ELSEVIER Mathematics and Computers in Simulation 38 (1995) 211-216

~ i MATHEMATICS AND

COMPUTERS N SIMULATION

Stochastic particle methods and approximation of the Boltzmann equation

Wolfgang Wagner Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D-IO117 Berlin, Germany

Abstract

Stochastic particle methods for the numerical treatment of the Boltzmann equation for dilute monatomic gases are considered. One particular stochastic model of particles with weights is introduced. Recent convergence results for this model are discussed. The introduction of certain degrees of freedom seems to be helpful for the reduction of the random fluctuations of the particle scheme.

1. Stochastic systems as a numerical tool

The Boltzmann equation for dilute monatomic gases is a nonlinear integrodifferential equation of the form

0 -~ f ( t , x, v) + (v, Vx) f ( t , x, v)

= f dw f deB(v,w,e)[f(t,x,v*(v,w,e)) f(t,x,v*(w, R3 S 2

v,e)) -- f ( t , x , v ) f ( t , x , w ) ] ,

(1)

t ~> to, x E G , v C /I~ 3 with appropriate initial and boundary conditions (cf. [4], or [10], for details). In Eq. (1) , G is a bounded domain in the three-dimensional Euclidean space ~3, V~ denotes the vector of the partial derivatives with respect to x, de denotes the uniform surface measure on the unit sphere S 2, and dw denotes the Lebesgue measure. The function B is called the collision kernel, and v* denotes the collision transformation

v * ( v , w , e ) = v + e ( e , w - v ) , v, wEIt~ 3, e c S 2, (2)

where ( . , . ) is the scalar product in ~3. Eq. ( 1 ) describes the time evolution of a distribution function

0378-4754/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 4 7 5 4 ( 93 ) E 0 0 8 4 - I

2 1 2 W. Wagner~Mathematics and Computers in Simulation 38 (1995) 211-216

f ( t , x , v ) > ~ O , f dx f d v f ( t , x , v ) = l , (3) G ~,3

where fA dxfB d v f ( t , x , v) represents the average relative number of gas particles with position x inside the volume A and velocity v inside the volume B. So the solution f ( t , x, v) of Eq. (1) depends on six variables (besides the time variable t). Problems with such a high dimension are a typical field of application of Monte Carlo techniques (numerical techniques using random numbers).

In many applications there is an interest not in the values of the solution f of the Boltzmann equation but in some integral functionals (e.g,, density, mean velocity, temperature) of the form

f dx f dv~(x,v) f(t,x,v), (4) G R3

where ~p is an appropriate test function. Therefore the basic objects to be evaluated are the measures

A(t, dx, dv) = f ( t , x, v) dx dv, (5)

where f is the solution of Eq. (1). Now particle systems

/1 (xi(t),vi(t))i=l , t >>, to, xi(t) E G, vi(t) E ~3, (6)

arise in a natural way as approximations to the solution of the Boltzmann equation in the sense that the corresponding empirical measures

n

~(") ( t, dx, dr) = _1 ~ 8( x,( O.vi( o ) ( dx, dr) , (7) 11. ~_

where 8 denotes the Dirac measure, are close to the measures (5). In particular, the particle system (6) may be a stochastic process.

The numerical method based on this approach consists in the simulation of the particle system (6) and in the approximation of the functional (4) by the integral with respect to the empirical measure (7), i.e.

W(x,v) A(t, dx, dv) ,~ - ¢ ( x i ( t ) , v i ( t ) ) . (8) 1"/ i=1

G R3

Stochastic particle methods of the type (6), (8) are a basic tool for the numerical treatment of the Boltzmann equation (cf. the survey articles [11,6,9,5]). The classical simulation scheme (cf. [2]) was derived on the basis of physical intuition, but there has been some progress in the mathematical foundation of various stochastic particle methods in recent years (cf. [ 1,14-16], concerning convergence results).

One of the main directions of the development of these numerical methods is the generalization to more realistic physical situations including internal degrees of freedom, chemical reactions, stochastic boundary conditions. However, from a mathematical point of view, another problem seems to be even more challenging - the study and the improvement of the efficiency of the schemes (cf., e.g., the discussion in [ 3] ). In particular, the reduction of the random fluctuations around the deterministic limit in (8) is of great importance. This problem has been tackled by the introduction of low

w. Wagner/Mathematics and Computers in Simulation 38 (1995) 211-216 213

discrepancy sequences instead of sequences of random numbers in some parts of the algorithm (cf. the description of the "finite pointset method" in [ 12] ).

A more general approach to the problem of variance reduction is to develop models with certain degrees of freedom, i.e. with such parameters that do not influence the limit but can be chosen in order to reduce the fluctuations. The purpose of this paper is to introduce one particular model of such type and to discuss recent convergence results for this model.

