9)n.l-F(9)i-l),

(9-4-4)

Numerical Method for the Compton Scattering

with the fluxes F(g)i+±

Operator

247

defined by (9.3.10) and have the structure

F(g)i+i=FLi+,(g)+FBi+i(g), with FLi+i(g) = ai+igi+i - bi+igi, FBi+^(g) = ci+Lgigi+i, and ai+i, bi+i and ci+1 are non-negative, FBi+i(g) is the Burgers flux and FLi+i(g) is the flux for the linear Kompaneets equation. The semi implicit scheme consists of treating all the fluxes FLi+i(g) implicitly and to implicit only the term gt+i in the Burgers fluxes FBi+i(g). That is, if we assume that g is known at time t, we compute g at time t + At as: Si = 9i +

T

(**+l® "

Fi iS)

-i

and Bi+i (g,g) = ci+igigi+i. i-e. Y,9i = Effi-

+ B

»ib,9)

~ Bi_t(g,gj)

(9.4.5)

One can verify that the density is preserved

This system can be written in the form (Id-AtM(g))g

= g,

(9.4.6)

where M(g) is a tridiagonal matrix and one can check easily that M(g) is also a so-called L matrix, that depends on g i.e. M(g)u > 0 and M(g)ij < 0 for all i T^ j . The main property of this semi-implicit scheme concerns its positivity: Lemma 9.5 If g is positive, then g defined by (9.4-5) is positive for all time steps At. Proof. Due to the special structure of M(g), it is easy to check that one can always construct X > 0 such that X e ker M(g) which is equivalent to find X > 0 such that Fi+i(X)

+ Bi+i(g,X)

= 0,

(9.4.7)

for all i: start from one index io by setting XJO arbitrary strictly positive and by virtue of the positivity of ai+i, bi+i, ci+i relation (9.4.7) generates positive strictly Xi. Then there exists X > 0 such that (Id - AtM(g))X = X, that is if we set D such that D{j = dijXi, thus (Id — AtM)D is a diagonal dominant

248

Lecture Notes on the Discretization

of the Boltzmann

Equation

matrix that is (Id—AtM) is a generalized diagonal dominant matrix, which is equivalent to the fact that M is an M-matrix or in other words it has a positive inverse, [1]. This means that the scheme is unconditionally positive (whatever the condensation occurs). • Note that X is related to equilibrium state in the proof. Concerning the equilibrium states we also have the following result Lemma 9.6

The scheme (9.4-5) preserves the equilibrium state.

Proof. If we write Q(g) the operator of the right hand side of (9.4.4), by construction Q(g) = 0 if g is an equilibrium state. Since M(g)g — Q{g), (9.4.6) reads

(Id-AtM(g))(g-g)=Q(g)=0. Since the matrix M(g) is an M-matrix, then we have g = g as we claim it. One should choose time step of the form C\h where C\ is such that the entropy decays. We are not able to exhibit a condition on the time step to ensure the decay of the discrete entropy. But, as we will see on the numerical examples, using a time step corresponding to the convective equation that is of the form At < CAk leads to a satisfactory behaviour of H even for singular initial data.

9.5

Numerical Tests for the Kompaneets Schemes

We will illustrate the scheme in the following two examples. The first one corresponds to a Planck distribution multiplied by 3/2. In this case, concentration near origin occurs after a few iterations since the initial density is greater than the critical one. On the second case, we consider an initial data with a lower density but we still observe a concentration when the initial density is close enough to 0, as expected after the analysis in [10].

9.5.1

Relaxation

of aB0

with a — 3 / 2

We plot the evolution in time of three macroscopic quantities, density for non-zero energy, energy and entropy. The runs correspond to the following three cases

Numerical Method for the Compton Scattering Operator

NteJ-g,

200

Fig. 9.1 Density in case 1.

N(kg)

200

Fig. 9.2 Energy in case 1.

• label "1600" is a reference computation with 1600 points of discretizations and
250

Lecture Notes on the Discretization

of the Boltzmann

Equation

Entropy -1,95 -1600 -60 100

-2

-2,05 -2,1 -2,15 -X

X

X

X

X—.

-2,2

50

0

100

150

200

time

Fig. 9.3

Entropy in case 1.

More precisely, in Fig. 1.1 we plot the quantity X)i>i 9ih ( t n e indices start at 1) i.e. the discrete version of jh° gdk. This quantity is constant when the solution remains smooth. It decays when concentration occurs. Fig. 2 (resp. 3) illustrates the evolution of the energy (resp. entropy). The differences of the three runs are in the number of discretization points and in the initial data (at zero energy). Indeed, as explained before (section 3.2.5) there are two ways to discretize the singularity. Either, go(0) then, g0(t) = 0, Vi > 0 and the distribution function converges to a generalized Bose-Einstein equilibrium (with fi < 0) that converges in a distributional sense toward the Planck distribution (B0) plus a Dirac measure at k = 0 or 0 toward the Planck distribution, and the remaining density concentrates on

fco=0. The presented results show that it is much better to discretize with <70(0) ^ 0, in particular in terms of entropy (the entropy is lower with only 60 points and go(0) ^ 0 than with 1600 points and go(0) = 0, i.e. generalized Bose-Einstein equilibrium).

9.5.2

Initial

Gaussian

distribution

In this case, we consider an initial Gaussian distribution with a sub-critical density N < N°. In the Compton case (or QBE), the solution goes toward its Bose-Einstein equilibrium. It has been proved in [10] that, for the

Numerical

Method for the Compton Scattering

Operator

251

distibution function

3,5

i

l

""r

3 2,5

2 r

f\

T\

t=0.00000e+00 — * — t=1.59278e-01 — i — t=2.63295e-01 —o—t=3.93318e-01 —•—t=4.68081e-01

1,5 1

'-

0,5

-

—i «

0

- t ^

-0,5

- . , . ,

/>,<

-^ j

'-_ 0,5

Fig. 9.4

^y

7

1,5

Evolution of the distribution function in case 2, k = 2.

Kompaneets equation, the solution may not be global in time provided that the initial density is close enough to the origin. Indeed, there is a balance between the Burgers term and the diffusion term : the Burgers part is a convective toward the origin that leads to concentration, whereas the diffusion spreads off the distribution that becomes smoother. In Fig. 1.4, we plot the distribution for different time when the initial data is a Gaussian centred at k = 2 with 3200 points of discretization. In this case, the diffusion dominates and the distribution function goes to its Bose-Einstein distribution (with v > 0). In Fig. 1.5, we plot the same quantities (distribution function versus k at different time) but with a Gaussian initial data (same total density, centered at k = 1). In this case, the Burgers term is stronger and the distribution function becomes singular near origin after a finite time T*. Clearly, the solution is no longer valid for t > T* but the simulation can be continued since the Kompaneets discretization can be interpreted as the discretization of QBE with a small parameter e linked to the discretization step h (see section 3.2.1) for which such a concentration is possible.

In Fig. 1.6, we plot the evolution in time of the density for non-zero energy and we observe the concentration at time T* : some part of the density goes into the zero energy part of the distribution function. In Fig. 1.7 (resp. 1.8), we plot the energy (resp. entropy) versus time.

252

Lecture Notes on the Discretization

of the Boltzmann

Equation

distribution function 3,5

t=0.00000e+00

3 r 2,5

—*— t=1.59278e-01

-

t=2.63295e-01 t=3.93318e-01 —»—t=4.68081e-01

;

2

;

1,5

i

!

1

/

'•tffif^

0,5

Ess^

ijy, J—< 0,5

Fig. 9.5

1,5

Evolution of the distribution function in case 2, k = 1.

N(g)-gi 1,95

1,85

1,75 8

Fig. 9.6

9.6

10

Density versus time in case 2 , k = 1.

Conclusions

The main improvements compared with the previous method investigated in [7; 16] are • This method does not require to construct an equilibrium state that usually required an iterative method (which is not always conservative).

Numerical Method for the Compton Scattering

Entropy

N(kg) 1,02

1

i

' ' ! ' ' ' 1600 105

x

; :

-

0,98

i

i

i

i

i

i

_

1600 —X— 105

0

-_

-0,5

-1

0,96

\

- \\

':

-1,5

\ X

0,92

253

Operator

; s

„

^-

V

„ „

W

"

0,9

2

4

6

8

10

-2

iV ,

-2,5

2

energy

6

8

time

time

Fig. 9.7

, . i , , ,

4

Fig. 9.8

entropy

• The semi-discretized system preserve all the properties entropy, conservation of density • The semi-implicit scheme is unconditionally positive • Scheme 1 is of order 2 in A;

decay of

Moreover, we are able to deal with condensation near the origin when the initial data have a density larger than the Planck one or is to close to the origin", both of these phenomena are illustrated by numerical results. There are two ways to describe this concentration at the discrete level : either go(0) = 0 and then, this remains equal 0 at all time - in this case, the equilibrium state are generalized Bose-Einstein state with fi < 0 that converge in a distribution sense to the Planck distribution plus a Dirac at k = 0, see section 3.2.5 and appendix B - or go(0) ^ 0 - in this case, the smooth part converges toward the Planck distribution and the remaining density concentrates on k = 0. The preliminary numerical results indicate that the second case is better and more precisely, and that the first case is unstable: when the initial data at k = 0 is not equal zero, the scheme converges toward the second representation of the singularity. Some questions remain to be addressed in further works : • Is the semi-implicit scheme entropic ? • How to construct a full implicit scheme (without iterative method)? • Can we insure the same properties for general cross sections with the (QBE).

10

254

Lecture Notes on the Discretization

of the Boltzmann

Equation

Appendix A : Proof of Lemma 2.2 Proof. We begin by the first part of the lemma. It is then easy to check that it suffices to prove the result only in the case ^ = 0, i.e. for BoFor k £ [0,log + (a)] with log + (x) = max(0, log(a;)), since the solution is non-negative and h(g,k) < 1 < aexp(-k)

— ah(Bo,k).

For k €]log(a),oo[, consider any function G(x) such that G(x) > 0 for x > 0, 0 < G'(x) < oo for x > 0 and G{x) = 0 for x < 0. Then, define Ha(x,k) as

for k e]log + (a),oo[ with x0 the unique positive solution of h(x0,k) = eexp(—k) ah{B0,k) (note that x0 = "^anexp(-k) ^S_% > 0) and Ha(x,k) = 0 for + k < log (a). Set now

E(t) = [

Ha(g(k),k)dk.

It can be checked that by construction E(t) > 0, E(t) is continuous and by the hypothesis E(0) = 0. Moreover, E(t) is C 1 and we have

dt

Jk>0

\h(B0,k)

J dt

Now using the weak form (9.3.5) of the QBE, it is easy to see that, since G ( h(Bk)-a ) 1S m o n o t o n e increasing in the variable g for each k. Thus, using the inequality {x — y) ((x) — (f>(x)) > 0, for any monotone increasing function (j), ^E < 0. For the K equation, using the weak form (9.2.12), it is also clear that jjE < 0. Thus, for both equations, E(t) = 0 for all time t, which is equivalent to h,g K < a almost everywhere in k 6]log + (a),co[.

Numerical

Method for the Compton Scattering

255

Operator

For the proof of the second part of the lemma we proceed as for the first part. With the function G denned above, we set H as

^--JA^h if g < x\ and H(x, k) = 0 for g > x\ and define E{t) = f H(g, k)dk. Jk>0 Proceeding as for the first part we obtain that E(t) is monotone decreasing, positive and E(0) = 0. Thus for any time E(t) = 0 that is h(g, k) > h(Bo, k) a.e. as we have claimed it. •

Appendix B : Proof of Proposition 3.3 Proof.

As explained above a is a function of N and ft.

