A new Monte Carlo approach for conservation laws and relaxation systems

Abstract

We present a Monte Carlo method for approximating the solution of conservation laws. A relaxation method is used to transform the conservation law to a kinetic form that can be interpreted in a probabilistic manner. A Monte Carlo algorithm is then used to simulate the kinetic approximation. The method we present in this paper is simple to formulate and to implement, and can be straightforwardly extended to higher dimensional conservation laws. Numerical experiments are carried out using Burgers equation subject to both smooth and nonsmooth initial data. © Springer-Verlag Berlin Heidelberg 2004.