Convergence of a particle Monte Carlo algorithm for scalar conservation laws

The subject of this paper is a Monte Carlo algorithm for scalar conservation laws proposed in [L. Pareschi and M. Sea & iuml;d, A new Monte Carlo approach for conservation laws and relaxation systems, Computational Science-ICCS 2004. Part II, Lecture Notes in Comput. Sci. 3037, Springer, Berlin 2004, 276-283]. The algorithm is a stochastic particle method based on a probabilistic interpretation of the Jin-Xin relaxation formulation of conservation laws. We prove convergence as the number of particles approaches infinity, and the spatial and temporal mesh sizes approach zero, assuming that the number of particles approaches infinity at a rate sufficiently high compared to the rate that the mesh size approaches zero. For the case where the solution can take either sign, our version of the algorithm is novel. We present two numerical examples as evidence of convergence.