Define N(a, h) as (for any h > 0 and a > —h) k2 dk. exp(fc + a) - 1

N(a,h)= f

Jhh

The equation N(a,h) = N determines implicitly a as a function of N and ft. If AT > TV0, since N° = N(0,0), a(N,h) 6 [-ft,0[ thus a ->• 0 as ft -» 0. Let <(> be a test function, we have /•OO

L{h)

=

j

(j>{k){Mh{k) -

B0(k))dk

%> em

=

Jh

dt _ r

exp(fc + a(ft)) - 1

J0

fcV(fc) dh exp(k) - 1

We write {k) = (/>(0) + kip(k) where ip is a C°°(0, oo). We know that a (ft) —> 0. Thus, we have

L(h) =

r

k2(f>(0) (

Jh

ex

+

1

———

\ P( Jh

k

h

*

m

_ r em

fc

+ "W) - !

-}-—-) dk exp(fc) -

\)

(exp(A; + «(/.)) - 1 " exP(A0 - l )

dh

Jo exp(fc) - 1

dk

256

Lecture Notes on the Discretization

of the Boltzmann

Equation

The first term gives

f°° k2 ( Jh

1

—

\exp(k + a(h)) -1

} dk = N - f

exp(fc)-iy ->• N

Jhh exp(fc) - 1 -N°.

dk

The second term can be written as r0

(1 - exp(a(/i))) r

°

k3ip(k)dk

*t{k\f,

-.

TTT-^dk

-• 0,

Jh (exp(fc + a ( / i ) ) - l ) ( e x p ( f c ) - l ) and the third term goes to 0. Thus, we proved that L(h) -> (TV - N°)<j>(0). This is equivalent to the convergence of B^ toward Bo + (N — NQ)SO in the distribution sense. Let us now consider the case TV < N°. There exists /? > 0 such that N(/3,0) = N. We want to prove that a(h) (N being fixed) goes to /? as h —> 0. The function a being decreasing with h and being positive for h small enough, one has 0 < a(h) < /3. Moreover, a is continuous, then limh-j.o ct(h) — c*(0) = /?• Then by dominated convergence theorem, we have that B% tends to B$ in i 1 ( 0 , oo) (and in the distributional sense). Let us now prove the convergence of the entropy. One has

H(9) = [ (siog(^) + e io g (^) + *,) dk. For the function B%, one easily checks that "• k2 +B*

= e-(fe+Q)

' k2 + B%

= l -

e

~(fc+a).

Then, using the expression of H and B^, one obtains H{B*) = -a

k2 log (1 - exp -{k + a)) dk.

B%+ JO

Jh

The first term is equal to —aN that goes to 0 because a -> 0. Let us note gh(k) = log (1 — exp — (k + a)). We know that gh converges to g0 almost everywhere and is integrable on [0,oo[. Note that Vfc > e since a < 0,

{9o-9h)(k)>0.

Numerical

Method for the Compton Scattering

Operator

257

to,-**)(*) = log' 1 — * exp— ^ (kik)+ a) t k2dk Jh exp-(fc + a ) - l Jh ,0

< exp(a)(exp(—a) — 1) /

Using log(l + h) < h. This proves that (flo -9h){k)dk

->0.

Jh

But, H{Bha)-H{B0)

= -aN+

(gh - g0)(k)dk g0(k)dk . Jh Jo The last term tends to 0 since go is integrable. This proves the convergence. In the second case JV < N° we know that a(h) is a decreasing function of h that converges toward /3 > 0 such that

f /o

Jo

k2dk exp(fc + P)-1

N.

The proof given for N > N° can be applied in this case. 9.7

•

References

[1] A. Berman, R.J. Plemmons, Nonnegative matrices in the mathematical sciences. Classics in Applied Mathematics, (1994). [2] N. Bose, Plancks Gesetz und Lichtquantenhypothese, Z. Phys., 26 (1924), 178-181. [3] C. Buet, S. Cordier, F. Filbet, Comparison of numerical schemes for Fokker-Planck-Landau equation. ESAIM Proa, 10 (2001), 161-181. [4] C. Buet, S. Cordier, Numerical analysis of conservative and entropy schemes for the Forkker-Planck-Landau equation., SIAM J. Numer. Anal., 36 (3) (1998), 953-973. [5] C. Buet, S. Cordier, P. Degond, M. Lemou, Fast algorithms for numerical, conservative, and entropy approximations of the Fokker-PlanckLandau equation. J. Comput. Phys., 133 (2) (1997), 310-322. [6] R.E. Caflisch, C D . Levermore, Equilibrium for radiation in a homogeneous plasma, Phys. Fluids 29 (1986), 748-752. [7] J.S. Chang, G. Cooper, A practical difference scheme for FokkerPlanck equation. J. Comput. Phys., 6 (1970), 1-16.

258

Lecture Notes on the Discretization

of the Boltzmann

Equation

[8] G. Cooper, Compton Fokker-Planck equation for hot plasmas, Phys. Rev. D 3 (1974), 2312-2316. [9] L. Desvillettes, C. Villani, On the Spatially Homogeneous Landau Equation for Hard Potentials. Part I: Existence, Uniqueness and Smoothness, CPDE vol. 25 (1-2) (2000), 179-259 and Part II: H Theorem and Applications , CPDE., 25 (1-2) (2000), 261-298. [10] M. Escobedo, M.A. Herrero, J.J.L. Velazquez, A nonlinear FokkerPlanck equation modeling the approach to thermal equilibrium in a homogeneous plasma, Trans. Amer. Math. Soc, 350 (1998), 38373901. [11] M. Escobedo, S. Mischler, Equation de Boltzmann quantique homogene: existence et comportement asymptotique, Note au C. R. Acad. Sci. Paris 329 Serie I, (1999), 593-598. [12] M. Escobedo, S. Mischler: On a quantum Boltzmann equation for a gas of photons, J. Math.Pures Appl, 80 (5) (2001), 471-515. [13] M. Escobedo, S. Mischler, M.A. Valle, On Boltzmann equation for a gas of quantum (and relativistic) particles, work in preparation. [14] A.S. Kompaneets, The establishment of thermal equilibrium between quanta and electrons, Soviet Physics JETP, (1957). [15] S. Kullback, On the convergence of discrimination information. IEEE Trans. Information Theory IT-14 (1968), 765-766. [16] E.W. Larsen, C D . Levermore, G.C. Pomraning, J.G. Sanderson, Discretization methods for one-dimensional Fokker-Planck operators. J. Comput. Phys., 61 (1985), 359-390.

Chapter 10

Discrete Models of the Boltzmann Equation in Quantum Optics and Arbitrary Partition of the Velocity Space F. Schiirrer Institute fur Theoretische Physik, Tech, Univ. Graz, Austria

10.1

Introduction

The famous Boltzmann equation [14], governing the spatio-temporal evolution of the velocity distribution of rarefied gases, can be solved exactly only for idealized special cases (see [8] and references herein). Realistic problems, however, require modified transport equations with a simpler mathematical structure to overcome the computational complexity. All of these so-called model Boltzmann equations must fulfill at least the fundamental properties of the full Boltzmann equation, namely the conservation laws and an i?-theorem. Such simple models are, e.g., the Boltzmann-like transport equations based on discrete velocity models as addressed in several chapters of this book. In these models the particle velocities can attain only a finite number of lattice points. A summary of the theory of discrete velocity models and their applications is given by Monaco and Preziosi [27] and Bellomo and Gustafsson [1]. Although the idea of discretizing the velocity space is rather old [26], it was not before the 60's that Broadwell [6; 7] succeeded in using a discrete velocity model of the Boltzmann equation. A decade later, Gatignol [20] and Cabannes [9] developed a systematic and rigorous foundation of the discrete kinetic theory. They analyzed in detail the relevant aspects of this theory such as modeling, analysis of thermodynamic equilibrium and 259

260

Lecture Notes on the Discretization

of the Boltzmann

Equation

the application to fluid dynamic problems. In their pioneering work only a few particle speeds were used, but a sufficient modeling of arbitrary velocity distributions was left open. To treat physically relevant problems, multispeed models [15] are needed in order to describe adequately macroscopic quantities such as pressure, temperature, etc. It has been proved by Bobylev, Palczewski, and Schneider in a series of papers [4; 5; 30] that the usual discrete Boltzmann equation [10; 20 ; 27] converges, but only very slowly, to the continuous Boltzmann equation on a regular grid in three dimensions. However, for the two-dimensional case, a suitable form of the cross section is needed to prove the convergence to the continuous one, as shown by Palczewski and Schneider [31]. Numerical solutions to the discrete Boltzmann equation on a regular grid can be found for the two- and three-dimensional case in the papers of Inamuro and Sturtevant [23], Rogier and Schneider [37], and Buet [8]. Discrete velocity models are well suited to treat the kinetics of quantum optic problems. In such scenarios particles of different species of a gas mixture collide elastically and inelastically with each other and interact additionally with monochromatic photons. To demonstrate the applicability of discrete velocity models in this field, the problem of laser-induced thermal acoustics [16; 30] is considered in Sec. 10.2. This study shows the formation and evolution of laser-pulse-driven thermal and density gratings by numerically solving the discrete kinetic equations based on the fractional step method. The results covering a wide scope of Knudsen numbers are then compared the fluid dynamic limit with results obtained from a fluid dynamic approach. When creating discrete velocity models, one has to pay attention to the conservation laws, which restrict the freedom of modeling dramatically. Furthermore, if one would like to simulate a continuous velocity distribution of real gases by means of discrete model Boltzmann equations, a partition of the continuous velocity space has to be introduced. In such a model each discrete distribution density, which corresponds to a certain discrete velocity, represents the average distribution density within the corresponding domain. In previous models [10; 27], discrete Boltzmann equations govern number densities without taking into account a partition of the velocity space. Consequently, it is an interesting goal to create an adequate discrete velocity model linked to the continuous Boltzmann equation. The resulting equations should permit the description of the evolution of a given continuous particle distribution by means of this discrete velocity model. This

Discrete Models of the Boltzmann

Equation in Quantum

Optics

261

new approach, presented in Sec. 10.3, can be seen as a generalization of the original discrete model Boltzmann equations. Section 10.4 deals with the scaling of discrete velocity models. It is shown that the speed of sound is a proper macroscopic quantity used to adjust the model to the given physical problem. The required scaling procedure is demonstrated for two different discrete velocity models applied to simulate an inelastically interacting gas mixture, which is affected by a laser-generated rediation field. The most significant differences between the models are discussed. Furthermore, the results reveal that the structure of the velocity model influences slightly the temporal of the density oscillations, while the evolution of the kinetic energy remains almost unaffected.

10.2

Laser-induced thermal acoustics: A discrete kinetic approach

The excitation of acoustic and thermal waves in strongly absorbing collisiondominated gas-phase systems with pulsed laser light was first reported as an artifact in degenerate four-wave mixing (DFWM) experiments [39]. A recently presented method based on this phenomenon is the laser-induced thermal acoustics (LITA) [16; 17]. It is an optical four-wave-mixing technique for measuring the physical and chemical properties of gases both remotely and non-intrusively. The process of laser-induced thermal acoustics can be subdivided into two steps. In the first one, light from a powerful pulsed laser is split into three phase-coherent beams. Two of them intersect at a shallow angle within a gas mixture. The interference of the laser beams generates an electric field intensity grating. Within one interval of periodicity, called fringe spacing, the photon intensity varies with space and time as plotted in Fig. 10.1. The laser frequency is tuned as to excite electronically a rare species of the mixture. However, in course of this nonlinear interaction, the excited species lose their internal energy due to inelastic collisions with the dominant species. Since this effect occurs periodically, a stationary acoustic wave is triggered. The second step involves scattering of the third laser beam into a coherent signal beam by opto-acoustic effects. The signal beam is modulated by the evolution of laser-induced perturbations. From these modulations, accurate physical properties of the medium may be inferred, including the

262

Lecture Notes on the Discretization

of the Boltzmann

Equation

Fig. 10.1 Temporal and spatial evolution of the photon intensity within one interval of periodicity.

sound speed, thermal diffusivity and other properties that affect the development of the laser-induced gratings. Therefore, deep insight and understanding of the dominant processes is very important. On physical grounds, the formation and evolution of these gratings has been treated using the equations of fluid dynamics in its linearized form [32]. This assumes uniform behavior of the medium and small perturbations. In addition, the assumption of linear hydrodynamic behavior eliminates a variety of phenomena which could affect the signals in real experiments. Moreover, using hydrodynamic equations assume scales of gradients of fluid properties to be small compared to molecular mean-free paths. This implies that for large Knudsen numbers (ratio between a laboratory length and the mean free path) or in situations in which nonlinear effects gain importance, a microscopic approach must be used. A model describing the medium on a microscopic level, which allows the investigation of the evolution of each component of the medium, consists of a set of Boltzmann equations for inelastically interacting gas mixtures [11; 19]. These very complicated coupled nonlinear integro-differential equations oppose direct analytical and numerical treatment [11]. One way to cope with this problem is the discretization of the velocity space [27]. A discrete velocity model of Shizuta type [38] with 36 velocities is well suited to describe the formation and dynamics of laser-induced density and thermal gratings. Section 10.2.1 presents the physical situation on a molecular level together with the dominant interaction processes. Section 10.2.2 discusses the most relevant features of the used discrete velocity model.

