Time Relaxed Monte Carlo Methods for the Boltzmann Equation

Abstract

A new family of Monte Carlo schemes is introduced for the numerical solution of the Boltzmann equation of rarefied gas dynamics. The schemes are inspired by the Wild sum expansion of the solution of the Boltzmann equation for Maxwellian molecules and consist of a novel time discretization of the equation. In particular, high order terms in the expansion are replaced by the equilibrium Maxwellian distribution. The two main features of the schemes are high order accuracy in time and asymptotic preservation. The first property allows to recover accurate solutions with time steps larger than those required by direct simulation Monte Carlo (DSMC), while the latter guarantees that for the vanishing Knudsen number, the numerical solution relaxes to the local Maxwellian. Conservation of mass, momentum, and energy are preserved by the scheme. Numerical results on several space homogeneous problems show the improvement of the new schemes over standard DSMC. Applications to a one-dimensional shock wave problem are also presented.