Improving the stochastic direct simulation method with applications to evolution partial differential equations

Abstract

The stochastic direct simulation method is a numerical scheme for approximating the solutions of ordinary differential equations by path simulations of certain associated Markov jump processes. Its particular features make it suitable especially when applied to ODE systems originating from the spatial discretization of PDEs. The present paper provides further improvements to this basic method, which are based on the predictor–corrector principle. They are made possible by the fact that in its context a full path of the jump process is computed. With this full set of data one can perform either Picard iterations, Runge–Kutta steps, or a combination, with the goal of increasing the order of convergence. The improved method is applied to standard test problems such as a reaction–diffusion equation modeling a combustion process in 1D and 2D as well as to the radiation–diffusion equations, a system of two partial differential equations in two space dimensions which is very demanding from the computational point of view. Further optimization aspects which are also discussed in this paper are related to the efficient implementation of sampling algorithms based on Huffman trees.