Discrete Models of the Boltzmann

Equation in Quantum

Optics

263

The corresponding discrete kinetic equations and some important aspects of equilibrium are presented in Sec. 10.2.3. In Sec. 10.2.4 the discrete kinetic equations are solved numerically by applying the fractional step method in order to demonstrate the formation and evolution of the laserinduced gratings in different situations. 10.2.1

Physical

situation

The region of intersection of two driver laser beams (Fig. 10.2) defines the sample volume of the gas mixture, which is composed of the molecules A, their electronically excited state A* and a second, usually dominant species B with masses TUA, TUA* and TUB, respectively. The laser frequency v is tuned to excite the electronic transition between the molecules A and A*. The energy gap of the transition is given by AE = E(A*) — E(A) = hi/. On a microscopic level, the molecules undergo the following types of interaction processes: (1) Elastic scattering between the molecules, M + N ^ M + N

for

M,N = A, A*,B;

(2) collisional excitation of molecules A, A + A -»• A* + A,

A + B^A*

+ B;

(3) collisional de-excitation of molecules A*, A* + A -» A + A,

A* + B->A

+ B;

(4) absorption of photons p by molecules A, A+p^

A*;

(5) spontaneous emission of photons p by molecules A*, A* ->• A+p; (6) and stimulated emission of photons p by molecules A*, A*+p->A+p

+ p.

The elastic particle-particle scattering processes are modeled by elastic hard sphere cross sections 0MN-*MN W ^ t n M,N = A, A*,B. The collisional excitation processes are indicated by the cross sections GJLA^AA*

264

Fig. 10.2 volume.

Lecture Notes on the Discretization

of the Boltzmann

Equation

The region of the intersection of the two driver laser beams defines the sample

and V1AB^>A'B- The inelastic cross sections O™A*-+AA anc * G%A*B^AB f° r the inverse processes (collisional de-excitation) are determined by a microreversibility condition [19]. Interactions between photons p of the pulsed driver laser and particle species A. and A* are taken into account by means of the Einstein coefficients a and /3 [29]. Spontaneous emission of photons is controlled by a. The magnitude of absorption and stimulated emission phenomena is proportional to 0. 10.2.2

Discrete

velocity

model

The physical situation sketched above in Sec. 10.2.1 is essentially one-dimensional in real space. However, when treating four-wave mixing in a two-dimensional phase space on a kinetic level, energy and momentum conservation rule out all nontrivial collisions. Therefore, we have to choose at least a two-dimensional discrete velocity model. For good relaxational behavior, multiple speed collisions in which three and four different speeds occur are crucial. This can be realized by means of the Shizuta type model [38] with 36 velocities and six speeds of relative magnitude a, y/5a, 3a, y/l3a, 5a, y/29a (Fig. 10.3). Later on the scaling parameter a is adapted to fit the physical situation. The Shizuta type model allows for elastic multiple speed collisions between all species, as for instance (V4,V22) ^

(V9,V16),

( v i , V 2 l ) i=i ( v 8 , V i 6 ) ,

(V3,V29) ^

(V7,V25),

( V 1 3 , V 3 2 ) ±=, ( V i 7 , v 2 7 ) -

At least 50% of all binary collisions involve more than two speeds. In the case of four-wave mixing, inelastic collisions represent collisional

Discrete Models of the Boltzmann 31

34

Fig. 10.3 speeds.

Equation in Quantum 26

28

Optics

265

38

33

Regular discrete velocity model of Shizuta type with 36 velocities and six

excitation and de-excitation processes. Thus, the model must provide such interactions for an appropriately fixed energy gap q. Shizuta type models meet this requirement for several possible values of q. For the energy gap q = 12mAd2 (assuming rriA = rriA*) exemplary collisions involving four different speeds are given by (v 5 ,v 3 3 ) i=i (v 4 ,vi 5 ) +q, (vio,v 2 9 ) <=; ( v i , v i 3 ) + q, (vi3,v 3 2 ) ^ (vi,v 2 0 ) + q. The above considerations imply that the collisional scheme shows all features necessary to meet the requirements of a kinetic description of fourwave mixing. It is clear that establishing a discrete velocity model consisting of a rather high number of velocities and three distinguishable species demands sophisticated computational techniques. The major problem to overcome is the high number (about 104) of admissible collisions inflating the collisional term of the kinetic equations. A computer program developed by Hanser [21] yields the complete collisional scheme. The input data of this program are the velocity vectors, the number of species, the masses of each species and the types of interactions taken into account.

266

10.2.3

Lecture Notes on the Discretization

Discrete

Boltzmann

of the Boltzmann

Equation

equations

The governing equations for the number densities of all three species of the gas mixture are a set of coupled nonlinear Boltzmann equations. Equipped with the complete collisional scheme of the system, the evolution equations

(& + v? • ^ ) N? = JtA[KM}

+ nf[NM,i},

+ v f • ^ ) KA' = JA' [NM] - K?[NM, I],

(jt

(|+vf-A)iVf

=^ [ ^ ]

(10.1)

(10.2)

(10.3)

can be formulated for the number densities N^(-x,t) of the molecules M (M = A, A*,B) with velocity vf1 (i = 1,...,36) at the point x at time t, where jV M is an abbreviation of (Nf,...,N$$). The right hand sides of Eqs. (10.1) - (10.3) are the collision terms due to the interaction processes described in Sec. 10.2.1. Interactions between the radiation intensity and the particle species A and A* are taken into account by Tl?[NM,I]

= (a + fiI)N? - (3INtA,

where J = I(x, t) is the laser intensity. The functionals jf\N_M], jf [N_M] and JiB[N_M] are the collision terms concerning interactions between the particles. Each particle-particle collision event (elastic scattering, collisional excitation and de-excitation) generates an expression of quadratic nonlinearity in the number densities of the molecules. The particular form reads as

Ak^{NfN^'-N^Nf), with M', N', M,N = A, A*,B. The transition rates A^j are defined in the following way: Ak,l A

_ el(in) ,, i,j —aMN-yM'N'\\vi

M

_

V

Nn, M _ V JV| *,J V j l^** j \ai,o'

The transition probability of each particle-particle event is taken into account by O j ' . For more detailed information about symmetry relations

Discrete Models of the Boltzmann

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267

of A*j see [27]. The complete expressions of the collision terms are very cumbersome and therefore not quoted here. The macroscopic quantities of interest are the number densities of each species nM, the total number density n, the mass density of each species pM, the mass density of the whole mixture p, the average gas velocity of each species u M , the average gas velocity of the whole mixture u, the average internal kinetic energy of each species eM and the average internal kinetic energy of the whole mixture e. The species related macroscopic quantities are denned in the usual way: 36

M

n

= Y1N^

for

M = A,A*,B,

(10.4)

for

M =

(10.5)

i=l

pM=mMnM -.

A,A*,B,

36

(10.6) P

3 6

1

eM =

4=1

1

lmM ,M _ UM)2 NM_

J_y

(10.7)

i=l

The quantities for the whole gas mixture are defined as follows: n = s£jnM,

(10.8)

M

M 36

1

u=-££m

36

M

v,

M

iVf,

(10.10)

i=l

M

1

M

1

8=1

It should be noted that all macroscopic quantities in Eqs. (10.4) - (10.11) are functions of space x and time t.

268

Lecture Notes on the Discretization

of the Boltzmann

Equation

The state of the whole gas mixture in the sample volume before the powerful driver laser pulse starts interacting with the gas species A and A* is assumed to be a thermodynamical equilibrium state. Equilibrium is established if all particle-particle collision terms of Eqs. (10.1) - (10.3) simultaneously vanish [11]. This means JiA[KA,NA',KB] Jf'lN*,^' A

A

= 0,

(10.12)

,K ] = o,

(10.13)

= 0.

(10.14)

B

B

jf[K ,K \K ]

Equilibrium concerning the elastic particle-particle scattering processes is fulfilled by the following expressions for the number densities: ' NtM = aM exp [c • v f - /3|vf| 2 ] ,

(10.15)

where aM, c and /3 are the so-called Maxwellian parameters. In addition, demanding equilibrium of the collisional excitation and de-excitation processes yields the condition aA' = aA exp [-/3q],

(10.16)

where the quantity q is the difference of the pre- and post-collisional kinetic energies. Since the laser frequency is assumed to be tuned to excite the electronic transition of species A and A*, the internal energy q is equal to hv. The Doppler effect, though essential for a variety of photon-gas interactions such as laser cooling [36], is of no importance in laser induced thermal acoustics and therefore neglected. The resulting independent Maxwellian parameters aA, aB, c, and P are related by a one-to-one map to the independent macroscopic quantities of the system [27]. It can easily be shown that c — 0 implies uM = u = 0 and vice versa. However, it is not possible, in general, to express the Maxwellian parameters as functions of the independent macroscopic quantities in terms of analytic functions. Despite this shortcoming, a thermodynamical equilibrium state can be fixed by simply specifying the independent Maxwellian parameters. 10.2.4

Simulations

The numerical solution of the set of coupled discrete Boltzmann equations, i.e., Eqs. (10.1) - (10.3), for suitable initial and boundary conditions reveals

Discrete Models of the Boltzmann

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269

the formation and evolution of the laser induced thermal and density gratings. Since the period of the intensity grating formed by the two driver laser beams (the fringe spacing A) is very small compared with the extension of the sample volume, the extension of the intensity grating and the sample volume is assumed to be infinite. Based on this assumption, it is sufficient to consider only one fringe spacing together with periodic boundary conditions. The fringe spacing depends on the laser frequency and the angle of intersection of the laser beams [30]. Typical values range from l ^ m to 30/xm. For the simulations presented here, this value is fixed to A = 6/mi. The scaling parameter a of the discrete velocity model is determined by resorting to the fact that a sufficiently small perturbation propagates with the speed of sound, which is around 330ms _ 1 for a gas like air at standard conditions. A value of a = 500ms _ 1 fulfills this requirement and adjusts the underlying discrete velocity model to our physical situation (see also Sec. 10.4). For the number density of molecules B, a typical value at standard condition is nB = 10 2 5 m - 3 . Molecules A and A* are considered to be of low concentration: nA + nA* = 1 0 _ 3 n B . Without loss of generality, the mass ratio is set to mAlmB = 1. This allows the use of only one discrete velocity model: wA = vf* = vf, which is a good approximation for NO, NO*,and02. The temporal and spatial profile of the intensity in the sample volume is described by the formula [30] T,

I(t,x)

