Time Splitting Error in DSMC Schemes for the Spatially Homogeneous Inelastic Boltzmann Equation

Abstract

The paper is concerned with the numerical treatment of the uniformly heated inelastic Boltzmann equation by the direct simulation Monte Carlo (DSMC) method. This technique is presently the most widely used numerical method in kinetic theory. We consider three modifications of the DSMC method and study them with respect to their efficiency and convergence properties. Convergence is investigated with respect to both the number of particles and the time step. The main issue of interest is the time step discretization error due to various splitting strategies. A scheme based on the Strang-splitting strategy is shown to be of second order with respect to time step, while there is only first order for the commonly used Euler-splitting scheme. On the other hand, a no-splitting scheme based on appropriate Markov jump processes does not produce any time step error. It is established in numerical examples that the no-splitting scheme is about two orders of magnitude more efficient than the Euler-splitting scheme. The Strang-splitting scheme reaches almost the same level of efficiency as that of the no-splitting scheme, since the deterministic time step error vanishes sufficiently fast.