Stochastic Models and Monte Carlo Algorithms for Boltzmann Type Equations

In this paper we are concerned with three typical aspects of the Monte Carlo approach. First there is a certain field of application, namely physical systems described by the Boltzmann equation. Then some class of stochastic models is introduced and its relation to the equation is studied using probability theory. Finally Monte Carlo algorithms based on those models are constructed. Here numerical issues like efficiency and error estimates are taken into account. In Section 1 we recall some basic facts from the kinetic theory of gases, introduce the Boltzmann equation and discuss some applications. Section 2 is devoted to the study of stochastic particle systems related to the Boltzmann equation. The main interest is in the convergence of the system (when the number of particles increases) to the solution of the equation in an appropriate sense. In Section 3 we introduce a modification of the standard “direct simulation Monte Carlo” method, which allows us to tackle the problem of variance reduction. Results of some numerical experiments are presented