N

T

2

t

= lo^-exp

cos

(H

where x G [—A, A] and t G [0, oo). A graphical representation is sketched in Fig. 10.1. The quantity T, fixed to 8 ns, indicates the duration of the laser pulse. The strength of the laser intensity is controlled by Jo- A chosen value for the product f3Io = 109 s _ 1 causes the ratio nA*/nA to be approximately 10~ 2 (weak pumping limit). The effect of spontaneous emission modeled by the Einstein coefficient a is for the formation and evolution of the gratings of minor importance. Nevertheless, this effect is taken into account by assigning a the value 105 s _ 1 . On a microscopic level, the different physical situations are characterized by the cross sections introduced in Sec. 10.2.1. To simulate these situations, six elastic and two inelastic cross sections are specified. The variety of

270

Lecture Notes on the Discretization

of the Boltzmann

Equation

possible variations is restricted by the assumption that all elastic and the two inelastic cross sections have the same value. This reads in detail as °el

= VMN-+MN

for

M, N = A, A*,

B,

and „in in " — "AA^AA*

—

„in a AB^>A-B-

To characterize the dominant gas species B, the Knudsen number defined by Kn =

XaelnB

is used. Based on the fractional step method [2; 25; 34], numerical simulations are performed for a wide range of Knudsen numbers by simultaneously varying the inelastic cross sections. The following three-dimensional plots show the temporal and spatial evolution of the relative particle and kinetic energy density deviations nB(t,x)-nB{0,x) nB(0,x)

eB(t,x)-eB(0,x) eB{0,x)

The two-dimensional graphs represent the temporal evolution of these quantities in the middle of the fringe spacing. Low Knudsen

numbers

In this range of Knudsen numbers, the gas mixture behaves as a fluid and can also be treated in the framework of fluid dynamics. This provides the possibility to compare the results gained by the discrete Boltzmann equation with those from the fluid dynamic approach. Since the perturbations are small, linearized fluid dynamic equations apply. In this approach, the laser energy deposit is taken into account by a gain term on the right of the energy equation [30]. Transport coefficients entering these equations, namely the thermal conductivity and the kinematic viscosity, are obtained by the Chapman-Ensk0g perturbation technique [22] for monatomic hard sphere molecules. Figure 10.4 presents some results of this comparison, which show a good agreement between the fluid dynamic and the kinetic model. In particular, the structures of the density and energy oscillations match. The small discrepancies between the two approaches are attributed

Discrete Models of the Boltzmann

(\ La: er pule

\

ft

A/ VA1 A/ *A V

d yV rN »\

Equation in Quantum

Optics

271

.J h i A/ 1 VNn / \ A ^

1 ^ V

V

v

»

T ,

T

"•

1 Li scr pi be 210

240

270

300

Time (ns)

120

150

ISO

210

240

270

300

Time (ns)

Fig. 10.4 Time evolution of the relative mass (left) and energy density deviation (right) in the middle of the fringe spacing. The dashed and solid lines show the results obtained by the fluid dynamic and the discrete kinetic model (Kn = 0.0033), respectively.

e spacing (^m)

Time (ns)

Fig. 10.5 Time-space evolution of the relative particle (left) and kinetic energy density deviation (right) inside the fringe spacing for a Knudsen number Kn = 0.0167 and an inelastic cross section
to the discrete character of the kinetic model. Furthermore, a faster energy deposit into the system is observed in the case of the fluid dynamic model, where the deposition of the laser energy is described by a one-step process. In the kinetic model, however, the physically more realistic two-step process, namely excitation of a rare species and deposit of kinetic energy by collisional de-excitation events, is taken into account. This explains the slower deposit of the laser energy in the latter case. Nevertheless, the simulations using the discrete velocity model give a valid picture of the physics of thermal gratings in the fluid dynamic limit. Figures 10.5 and 10.6 (left) show interesting results of the discrete veloc-

272

Lecture Notes on the Discretization

/^

erpul e

h" \

of the Boltzmann

1

(VA

D ffiN d JwW W 11t (

/v

30

60

90

120

150

Lase pulse

My/

V1

1

^ \j

ill i / V/

(b)

0

Equation

H

V^

180

210

240

270

Tune (ns)

300

0

10

20

30

40

50

60

70

80

90

100

Time (ns)

Fig. 10.6 Relative density deviation in the middle of the fringe spacing for an inelastic cross section atn = l n m 2 . The curves (a), (b), and (c) in the left plot correspond to Knudsen numbers Kn with values 0.0033, 0.0083, and 0.0167, respectively. The curves (a), (b), (c), and (d) in the right plot refer to Knudsen numbers Kn with values 0.021, 0.033, 0.083, and 0.167, respectively.

Fig. 10.7 Relative kinetic energy deviation in the middle of the fringe spacing for a Knudsen number Kn = 0.0167 (left) and Kn = 0.167 (right), respectively. The curves (1), (2), (3), (4), (5), (6) and (7) correspond to inelastic cross sections <xm with values l n m 2 , 0.1 n m 2 , 0.05nm 2 , 0.02nm 2 , 0.01 n m 2 , 0.005 nm 2 and 0.001 nm 2 , respectively.

ity model for Knudsen numbers ranging from Kn = 0.0033 to Kn = 0.0167 and Fig. 10.7 (left) for inelastic cross sections from alH = 0.005 nm 2 to am = l n m 2 . The gas mixture responds to the laser pulse with damped oscillations for both the number density and kinetic energy. The damping rate of the oscillations depends strongly on the Knudsen number as can be seen in Figures 10.6 (left). For a Knudsen number Kn = 0.003, the gas mixture oscillates more strongly and much longer than for Kn = 0.0167. For high values (strong quenching) the increase of the kinetic energy is

Discrete Models of the Boltzmann

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273

accompanied by oscillations, whereas for low values the energy increases monotonously and spatially uniformly. This is demonstrated in Figs. 10.5 (right) and 10.7 (left). Medium

Knudsen

numbers

In this range the equations of fluid dynamics gradually lose their validity. An unexpected behavior of the gas mixture exposes Fig. 10.6 (right). The temporal density oscillations disappear almost completely with increasing Knudsen numbers. In other words, at Kn = 0.083 the mechanisms controlling the damping effects reveal their strongest influence. Figure 10.7 (right) shows the increase of the kinetic energy at Kn = 0.167 for several inelastic cross sections. For low values of am the deposition of the internal energy of the gas species A* cannot take place rapidly, and therefore, the kinetic energy of the whole gas mixture increases very smoothly. High Knudsen

numbers

This is the region where a molecular gas kinetic model must be used instead of a fluid dynamic approach. As mentioned before, the density oscillations disappear until a Knudsen number Kn = 0.083 is reached. However, the density oscillations regenerate with increasing Knudsen numbers as demonstrated in Fig. 10.8. For Kn = 1.67, the gas mixture is a Knudsen gas, which means that the gas behavior is mainly determined by the periodic boundary conditions. This explains the re-appearance of the density oscilla-

Fig. 10.8 Time-space evolution of the relative density deviation inside the fringe spacing for a Knudsen number Kn — 1.67 and an inelastic cross section aln = l n m 2 .

274

Lecture Notes on the Discretization

of the Boltzmann

Equation

tions. Moreover, the character of the oscillations gains more and more nonlinear features. Even the kinetic energy no longer increases monotonously as can be seen in Fig. 10.9. 6

A

5.5

e © •-a .3

t

erg

•o

.H

a

.a .* -: & _,©

5 4.5

(b)

(a)

4

\, /

3.5

3
2.5

(c)

>

2 1.5

]

\L

i - -

OJ 0

)

10

20

be

30

40

50

60

70

80

90

100

Time (ns)

Fig. 10.9 Relative kinetic energy deviation in the middle of the fringe spacing for an inelastic cross section cln = l n m 2 . The curves (a), (b), (c), and (d) correspond to Knudsen numbers Kn with values 0.21, 0.33, 0.83,and 1.67, respectively.

To sum up, the study shows that a regular discrete velocity model of Shizuta type with 36 velocities and six speeds proves capable of describing gas mixtures that interact with monochromatic photons. For low Knudsen numbers, the model provides the same results as obtained by numerically solving the linearized equations of fluid dynamics. Moreover, the discrete kinetic approach provides the description of the evolution of such scenarios also for high Knudsen numbers.

10.3

Modified discrete Boltzmann equation

This section addresses the connection between the continuous and the discrete theory and set up the corresponding discrete Boltzmann equations. By introducing appropriate velocity domains to cover the whole physically relevant velocity space a new generalized discrete Boltzmann equation is derived from the continuous one. Establishing an J?-theorem requires certain symmetry relations concerning the transition rates. This in turn must be consistent with the fact that the size of the cells is not uniform. The new discrete Boltzmann equation is then applied to the two-dimensional "Union Jack" model. This two-dimensional hierarchical discrete velocity model is simple, however, regular since it provides sufficiently many

Discrete Models of the Boltzmann

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275

mixing speed collisions to ensure energy transfer among particles of different speeds. Finally, by assuming the Krook-Wu scattering model, a comparison is given by the temporal evolution of the distribution function obtained from this discrete velocity model with an exact solution to the continuous Boltzmann equation, namely, the famous BKW mode.

10.3.1

The continuous

Boltzmann

equation

For rarefied gases the temporal and spatial evolution of the distribution function / ( x , v, t) is governed by the nonlinear Boltzmann equation [11;12]

ijt + V ' V ) /(X'V'<} = °[f] ~ L[f]'

(1 17)

°-

where the loss term L[f] is given by L[f] = [ dw\v - w| J dv' / dw' v', w -> w')/(x, v, i ) / ( x , w, t), (10.18) and the gain term G[f] reads G[f] = fdw

fdv'

f dw'|v' - w > ' ( v ' -^ v, w' ^ w ) / ( x , v', t ) / ( x , w', t) (10.19)

in probabilistic formulation. Here a(\ -¥ v ' , w -> w') denote the differential cross sections for the direct collision and er'(v' —> v, w' -> w) the corresponding inverse one. The total cross section, 5 ( | v - w | ) = J dv' J dw'a(v -> v ' , w ^ w') = f dv' f d w V ( v ' -^ v, w' -)> w ) ,

(10.20)

depends only on the relative speed of the particles. Due to conservation of momentum and energy, the velocities before and

276

Lecture Notes on the Discretization of the Boltzmann Equation

after the collision are connected by v'=i(v+w) + i|v-w|fi',

(10.21a)

w ' = I ( v + w )_I| v _ w |n\

(10.21b)

| v - w | = |v'-w'|.

(10.21c)

The unit vectors „

V—W

fi = i

,

_,

r and SI'

|v'-w'|

indicate the direction of the relative velocities of the particles before and after the collision. 10.3.2

Connecting

the continuous

and discrete

theory

To establish a discrete velocity model for the continuous Boltzmann equation, a set of velocities v; (i = 1,2,..., M) must be chosen. Then the physically relevant velocity space V C Rd is subdivided into a disjoint set of M arbitrary domains AVi with the properties Vj € AVi and V = U f c i ^ V j . Physically relevant means that only velocities up to a maximal speed t>max are considered. The number of particles with speeds greater than i»max is neglected. The approximation f(x,vi,t)Avi

= fiAvi = Niiit /

dv/(x,v,i)

JAVi

must be applied within each domain AVi, where fa denotes the discrete distribution function / ( x , Vj,t), Nj the number of particles in AV,, and Avi = J"AV. dv represents the size of the cell AVi. By integrating Eq. (10.17) with respect to v over AVi, the left side results in /

d V A V j

(^

+ V

-

V

)

/ ( X

'

V

'*

) W

( s

+ V i

'

V

)

M

^

(10 22)

'

Performing the same integration for the loss term, Eq. (10.18), yields M

/

dvL[f] « Avi^Avjfri

- Vj\

dv' dw'a(vi -> v',Vj -> w ' ) / i / j , (10.23)

Discrete Models of the Boltzmann

Equation in Quantum

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277

where
_,

v

> . -> w') = £ S{\Vi - Vj\) O J J ^ V ' - Vk)S(w' - V,). (fc,0 (10.24)

The symbol J2
= 1

-

(10-25)

(M) In this way, the loss term .

