Monte Carlo Methods and Numerical Solutions.

The purpose of this paper is to illustrate that direct simulation Monte Carlo methods can often be considered as rigorous mathematical tools for solving nonlinear kinetic equations numerically. First a convergence result for Bird’s DSMC method is recalled. Then some sketch of the history of stochastic models related to rarefied gas dynamics is given. The model introduced by Leontovich in 1935 provides the basis for a rigorous derivation of the Boltzmann equation from a stochastic particle system. The last part of the paper is concerned with some recent directions of study in the field of Monte Carlo methods for nonlinear kinetic equations. Models with general particle interactions and the corresponding limiting equations are discussed in some detail. In particular, these models cover rarefied granular gases (inelastic Boltzmann equation) and ideal quantum gases (Uehling-Uhlenbeck-Boltzmann equation). Problems related to the order of convergence, to the approximation of the steady state solution, and to variance reduction are briefly mentioned. © 2005 American Institute of Physics [ABSTRACT FROM AUTHOR]