On the Stochastic Weighted Particle Method
The stochastic weighted particle method is one of the particle methods recently developed to approximate the solution of the Boltzmann equation, one of the well known kinetic equations. The main idea is to use random weight transfer between particles duri
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1
Introduction
Let the function of interest be
that defines the probability density function of particles, which explains the probability that a gas particle at time t ~ 0 can be observed at position x Efland moving with velocity v E lR 3 . The equation to be discussed is
of + (v, \7 xf) = QU, f) at
(1.1)
known as the Boltzmann equation (cf. [1], [2], [3]). In this case QU, f) denotes the collision integral and has the following form
QU, f)(v)
=
r rB(v, w, e) [f(v')f(w') - f(v)f(w)] dw de,
jIR3 j 82
where e is the direction vector defined on the unit sphere 8 2 , the v', w' are the post-collision velocities, and B(v, w, e) is the collision kernel, which depends on the model. In order to study the behaviour of the gas more closely, let the system of N particles Pj, j = 1, 2, ... , N be defined as (1.2) M. Griebel et al. (eds.), Meshfree Methods for Partial Differential Equations © Springer-Verlag Berlin Heidelberg 2003
E. H. Nugrahani, S. Rjasanow
320
where g(t) E IR+ is the weight, x(t) E D C IR3 is the position, v(t) E IR3 is the velocity, and t 2: 0 is the time variable. The distribution function f(t,x,v) in the low density regions can be resolved mOre accurately by producing many particles of low weight. So instead of using the direct simulation Monte Carlo (DSMC) method in simulation, which considersparticles with constant weights, the stochastic weighted particle method (SWPM) was introduced [3]. The main difference between SWPM and DSMC is the idea of a random weight transfer between particles during collisions in the SWPM. This procedure usually leads to an increase in the number of particles in the system. But if we cannot compensate this increase in some natural way, then we might have to reduce the number of particles
[3]. In the following sections we will discuss the theoretical and numerical aspects of reducing the number of particles in solving the Boltzmann equation using SWPM.
2
Description of the Method
In the particle simulation procedures, the particles are assumed to be in one of two conditions, namely the free flow Or the collision stage. In the free flow step, the particles move according to their velocity and their weights remain constant. In the collision step, the behaviour of the particles will be described as follows. First it is assumed that the particles are sorted in spatial cells De,£=1, ... ,£c
that define a non-overlapping decomposition of the computational domain D. In each cell De collisions are then simulated. The collision simulation step in one spatial cell corresponds to the spatially homogeneous Boltzmann equation [3]
{){)f t
= Q(j, f)(v) =
r r B(v, w, e) [f(v/)f(w /) -
}1R3}52
f(v)f(w] dw de. (2.1)
The stochastic process of the collisions can be defined as
where k is the simulation step index. The infinitesimal generator of the process is given by
A(p)(z) =
L
l~i#j~n
~
12 5
q(z,i,j,e) [p(J(z,i,j,e)) - p(z)] de,
Stochastic Weighted Particle Method
321
where P is a measurable test function and q(z, i, j, e) is an intensity function. The jump
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