M

/AVi dvL[/\« J2

Av AV

* Jiv< - v ^ 5 (i v ^ - v^i) E ° ( S t o

(10-26)

(*,*)

J=I

is finally obtained. The integration of the gain term, Eq. (10.19), results in / dvG[f] « Avi y~] AVJ |Vj - Vj | / dv' / rfwV(v' ->• v*, w' -> Vj) JAVi ~[ J J x/(x,v',t)/(x,w',*), (10.27) where a'(v' —> v ; , w ' —>• v,-) indicates that the velocities v and w of the scattered out particles have already been discretized. To express the discretization of the pre-collisional velocities v' and w', it is again necessary to represent the differential cross section with the help of Dirac's J-function as a'(v' -»• vit w' -»• v,) = £

S(Wi - Vj\)a'{{;p(v'

- v*)«J(w' - v , ) ,

(W)

where X)(jt,() a(k'i) ~ 1' w m c h is consistent with Eq. (10.20). The gain term reads . M

/AVi dvG[f] « E

'

J=I

Au

< A ^l v * - v ;l 5 d v * - v^l) E (*,o

a

'lk1)fkfh

(10-28)

278

Lecture Notes on the Discretization

of the Boltzmann

Equation

and the discrete Boltzmann equation using Eqs. (10.22), (10.26) and (10.28) can now be written in the form

^

'

j=i

(*,0 The same result can be obtained when starting from the kinetic formulation of the Boltzmann equation [35]. The probabilities a L ' j and o'A'n have yet to be determined. Being aware that £ V ^ is equivalent to | J2k=i ^2i=i and introducing the transition coefficients A\^]

:= AviAvjK

-

Vj-15(|Vi

- v, |) ajfjj,

(10.30a)

A'g$

:= A ^ A ^ v , - v , | S ( | V i - v,-|) af$,

(10.30b)

Eq. (10.29) can be expressed in the compact form Ft

\

i

M

(10-31)

| + v, • Vj fiAVi = i J ] W5?/** - ^(fjj/i/i) • j,k,l=l

The coefficients -A^'-j and the probabilities a^'.l have the property that ^j*jj ^ 0 and ajjjj ^ 0 if and only if V i + V j = v^+v, and v f + v | = vf+vf. Furthermore, the relations A^'1) = A{,k'1} = A?.'k). = A(}:h} imply that oL'-\ = a(,-'i) = a (i',)

= a

(i'i)- For t n e term A'A1'^ analog relations hold. It

should be noted that a relation between A\t '.l and A^l has not yet been established. Such a symmetry is necessary to obtain an .//-theorem. 10.3.3

H-theorem

Before proving an .ff-theorem for Eq. (10.31), it is convenient to introduce a diagonal matrix B with entries Ba := v; • V as well as a vector F with components 1

M

F,(U,U):=- £ j,k,l=l

(A'ltfjUM-AWUiUi).

(10.32)

Discrete Models of the Boltzmann

Equation in Quantum

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279

Let N be the vector with components Ni and f the vector with components /;. Then, the discrete Boltzmann equation (10.31) can be expressed in vector notation: f

+

BN = f

= F(f,l).

(10.33)

The scalar product M

-.

M

toF(U,U)>:=5>*i(U,U) = - Y, i=l

^{A'Si)U"U'-AliSU*Ui)'

i,j,k,l=l

(10.34) denned for an arbitrary vector

1

<0,F(U,U)>=- Y, ^i + ^iKtjJUkUt-A^UiUj) i,j,k,l=l M

1

- g E (^ + ^ ) K , ? ) ^ ^ - ^ f f i ! ) ^ - (10-35) i,j,fc,(=l

Now, by requiring A

(k,i)

-

A

(k,i)

a n a

A

(i,j)

_/1

(i,i)'

uu.dbj

Eq. (10.35) reduces to 1

(0, F(U, U)> = -

M

E

(& +fc- ^* - 0i) ( A < $ 17*J7, - A\^]UiUj) . (10.37)

Next, the H-function is introduced in the special form M N ln

H = Y i fi-

( 10 - 38 )

i=l

With the help of Eqs. (10.33) and (10.37) and by setting & = 1 + In/*, its time derivative reads as dH dt

1

M

= I i E_x (ln H ) W > * - 4'5^/i) •

dO-39)

280

Lecture Notes on the Discretization

of the Boltzmann

Equation

If it is additionally required that

then it follows from Eq. (10.30) that AviAvj aj*;!} = AvkAvt

agj,

(10.41)

and Eq. (10.39) can be written in the form M

dt

sJ-LC-sfcX'-w)^™-

iM2

< >

Equation (10.40) is a consequence of microreversibility and central interaction laws typical for monatomic gases. Hence, the reverse collisions can be disposed of. By introducing the quantities m

a(M) = jM_t

m

=

a(i,i) ^ L . ,

( 1 0 .43)

and applying Eqs. (10.30), (10.36), and (10.40), the discrete Boltzmann equation (10.31) now becomes M

Q-t + Vi • V J fiAvi = - ^

AviAvjAvkAvilvi

- v^Sflvi -

x ftjjjj (/ fc /, - /
Vj|)

(10.44)

and Eq. (10.41) simply reads

As the probabilities aL'-j have to fulfill the condition given by Eq. (10.25), the coefficients b).'.{ must obey the relation Y, b$]AvkAv, = 1. (fc,0

(10.46)

Since the product (In y-j-) (1 — 7 / ) < 0 , t n e equilibrium is given by fofj = fkflt

which implies the discrete Maxwell Boltzmann distribution fi = Cexp(-E(vi-D)2),

(10.47)

Discrete Models of the Boltzmann

Equation in Quantum

Optics

281

where C = C(x), D = D(x), and E — E(x) are constants to be determined from the macroscopic quantities defined by

M

n(x,t) = ^2f(x,Vi,t)Avi,

(10.48a)

i=l

1 M u(x, t) = - V v 4 /(x, Vi, t) A«i, n z—'

(10.48b)

4=1

eth = - £

m p (

V

f(^ut)AVi.

(10.48c)

«=l

Here, n(x, t) is the total number density of the particles, u(x, t) the bulk velocity, eth the average thermal energy and mp denotes the particles mass. It should be noted that the concept of temperature in the kinetic theory has not a clear status due to the lack of Galileo invariance of discrete velocity models as discussed in detail in [13]. For this reason, the term average thermal energy eth is used instead.

10.3.4

The hierarchical

"Union Jack"

model

The new modified discrete Boltzmann equation (10.44) is now applied to the two-dimensional 'Union Jack' model [28] shown in Fig. 10.10. It is the simplest hierarchical model that allows mixing speed collisions, where at least one of the post-collisional speeds differs from both pre-collisional ones. This means that energy can be transferred among different hierarchies. The eight possible directions are indicated with the subscript m. Along the vx- and Vy-ax.es, the speeds are given by u j , = 1 357 = hAv with Av = "max/^max and h = 1,2,..., /i m a x , whereas in the diagonals, the speeds are ^=2,4,6,8 = y/2hAv. The hierarchy h = 0 consists only of the velocity vR = 0. The maximum hierarchy /i m a x is associated with the maximum discrete speed u m a x (along the axes m = 1,3,5,7) which corresponds to a maximum energy of 50 eth- Thus the maximum speed is 10 if v is measured in units of y/eth/m^.

282

Lecture Notes on the Discretization

of the Boltzmann

Equation

•m—\

m=2

m=8

*•

\

m=3

AV« 771=4

771=6 771=5

Fig. 10.10

The two-dimensional hierarchical Union Jack model.

The Union Jack model permits the following bimolecular collisions: (v?,v£)o(v£,v?)

(10.49a)

ft

(v2 ,v6VK,vJ)

(10.49b)

h

(vj\v£)o(v2\v5 )

(10.49c)

(vf,vj)«(vf,vj)

(10.49d)

(vf,vj)^(v*,vg)

(10.49e)

fc

(v5,v*)*»(vl ,v°) (vi,v»)^(v^

+9)/2

(10.49f) ,v^

_9)/2

)

with

l
(10.49g)

For Eqs. (10.49c)-(10.49g) the corresponding symmetric collisions must also be considered. The number of possible collisions grows moderately with

Wax). With the above collision scheme, the model is regular for /i m a x > 1. It possesses 4 collisional invariants, which correspond to preservation of mass,

Discrete Models of the Boltzmann

Equation in Quantum

Optics

283

Fig. 10.11 Relaxation time T r e j a x obtained from the discrete Boltzmann equation Eq. (10.44) for the BKW mode Eq. (10.60) depending on the cell size ratio r.

momentum in x- and y-direction and energy. Consequently, the model does not decompose into independent subsystems. An appropriate partition of the velocity space is indicated by the dotted lines in Fig. 10.10. The domains AV^ are radially centered to each discrete velocity vector v ^ . For the domain sizes Av^ the conditions Auf = Av$ = Avi> = Av? and Av% = Av% = Av$ = Avg hold because of the symmetry of the model. The chosen partition yields the following relations between the domain angles -

7T

r

TV

a = r + 2' and the ratio of the domain sizes A^m=2,4,6,8

(10.50)

A

<=1,3,5,7'

In detail, the domain sizes are given by (At;) 2 r + 1 A«g = ir2 r + 2'

•K

h

A <1>3 3 j 55i 77 = v (AvY '' ' r + 2'

A

<4,6,8=rA<3,5,7"

(10.51) The study of the temporal evolution of a non-equilibrium distribution towards equilibrium obtained by applying the discrete Boltzmann equation (10.44) as well as the continuous one shows that the relaxation time r r e i a x for the discrete model is slower (cf. 'BKW Mode' and 'DVM ( K = l ) ' in Fig. 10.13). In Fig. 10.11 the relaxation time (defined by the time that has

284

Lecture Notes on the Discretization

of the Boltzmann

Equation

elapsed until deviation from equilibrium is smaller than 2 percent) obtained from the discrete Boltzmann equation Eq. (10.44) is shown for the twodimensional BKW mode Eq. (10.60) depending on different cell size ratios r. For the selected value of r = 1.7, Treiax becomes a minimum. This choice of r corresponds to the domain angles a = 48.65° and (1 = 41.35°. It should be noted that this minimum of r re i a x at r = 1.7 can also be observed for other initial conditions. The reason for applying the partition shown in Fig. 10.10 is that for a normalized continuous two-dimensional Maxwell Boltzmann distribution with u = 0, /(v)rfv = ^ exp (- y ) dv,

(10.52)

the corresponding discrete equilibrium distribution reads for the Union Jack model

^ = C l 27 e X P ( - C , 2 f)'

(1 53)

°-

with coefficients (At;)2 + O {(Av)*) ,

Ci = l-

g^y

C2 = l~

r + 1 (A,;)2 + 12 (r + 2)

°

((AV)4)

(10.54a) (1

°"54b)

obtained from Eqs. (10.48) after some tedious calculation. If the interval Av approaches zero, then both C\ and C
exp ( ~ e ^ ( K * - iuxf

+ «?„)) .

