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. DSMC AND THE BOLTZMANN EQUATION Direct simulation Monte Carlo (DSMC) has been the most widely used numerical algorithm in kinetic theory. We refer to G.A. Bird’s monograph [1, Sections 9.4, 11.1] concerning remarks on the historical development. The history of the subject is also well reflected in the proceedings of the bi-annual conferences on “Rarefied Gas Dynamics” ranging from [2] to the present volume. The method is based on systems of particles x1(t); v1(t); : : :;xn(t); vn(t) ; t 0 ; n 1 ; (1) imitating the behaviour of gas molecules in a probabilistic way. It includes several numerically motivated approximations. Independent motion (free flow) of the particles and their pairwise interactions (collisions) are separated using a splitting procedure with a time increment t : During the free flow step, particles move according to their velocities, xi(t+ t) = xi(t)+Z t+ t t vi(s)ds ; i = 1; : : :;n ; and do not collide. At this step boundary conditions are taken into account. During the collision step, particles do not change their positions. At this step some partitionD = [lc l=1Dl of the spatial domain into a finite number lc of disjoint cells is introduced. In each cell a certain amount of binary collisions between particles is performed, vi; vj ) v i ; v j : The probabilistic rules of these transformations depend on the interaction potential between gas molecules. The interest in studying the connection between stochastic simulation procedures in rarefied gas dynamics and the Boltzmann equation was stimulated by K. Nanbu’s paper [3] (cf. the survey papers [4, 5]). Convergence for the Nanbu scheme and its modifications was studied in [6] (spatially homogeneous case) and [7] (spatially inhomogeneous case). Convergence for Bird’s method was proved in [8]. It was established that the empirical measures related to the particle system (1), (n)(t;dx;dv)= 1 n n Xi=1 Æxi(t);vi(t)(dx;dv) ;