2. Stochastic systems of particles with weights

We consider the spatially homogeneous case, in which the Boltzmann equation (1) takes the form

- ~ f ( t , v ) = dw d e B ( v , w , e ) [ f ( t , v * ( v , w , e ) ) f ( t , v * ( w , v , e ) ) - f ( t , v ) f ( t , w ) ] , (9) R3 S 2

t ~ to, v E Il~ 3. Instead of (6), we consider particle systems of the form

(gi( t) . ~. ~m~,) ,vitt))i=t , t >~ to, gi(t) E [0, 1] , wi(t) E/~3, (10)

where the quantities gi( t ) are interpreted as "weights" of the particles, and m ( t ) is the number of particles in the system. Instead of (7), the empirical measures corresponding to the system (10) take the form

re(t)

~'(") ( t, dr ) = ~ gi( t) 8~,(t)( dr) . i=I

(11)

A stochastic system of the type (10) has been introduced in [8] and called "random discrete velocity model". In this model, a time-independent set of "discrete velocities" (vi)i"=~ was used. Moreover, the model was given on a time discretization ( tk) , k = 0, 1 . . . . . The random evolution of the weights (gi(t))i~=~ was defined on the basis of a Broadwell-type equation. It has been shown that the empirical measures (11) converge to the solution of a time-discretized version of Eq. (9). Weighted particles have been considered in connection with the finite pointset method in [ 13 ].

Now we will describe a stochastic system of the type (10) (more precisely, a Markov jump process) such that the empirical measures (11) converge to the solution of Eq. (9) . Suppose that m( to) = n and

gi(to) E [ O , 2 ] , i = 1 ..... n. (12)

Suppose that the collision kernel B is bounded, i.e.

O ~ B ( v , w , e ) < ~ B m a x , v, w E ~ 3, e E S 2. (13)

Given a state z = (g~, vl . . . . . gin, Vm), the process waits a random time having exponential distribu- tion with the parameter

2 1 4 W. Wagner~Mathematics and Computers in Simulation 38 (1995) 211-216

m ( m - 1) fl = 4 7"r Bmax (14)

n

The term 4 rr is the surface measure of the unit sphere. After that time, the process jumps into a state f . If m ~> 2 n - l , then ~ = z. If m ~< 2 n - 2 ,

then ~ is calculated as follows. Two indices 1 ~< i < j <~ m are chosen at random with uniform distribution. A random element e is chosen uniformly distributed on the unit sphere S 2. Two new particles 5m+l = v*(vi, Uj, e) and Urn+2 = V*(Vj, Vi, e) are generated, where v* is defined in (2). The weights are changed according to the formulas

gi = gi -- G ( z , i , j , e ) ,

~j= g j - G ( z , i , j , e ) , (15)

g,,+l = g,,+2 = G(z, i , j , e),

where

n B(vi, v~, e) G(z, i, j, e) - gi gj.

2 Bmax

Thus, the evolution of the process consists of random collisions of particles. During a collision the pre-collisional particles do not disappear. They create post-collisional particles and give them a part of their weights. Notice that m( t) <~ 2n, t >~ to, and

gi(t) E [ 0 , 2 ] , i = l ..... m ( t ) , t >~ to. (16)

Let p denote the bounded Lipschitz distance between two probability measures/zl and/x2 defined a s

: supi [ [ q~( v ) l-t2( dv ) I , ~ED d d

where

D = {~o:II~3--, [0,1]; I~o(x) -q'(Y)I ~< Ilx-Yll}.

Consider the measures A(t) related to the solution of Eq. (9) via (5), and the empirical measures ( 11 ). The following convergence result holds (cf. [ 17] ).

T h e o r e m . Suppose that

sup g (") f Ilvllh'.(t0, dr) n

R 3

and

< o c

lim E(n)p(Vn(to), A(t0) ) = 0 , n - - - - 4 0 0

where E (n) denotes the mathematical expectation. Then,

W. Wagner~Mathematics and Computers in Simulation 38 (1995) 211-216 215

1 l im E <n) sup p ( v , ( t ) , A( t) ) = 0 , VAt <

n---*oo tE[to,to+At] 16"rr Bmax "

In more general physical situations (gas mixtures, chemical react ions) , different types o f parti- cles appear in a natural way. In our situation (spatial ly homogeneous , one type of part ic les) , the introduction of weights is complete ly ar t i f ic ia l - -a feature of the numerical scheme.

Consider ing the particle sys tem (10) as a numerical scheme, the velocities v i ( t ) , i= 1 . . . . . r e ( t ) , can be interpreted as a t ime-dependent grid in the space I~ 3, and the weights gi ( t ) , i= 1 . . . . . re ( t ) , as a function on this grid. In the case of the " random discrete velocity model" there was a fixed t ime discretization, so a discretization error remained in the limit. In the Markov model there is no fixed t ime discretization. Thus, the approximat ion error disappears in the limit as the number of grid points tends to infinity.

There is a random blow-up of the grid in t ime limiting the convergence result to small t ime intervals. After a certain time, it is necessary to reduce the number of particles in the system. One opportuni ty was proposed in [8] (cf. also [7] ), but this p rob lem still needs some effort. Also, for a practical implementat ion, one has to approximate the random waiting t ime in the Markov model , e.g., by its expec ta t ion /3 -1, w h e r e / 3 is given in (14) .

References

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[ 16] W. Wagner, Theoretical comparison of stochastic particle methods in rarefied gas dynamics, in: B.D. Shizgal and D.P. Weaver, eds., Rarefied Gas Dynamics: Theory and Simulations, Progress in Astronautics and Aeronautics (AIAA, Washington, DC, 1994) 353-360.

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