(10.55)

The terms a, e, and 7 are a measure for the deviation from the continuous Maxwellian. These three factors are calculated from the macroscopic quantities Eqs. (10.48) with a multidimensional, globally convergent Newton method [34] for 200, 2 000 and 20 000 hierarchies. This study shows that a, e, and 7 barely depend on the number /i m a x of hierarchies if /i m a x is large. Their dependence on average thermal energy eth is shown in Fig. 10.12. As expected, all three values tend to 1 if eth becomes large compared to the

Discrete Models of the Boltzmann

Equation in Quantum

Optics

285

elh/(m„u, )

Fig. 10.12

Factors a, e, 7 in Eq. (10.55) dependent on average thermal energy e t h-

energy m p w£/2. However, for relatively small values of e t h, the deviation from the continuous equilibrium distribution is significant. It should be noted that the value of 7 is not far from 1 which means that the value of D in Eq. (10.47) is almost the bulk velocity u. 10.3.5

Comparison

with the BKW

mode

This section presents a test of the Union Jack model as well as the new modified discrete Boltzmann equation by applying them to the Krook-Wu scattering model [3; 24]. It is now useful to introduce the abbreviation / £ := / ( x , v ^ , i ) . In this and the following section, only a spatial homogeneous gas with bulk velocity u = 0 is considered, which implies //* = / £ = f£ = f^ and f£ — f£ = fe = fit- I n t m s case > o n l y t n e three collision types Eqs. (10.49e), (10.49f) and (10.49g) contribute in a nontrivial manner to the discrete Boltzmann equation (10.44). The total number of nontrivial collisions is given by max*

Assuming Maxwell molecules, then in the two-dimensional case, d — 2, the differential cross section reads

or c (|v-w|,n-n') =

K 27TV-

w

which leads to the total cross section

5(|v-w|) =

K v —w

(10.56)

286

Lecture Notes on the Discretization

of the Boltzmann

Equation

by applying Eq. (10.20). Setting K = 1, the Boltzmann equation (10.17) simplifies in the spatial homogeneous case to d_ f(v, t) = jdwj dt

d n ' ^ [f(v',t)f(w',t)

- / ( « , t)f(w, t)].

(10.57)

In accordance with Ernst [18], an exact solution of Eq. (10.57), namely the two-dimensional BKW mode, can be written as

/&V(«)dv = ^ « p

2g

2s - 1 1 - s 2 dv, + ~^~ 2s 3 V

(10.58)

with s(t) = l + riexp(-£j,

-^<J?<0,

\<s
(10-59)

It is assumed in Eq. (10.58) that the microscopic and macroscopic quantities are measured in units such that m p = 1, n — 1 and eth = 1The discrete analog of the continuous BKW mode is given by 2N

/^VK) = | e x p ( - C 2 | )

2 s _- 1

1 -_s ^

o

(10.60)

where C\ and C2 are normalization constants. By applying Eqs. (10.48), Cl

=l ~

2J

6^VT\{Av)2

+

° ((Aw)4)'

(10 61a)

-

9« — 1 r 4- 1

C2 = 1

- -T2^7h{Av?+°{{Av)4)

(10 61b)

-

are found after some algebra. It should be noted that Eqs. (10.54) are special cases of Eqs. (10.61) for s = 1. For t -»• 00 and consequently s -> 1, the BKW mode approaches the Maxwell Boltzmann distribution. Since in the limiting case, Av -> 0, both C\ and C2 tend towards 1, the discrete distribution function, Eq. (10.60), approaches the continuous solution, Eq. (10.58), asymptotically. The discrete version of Eq. (10.57) can be simply obtained by inserting the total cross section, Eq. (10.56), into Eq. (10.44): 8 -fiAvi

1 M = 5 E ^AvjAvkAv, j,k,t=l

K6<*jj (/*/, - / , / , - ) .

(10.62)

Discrete Models of the Boltzmann

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287

In the next step the probabilities a;*'!? needed for evaluating the coefficients &[*'!v by means of Eq. (10.43) are determined. The collision scheme, Eqs. (10.49), shows that two possible outputs always exist for a collision (VJ,VJ) -» < [(v*,Vj)

and the inverse collision

(v*, vj) ->• < UVi.Vj)-

The probabilities a have to fulfill the relations given by Eqs. (10.25) and Eqs. (10.41):

(10.63) These are three equations with four unknowns. Since the scattering is isotropic, the probabilities a do not depend on the domain sizes of the pre-collisional velocities but only on the domain sizes of the velocities after collision. Hence, setting (ij)

= const

i AvjAuj,

a

(ki)

= const

2 AviAvj,

a)k'J. = const 2 AujtAuj

a

results in consti = const2 = it follows that h(k,l)

_

h(i,j)

\ij) — consti AvfcAuj,

1/(AUJAUJ

_ u(iJ) _

h(k,l)

+ AiifcA?;;), and from Eq. (10.43) _

%,j) ~ °(k,i) ~ °(i,j) ~ °(k,i) ~

1 Av.Av.

+

AvkAVl'

By integrating Eq. (10.62) numerically, the temporal evolution of the discrete distribution function shown in Fig. 10.13 is obtained. Setting T) = — \ in Eq. (10.59) corresponds to an initial condition with / ( v = 0, t = 0) = 0, showing a maximum deviation from the Maxwell Boltzmann distribution. The discrete velocity distribution splits into two branches corresponding to particles with velocities along the axis and diagonals, respectively. This means that the relaxation differs for those two directions. Moreover, the average relaxation for K = 1 is too slow compared with the BKW-mode. This is a general problem of two-dimensional models as has been proved by Bobylev et al. [4; 5; 30]. To overcome this shortcoming, the total cross section is adjusted in such a way that the discrete macroscopic quantities, e.g. the .^-function, show the same relaxation behavior as the continuous macroscopic quantities. This is demonstrated for various total cross sections in Fig. 10.14. For a value of K = 6, the .H-function obtained

288

Lecture Notes on the Discretization

of the Boltzmann

Equation

,

- BKW Mode DVM (K=l) 0 - - - DVM (K=6)

t =0 0.16 « 0

0.12

BKW Mode DVM(K=1) DVM (K=6)

0.16

v 0.08

0.08 '

S?0

\\

/' /

*s I 0.04

0.04

• X .•/

\

2 v t=4 0.16

>- \

x 0

0.12 0.08 0.04

• • - . - . /

\

2 v 1=16 BKW Mode DVM(K=1) DVM (K=6)

0.16< X

V

0.04

x 0

\^

0.12 "Vv 0.08

\

.

>

\ x -A

BKW Mode DVM(K=1) DVM (K=6)

V\ \ \ \\ \

Fig. 10.13 Temporal evolution of the two-dimensional BKW mode given by Eq. (10.58) and of the numerical solution of the corresponding discrete Boltzmann equations (10.62) applied to the Union Jack velocity model (DVM).

from the analytical solution, Eq. (10.58), and the discrete one obtained from integration of Eq. (10.62) show the same temporal behavior. The correction factor K — 6 is also the best match for other values of r) in Eq. (10.59), which correspond to different initial distributions. The numerical calculations have been performed with a maximum hierarchy /i m a x = 200 using a Cash-Karp Runge Kutta routine as well as a Bulirsch-Stoer integration routine taken from [34]. To conclude, this new approach permits relating results of the discrete Boltzmann equations to continuous velocity distributions of real gases. The discrete transport equations are derived from the continuous Boltzmann equation and conserve mass, momentum and energy. The subdivision of the velocity space requires a representation of the differential cross section by means of generalized functions, where the occurring discrete scatter-

Discrete Models of the Boltzmann

A. / Z

Equation in Quantum

'

BKWMode DVM

\ \\

\

\ N

i\

\ M

^x

ii

2.8

\ V \V s

2.84

289

'

v 2.76

Optics

v

N

-

*-

\ < ~

K = 11 ~

v

*"

K

=

^~ := "

~"~.

K=l

5

'

" "

10

:

15

Fig. 10.14 The .ff-functions of the two-dimensional BKW mode given by Eq. (10.58) and of the numerical solution of the corresponding discrete Boltzmann equations (10.62) for different total cross sections (DVM).

ing probabilities must obey symmetry relations in order to obtain an Htheorem in the form H = J2i=i -W» m / » - This if-theorem corresponds to a Maxwellian for the discrete distribution function /;. The partition of the velocity space allows the relaxation time to be minimized by choosing the domain angles properly. The discrepancy of the solutions to the discrete Boltzmann equations from the solutions to the continuous one requires the adjustment of the total cross section for the discrete model, which leads to the same temporal behavior in both cases. Using two properly scaled 'Union Jacks', the new model can be extended to binary mixtures as well.

10.4

Scaling of discrete velocity models

When applying discrete velocity models to simulate sophisticated physical situations, the problem of choosing appropriate values for certain model parameters is encountered. The adjustment of a discrete velocity model to a given physical problem by scaling the velocity moduli and the total cross section is one of the most crucial points in the discrete kinetic theory. It is a good choice to use the speed of sound to scale the velocities of a discrete velocity model.

290

Lecture Notes on the Discretization

10.4.1

The problem

of the Boltzmann

Equation

of scaling

For a wide range of kinetic problems, the resulting discrete kinetic equations are of the following form:

{jt+v? • J L ) K { t ' x ) = j m ] + Q m + n m ' j ] '

(io 64)

-

for i = 1 , . . . , n and a = 1 , . . . , M. The nonlinear functional J?[N] takes into account all elastic scattering processes among the particles and consequently dominates the relaxation towards equilibrium. The term Qf [N] describes inelastic interactions. Examples of this are chemical reactions and collisional excitation or collisional de-excitation events. Supposing that a monochromatic radiation field is present and one particle species absorbs and emits photons, a third term, denoted by 7£"[JV, I], appears in Eqs. 10.64. The quantity / indicates the intensity of the radiation field. Triple and multiple collisions are not considered in these investigations. They are used to break spurious invariants in order to make a discrete velocity model regular, or for other physical reasons. When radiation transport becomes important, the discrete kinetic equations have to be combined with the equation of radiation transport [33]:

! + cfl. £ )/(*,*) = cfo, (a + /3/)£A7-/3/5>f i=l

(10.65)

t=l

Absorption and emission of photons are modeled by means of the Einstein coefficients a and /?. The particle species, described by the number densities Nf, indicates atoms at the ground state. Conversely, atoms at the excited state are denoted by Nf. The energy of the radiation field hv equals the energy gap between the excited and the ground state. The quantity c denotes the speed of light and fl is the unit vector in the direction of motion. Inspection of the streaming and the collision terms of Eqs. 10.64 reveals the problem concerning which magnitudes should be chosen for the velocities v? to match the physical situation as realistically as possible. There is no general procedure based on first principles which enables the selection of these magnitudes. To ensure that a hierarchical model is regular, the magnitudes of the velocities have to fulfill a certain law of growth. However, there is in general one parameter left to scale the system. This scaling parameter influences the "performance" of a discrete kinetic model, since

Discrete Models of the Boltzmann

Equation in Quantum

Optics

291

some macroscopic quantities, especially the speed of sound, depend on it. Hence, an optimal value for this parameter is crucial. Such a scaling can be performed by resorting to the fact that in the case of small Knudsen numbers (fluid dynamic limit) a sufficiently small perturbation propagates with the speed of sound. The most important feature, which a regular discrete velocity model has to meet, is a sufficiently good relaxation behavior. This requirement is fulfilled when the number of admissible elastic collisions compared to the number of velocities is high and mixing-speed collisions exist. Mixing speed collisions ensure that at least one of the post-collisional speeds differs from both pre-collisional ones. This means that energy can be transferred to several hierarchies. Another important point is to provide a certain amount of kinetic energy corresponding to the internal energy gap by implementing appropriate inelastic collisions. The existence of a variety of collisions with various speeds yielding the same amount of kinetic energy release is essential. Shizuta [38] and Union Jack type models [28] can be quoted as examples (Fig. 10.15). Both models prove regular, i.e. they do not display spurious collision invariants. Table 10.1 lists data for a Shizuta model of 31

26

30

34

28

35

Fig. 10.15 Hierarchical and regular discrete velocity models: Shizuta (left) and Union Jack type (right).

36 velocities and a Union Jack one for 25 velocities. It reveals differences in this special case in favor of the Shizuta model because the ratio between the total number of collisions and the number of velocities is much higher

292

Lecture Notes on the Discretization

of the Boltzmann

Equation

Table 10.1 Elastic and inelastic collision scheme for the Shizuta and Union Jack model. The numbers without brackets hold for the particles of one species. The numbers in brackets belong to the inelastically interacting particles of two species. One-, two-, three-, and four-speed collisions denote collision events where one, two, three and four different speeds, respectively, are involved.

