Numerical approximation of stochastic conservation laws on bounded domains

This paper is devoted to the study of finite volume methods for the discretization of scalar conservation laws with a multiplicative stochastic force defined on a bounded domain D of R with Dirichlet boundary conditions and a given initial data in L(D). We introduce a notion of stochastic entropy process solution which generalizes the concept of weak entropy solution introduced by F.Otto for such kind of hyperbolic bounded value problems in the deterministic case. Using a uniqueness result on this solution, we prove that the numerical solution converges to the unique stochastic entropy weak solution of the continuous problem under a stability condition on the time and space steps.