Features Number of velocities Number of speeds Number of directions Zero velocity One-speed collisions Two-speed collisions Three-speed collisions Four-speed collisions Total number of collisions

Shizuta 36 6 28 no 21 92 (144) 32 (240) 64 (96) 209

Union Jack 25 6 8 yes 6 20(8) 20 (24) 0(0) 46

for this model. The Union Jack model does not enable collisions with four different speeds for elastic and inelastic interactions. The high number of multiple-speed collisions implying a more satisfactory relaxation behavior is especially remarkable for the Shizuta model. 10.4.2

The scaling

procedure

To demonstrate the scaling procedure for the two discrete velocity models, the following physical situation at a molecular level is considered. A gas mixture composed of three particle species undergoes all types of elastic scattering. Species 1 and 2 (excited state of 1) interact with the photons p of a radiation field by the following events: - absorption: - spontaneous emission: - stimulated emission:

1 + p —> 2, 2 —> 1 + £>, 2 + p —> 1 +p + p.

The linear momentum transfer from photons to particles is neglected. In addition, atoms 1, 2, and 3 give rise to: - collisional excitation: - collisional de-excitation:

1 + 1 —> 1 + 2, 1 + 2 —> 1 + 1,

1 + 3 —> 2 + 3, 2 + 3 —> 1 + 3.

Discrete Models of the Boltzmann

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293

To implement these inelastic interaction processes in the discrete velocity models, the following inelastic collisions are assumed: Shizuta: Shizuta: Union Jack: Union Jack:

K,vJ3) (vio>v|9) (V2»VM)

(V4,v? 2 )+gt/j

(V20> v !)

including those obtained by simple symmetry operations. The difference between the pre-collisional and post-collisional kinetic energies are given by qs = 1 2 m a | and quj = 12ma\,j. For simplicity, equal particle masses mi = m,2 = mz = m are assumed implying v | = v? = vf. The particleparticle interaction terms are tremendously inflated and cannot be quoted here. Shizuta

Union Jack

€ ^

Rel. dens

•s

—

eviat

i.:i '

A/ yy V />-'

\A M\A

(A r lr

r

60

90

/y

V"

/"-1

*/

i

V

30

'N a ' \ © A

r

•a

VV , nr 120 150 180 210 240 270 300

Time [ns]

0

30

«0

90

120 150 180 210 240 270 300

Time [ns]

Fig. 10.16 Relative density deviations in the middle of the slab. The density oscillates with the frequency of the sound wave. The temporal evolution of the laser pulse intensity in arbitrary units is indicated on top of the figures.

Equipped with a system of discrete model Boltzmann equations (10.64), the procedure of velocity scaling for the two discrete velocity models can be performed. To excite the system, a sufficiently small perturbation is imposed to obtain sound waves. This can easily be achieved by a pulsed and spatially periodic photon field with the intensity I(t,x) = IQ exp{—(t/r)2} cos2(irx/X), x £ [—A/2, A/2], where r = 8 ns is a measure for the duration of the laser pulse and A = 6 /xm indicates the spatial period. It is an advantage of a periodic field that only one spatial period has to be considered. Thus, one has to treat a one-dimensional slab problem with periodic boundary conditions. The number density of the dominant species 713 is assumed to

294

Lecture Notes on the Discretization

of the Boltzmann

Equation

be 1 0 2 5 m - 3 . In order to ensure a small perturbation by the laser field, a low-concentration value of 10 22 m - 3 for the sum of the number densities of species 1 and 2 is selected. The cross section for all collisional de-excitation events is a^ = l n m 2 . The cross sections for the inverse interactions are determined by microreversibility conditions. The first results obtained by solving the initial boundary value problem by means of the method of operator splitting are displayed in Fig. 10.16. The graphs show oscillations of the relative density deviation from the equilibrium of species 3. These damped density oscillations correspond to stationary sound waves. In this simulation a scaling parameter a = 500 m s - 1 has been used, leading to a speed of sound of 318 m s - 1 and 314 m s - 1 for the Shizuta and Union Jack model, respectively. The dependence of the speed of sound on the scaling parameters, theoretically expected to be linear [27], is shown in the left graph in Fig. 10.17. A value of 3 3 1 m s - 1 for the speed of sound yields values for the scaling parameters of as = 520ms _ 1 (Shizuta) and auj = 524 m s - 1 (Union Jack). 360 ISO

s

340 1& vtn

u

3 a > •O e s

o EC

Shi zut i

.4 *

320 310 OKI

im

f -^

/'

'S
s

S 'S *>

Un ion Jat k

450 460 470 480 490 500 510 520 530 540 550

Scaling parameter fm/s] Fig. 10.17

Speed of sound at Kn = 1/300 versus the scaling parameter.

Based on experience, the Union Jack model relaxes more slowly than the Shizuta model because of its smaller number of possible collisions. This discrepancy can be removed approximately by adjusting the cross sections. Such a scaling of cross sections to match the relaxation behavior according to an analytic solution of the Boltzmann equation is shown in Sec. 10.3.5, where the temporal evolution of the .ff-function of the two-dimensional BKW-mode [3; 24] is compared to the H-function of the corresponding Union Jack model. It turned out that the same temporal behavior can be obtained by increasing the total elastic cross section for the Union Jack

Discrete Models of the Boltzmann

Equation in Quantum

S > ^0.5

I;

f\ /v

r

V.T 1 •S

3

30

60

*\ /",

, r

-4 I A r-4-Vr \\ I 0,

\ \ r \>

nrz 1 !

90

295

Union Jack

Shizuta 1.5

Optics

h A \

h

\

V

120 150 180 210 240 270

180 210 240 270

Time [ns]

Time [ns]

Fig. 10.18 Relative deviation of the density in the middle of the slab for a Knudsen number Kn = 0.17. The peaks on top of the figures indicate the temporal evolution of the laser intensity in arbitrary units. The comparison of the two diagrams shows a slightly delayed reaction of the gas on the laser pulses for the Union Jack model.

Union Jack

Shizuta

r, r

r\ / \

t>

r>

r

"0

<——

J f

'

30

_J_,

90 120 150 180 210 240 270

Time [ns]

r

f\r

'd

(

»!,

A." 60

'N / \

a

rJ 1

•

V.

etic

/

r\r

t

f~ 1 r^

a o 1 30

60

f

I

r

90

120 150 180 210 240 270

Time [ns]

Fig. 10.19 Relative deviation of the kinetic energy in the middle of the slab at a Knudsen number Kn = 0.17. The peaks on top of the figures indicate the temporal evolution of the laser intensity in arbitrary units. Each laser pulse increases the kinetic energy of the gas by the same amount, as expected.

model by a factor of seven. The results concerning a more complex simulation based on the above chosen scaling parameters, where the system is excited by an irregular series of six pulses, are shown in Figs. 10.18 and 10.19. Despite the scaling of the cross sections the magnitudes of the oscillations of the density, evaluated with the Union Jack model, are smaller and more damped compared to those obtained with the Shizuta model. Obviously, it is not possible to

296

Lecture Notes on the Discretization

of the Boltzmann

Equation

compensate completely for the better relaxation behavior of the Shizuta model by this scaling of the elastic cross section of the Union Jack model. The laser pulses not only give rise to density oscillations but also to an increase of the kinetic energy of the particles. Figure 10.19 shows that each laser pulse transmits radiation energy to the gas. Compared to the density, the evolution of the energy coincides qualitatively for the two applied discrete velocity models. However, corresponding to the delays in the density deviations the same delay of approximately 5 to 10 ns for the energy jumps is observed. The results show that discrete velocity models are capable of simulating rather complicated effects. However, a suitable scaling of the velocities and the cross sections is necessary to adjust the model to the given physical situation.

10.5

References

[1] N. Bellomo, T. Gustafsson, On the initial and initial-boundary value problem for the discrete Boltzmann equation, Review Math. Phys., 3 (1992), 137-162. [2] N. Bellomo, Lecture N o t e s on the Mathematical Theory of the Boltzmann Equation, World Scientific (1995). [3] A. V. Bobylev, Exact solutions of the Boltzmann equation, Sov. Phys. Dokl, 20 (1976), 822-824. [4] A. V. Bobylev, A. Palczewski, J. Schneider, Discretization of the Boltzmann equation and discrete velocity models, in Rarefied Gas Dynamics 19, J. Harvey and G. Lord, eds., Oxford University Press (1995), 857-863. [5] A. V. Bobylev, A. Palczewski, J. Schneider, On approximation of the Boltzmann equation by discrete velocity models, C. R. Acad. Set. Paris 320 (1995), 639-644. [6] J. E. Broadwell, Study of rarefied shear flow by the discrete velocity method, J. Fluid Mech., 19 (1964), 401-414. [7] J. E. Broadwell, Shock structure in a simple discrete velocity gas, Phys. Fluids 7 (1964) 1243-1248. [8] C. Buet, A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics, Trans. Th. Stat. Phys., 25 (1996), 33-60. [9] H. Cabannes, Etude de la propagation des ondes dans un gaz a 14 vitesses, J. de Mecanique 14 (1975), 705-744.

Discrete Models of the Boltzmann

Equation in Quantum

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[10] H. Cabannes, The Discrete Boltzmann Equation (Theory and Application), University of California, Berkeley (1980). [11] C. Cercignani, The Boltzmann Equation and Its Applications, Springer (1988). [12] C. Cercignani, R. Illner, M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer (1994). [13] C. Cercignani, Temperature, entropy, and kinetic theory, J. Stat. Phys., 87 (1997), 1097-1109. [14] C. Cercignani, Ludwig Boltzmann. The Man W h o Trusted Atoms, Oxford University Press (1998). [15] H. Cornille, Nested-squares discrete Boltzmann models with arbitrary number of velocities satisfying a continuous theory relation, Trans. Th. Stat. Phys., 26 (1997), 359-371. [16] E.B. Cummings, Laser-induced Thermal Acoustics, Ph.D. thesis, California Institute of Technology, Pasadena (1995). [17] A.C. Eckbreth, Laser Diagnostics for Combustion Species and Temperature, Abacus Press (1988). [18] M.H. Ernest, Nonlinear model Boltzmann equations and exact solutions, Phys. Rep., 78 (1981), 1-171. [19] C.R. Garibotti, G. Spiga, Boltzmann equation for inelastic scattering, J. Phys. A 27 (1994), 2709-2717. [20] R. Gatignol, Theorie Cinetique des Gaz a Repartition Discrete de Vitesses, Springer (1975). [21] F. Hanser, Applications of Discrete Velocity Models to Rarefied Gas Mixtures Interacting with Monochromatic Photons, Ph.D. thesis, Graz University of Technology, Graz (2000). [22] J.O. Hirschfelder, C.F Curtiss , R.B. Bird, Molecular Theory of Gases and Liquids, Wiley (1964). [23] T. Inamuro, B. Sturtevant, Numerical study of discrete-velocity gases, Phys. Fluids A 2 (1990), 2196-2203. [24] M. Krook, T.T. Wu, Formation of Maxwellian tails, Phys. Rev. Lett. 36 (1976), 1107-1109. [25] L. Lapidus, G.F. Pinder, Numerical Solutions of Partial Differential Equations in Science and Engineering, Wiley (1982). [26] J. C. Maxwell, Scientific Papers II, Cambridge University Press (1890). [27] R. Monaco, L. Preziosi, Fluid Dynamic Application of the Dis-

298

[28]

[29] [30]

[31]

[32]

[33] [34] [35]

[36] [37] [38] [39]

Lecture Notes on the Discretization

of the Boltzmann

Equation

crete Boltzmann Equation, World Scientific (1991). H. Nievoll, P. Griehsnig, P. Reitre, F. Schiirrer, A general discrete velocity model including internal degrees of freedom, Complex Systems 10 (1996), 417-435. J. Oxenius, Kinetic Theory of Particles and Photons, Springer (1986). A. Palczewski, J. Schneider, A. V. Bobylev, A consistency result for a discrete-velocity model of the Boltzmann equation, SIAM J. Numer. Anal, 34 (1997), 1865-1883. A. Palczewski, J. Schneider, Existence, stability and convergence of solutions of discrete velocity models to the Boltzmann equation, J. Stat. Phys., 91 (1998), 307-326. P.H. Paul, R.L. Farrow, P.M. Danehy, Gas-phase thermal-grating contributions to four-wave mixing, J. Opt. Soc. Am., B 12 (1995), 384392. G.C Pomraning, The Equations of Radiation Hydrodynamics, Pergamon Press (1973). W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes in C, Cambridge University Press (1995). P. Reiterer, C. Reitshammer, F. Schiirrer, F. Hanser, T. Eitzenberger, New discrete model Boltzmann equations for arbitrary partitions of the velocity space, J. Stat. Phys., 98 (2000), 419-440. C. Reitshammer, F. Schiirrer, A. Rossani, Atomic beam slowing and cooling: Discrete velocity model, Phys. Rev. E 58 (1998), 3964-3970. F. Rogier, J. Schneider, A direct method for solving the Boltzmann equation, Trans. Th. Stat. Phys., 23 (1994), 313-338. Y. Shizuta, S. Kawashima, The regular discrete models of the Boltzmann equation, J. Math. Kyoto Univ., 27 (1987), 131. S. Williams, L.A. Rahn, P.H. Paul, J.W. Forsman, R.N. Zare, Laserinduced thermal-grating effects in flames, Opt. Lett., 19 (1994), 16811683.

LIST OF CONTRIBUTORS

Babovsky H . TU Ilmenau, Fakultat, Mathematik/Naturwiss, Postfach 10 0565, D-98684 Ilmenau, Germany; e-mail: babovskyQmathematik. tu-ilmenau. de.

Bellomo N . Department of Mathematics, Politecnico Torino, Coso Duca degli Abruzzi 24, 10129 Torino, Italy; e-mail: b e l l o m o Q p o l i t o . i t . Bernhoff N . Department of Engineering Sciences, Karlstad University, Universitegatan 1, 61188 Karlstad, Sweden; e-mail: BerrnhoffQkau.se. Bobylev A. Department of Engineering Sciences, Karlstad University, Universitegatan 1, 61188 Karlstad, Sweden; e-mail: Alexander. BobylevQkau. se. Buet C. Commissarial a l'Energie Atomique, B.P. 12, 91680 Bruyeres le Chatel, France; e-mail: christophe.buetQcea.fr

Cercignani C. Milano;

Department of Mathematics, Politenico Milano, 10122

e-mail: CarcerQmate.polimi.it.

Cordier S. Laboratoire Analyse Numerique, URA CNRS 189, University Pierre et Marie Curie, 4 Place Jussieu, 75 252 Paris Cedex 05, France; e-mail: c o r d i e r Q a n n . j u s s i e u . f r .

299

300

List of

Contributors

Gatignol R. Laboratoire Modelisation Mecanique k, CNRS, Univ. Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cedex 5, France; e-mail: g a t i g n o l O c i c r p . j u s s i e u . fr. Gorsch D . TU Ilmenau, Fakultat, Mathematik/Naturwiss, Postfach 10 0565, D-98684 Ilmenau, Germany; e-mail: GoerschOmathematik. tu-ilmenau. de.

Koller W . Institute fur Theoretische Physik, Tech. Univ. Graz, Austria; e-mail: kowinQitp. t u - g r a z . a c . a t . Pareschi L. Department of Mathematics, University of Ferrara, Italy; e-mail: [email protected]. Preziosi L. Department of Mathematics, Politecnico Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy; e-mail: p r e z i o s i Q p o l i t o . i t . Rondoni L. Department of Mathematics, Politecnico Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy; e-mail: rondoniQcalvino . p o l i t o . i t . Schilder F. TU Ilmenau, Fakultat, Mathematik/Naturwiss, Postfach 10 0565, D-98684 Ilmenau, Germany; e-mail: SchilderSmathematik. tu-ilmenau. de.

Schurrer F . Institute fur Theoretische Physik, Tech. Univ. Graz, Austria; e-mail: SchuerrerQitp. t u - g r a z . a c . at.

Series on Advances in Mathematics for Applied Sciences Editorial Board N. Bellomo Editor-in-Charge Department of Mathematics Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino Italy E-mail: [email protected]

F. Brezzi Editor-in-Charge Istituto di Analisi Numerica del CNR Via Abbiategrasso 209 1-27100 Pavia Italy E-mail: [email protected]

M. A. J. Chaplain Department of Mathematics University of Dundee Dundee DD1 4HN Scotland

P. L. Lions University Paris Xl-Dauphine Place du Marechal de Lattre de Tassigny Paris Cedex 16 France

C. M. Dafermos Lefschetz Center for Dynamical Systems Brown University Providence, Rl 02912 USA

B. Perthame Laboratoire d'Analyse Numerique University Paris VI tour 55-65, 5ieme etage 4, place Jussieu 75252 Paris Cedex 5 France

M. A. Herrero Departamento de Matematica Aplicada Facultad de Matematicas Universidad Complutense Ciudad Universitaria s/n 28040 Madrid Spain S. Kawashima Department of Applied Sciences Engineering Faculty Kyushu University 36 Fukuoka 812 Japan M. Lachowicz Department of Mathematics University of Warsaw Ul. Banacha 2 PL-02097 Warsaw Poland S. Lenhart Mathematics Department University of Tennessee Knoxville, TN 37996-1300 USA

K. R. Rajagopal Department of Mechanical Engrg. Texas A&M University College Station, TX 77843-3123 USA R. Russo Dipartimento di Matematica Universita degli Studi Napoli II 81100 Caserta Italy J. C. Willems Mathematics & Physics Faculty University of Groningen P. O. Box 800 9700 Av. Groningen The Netherlands

Series on Advances in Mathematics for Applied Sciences Aims and Scope This Series reports on new developments in mathematical research relating to methods, qualitative and numerical analysis, mathematical modeling in the applied and the technological sciences. Contributions related to constitutive theories, fluid dynamics, kinetic and transport theories, solid mechanics, system theory and mathematical methods for the applications are welcomed. This Series includes books, lecture notes, proceedings, collections of research papers. Monograph collections on specialized topics of current interest are particularly encouraged. Both the proceedings and monograph collections will generally be edited by a Guest editor. High quality, novelty of the content and potential for the applications to modem problems in applied science will be the guidelines for the selection of the content of this series.

Instructions for Authors Submission of proposals should be addressed to the editors-in-charge or to any member of the editorial board. In the latter, the authors should also notify the proposal to one of the editors-in-charge. Acceptance of books and lecture notes will generally be based on the description of the general content and scope of the book or lecture notes as well as on sample of the parts judged to be more significantly by the authors. Acceptance of proceedings will be based on relevance of the topics and of the lecturers contributing to the volume. Acceptance of monograph collections will be based on relevance of the subject and of the authors contributing to the volume. Authors are urged, in order to avoid re-typing, not to begin the final preparation of the text until they received the publisher's guidelines. They will receive from World Scientific the instructions for preparing camera-ready manuscript.

SERIES ON ADVANCES IN MATHEMATICS FOR APPLIED SCIENCES Vol. 17

The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solution by C. Soize

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Calculus of Variation, Homogenization and Continuum Mechanics eds. G. Bouchitte et al.

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Advances in Kinetic Theory and Computing: Selected Papers ed. B. Perthame

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Waves and Stability in Continuous Media eds. S. Rionero and T. Ruggeri

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Impulsive Differential Equations with a Small Parameter by D. Bainov and V. Covachev

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Mathematical Models and Methods of Localized Interaction Theory by A. I. Bunimovich and A. V. Dubinskii

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Recent Advances in Elasticity, Viscoelasticity and Inelasticity ed. K. R. Rajagopal

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Nonstandard Methods for Stochastic Fluid Mechanics by M. Capinski and N. J. Cutland

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Impulsive Differential Equations: Asymptotic Properties of the Solutions by D. Bainov and P. Simeonov

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The Method of Maximum Entropy by H. Gzyl

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Lectures on Probability and Second Order Random Fields by D. B. Hernandez

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Parallel and Distributed Signal and Image Integration Problems ecte. R. N. Madan et al.

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On the Way to Understanding The Time Phenomenon: The Constructions of Time in Natural Science - Part 1. Interdisciplinary Time Studies ed. A. P. Levich

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Lecture Notes on the Mathematical Theory of the Boltzmann Equation ed. N. Bellomo

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Singularly Perturbed Evolution Equations with Applications to Kinetic Theory by J. R. Mika and J. Banasiak

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Mechanics of Mixtures by K. R. Rajagopal and L Tao

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Dynamical Mechanical Systems Under Random Impulses by R. Iwankiewicz

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Oscillations in Planar Dynamic Systems by R. E. Mickens

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Mathematical Problems in Elasticity ed. R. Russo

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On the Way to Understanding the Time Phenomenon: The Constructions of Time in Natural Science — Part 2. The "Active" Properties of Time According to N. A. Kozyrev ed. A. P. Levich

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Advanced Mathematical Tools in Metrology II eds. P. Ciarlini et al.

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Mathematical Methods in Electromagnetism Linear Theory and Applications byM. Cessenat

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Constrained Control Problems of Discrete Processes by V. N. Phat

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Motor Vehicle Dynamics: Modeling and Simulation by G. Genta

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Microscopic Theory of Condensation in Gases and Plasma by A. L Itkin and E. G. Kolesnichenko

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Advanced Mathematical Tools in Metrology III eds. P. Cianlini et al.

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Mathematical Topics in Neutron Transport Theory — New Aspects by M. Mokhtar-Kharroubi

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Advances in Nonlinear Partial Differential Equations and Stochastics eds. S. Kawashima and T. Yanagisawa

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Propagation and Reflection of Shock Waves by F. V. Shugaev and L S. Shtemenko

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Homogenization eds. V. Berdichevsky, V. Jikov and G. Papanicolaou

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Lecture Notes on the Mathematical Theory of Generalized Boltzmann Models by N. Bellomo and M. Lo Schiavo

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Plates, Laminates and Shells: Asymptotic Analysis and Homogenization by T. Lewihski and J. J. Telega

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Advanced Mathematical and Computational Tools in Metrology IV eds. P. Ciarlini et al.

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Differential Models and Neutral Systems for Controlling the Wealth of Nations by E. N. Chukwu

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Mechanical and Thermodynamical Modeling of Fluid Interfaces by R. Gatignol and R. Prud'homme

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Numerical Methods for Viscosity Solutions and Applications eds. M. Falcone and Ch. Makridakis

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Stability and Time-Optimal Control of Hereditary Systems — With Application to the Economic Dynamics of the US (2nd Edition) by E. N. Chukwu

Vol. 61

Evolution Equations and Approximations by K. Ito and F. Kappel

Vol. 62

Mathematical Models and Methods for Smart Materials eds. M. Fabrizio, B. LazzariandA. Mono

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Lecture Notes on the Discretization of the Boltzmann Equation eds. N. Bellomo and R. Gatignol

Lecture notes on the mathematical theory of the Boltzmann equation

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Lecture Notes on Jurisprudence (Lecture Notes Series)

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Lecture Notes On The Mathematics Of Acoustics

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Lecture Notes On The Mathematics Of Acoustics

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Lecture Notes on Cryptography

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Notes on Dynamical Systems (Courant Lecture Notes)

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Lecture Notes on Intermediate Thermodynamics

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Lecture notes on maximum principles

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Lecture notes on general relativity

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Lecture Notes on Diophantine Analysis

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Lecture notes on motivic cohomology

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Lecture Notes on General Relativity

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Lecture Notes on Forensic Medicine

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Lecture Notes on Complex Analysis

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Lecture notes on nilpotent groups

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Lecture notes on weak interactions

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