Numerical approximation of stochastic conservation laws on bounded domains
1 LMA, Aix-Marseille Univ, CNRS, UPR 7051, Centrale Marseille, 13402 Marseille cedex 20, France.
2 I2M, Aix-Marseille Univ, CNRS, UMR 7373, Centrale Marseille, 13453 Marseille, France.
Received: 11 September 2015
Accepted: 11 March 2016
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 Rd 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.
Mathematics Subject Classification: 35L60 / 60H15 / 35L60
Key words: Stochastic PDE / first-order hyperbolic equation / multiplicative noise / finite volume method / monotone scheme / Dirichlet boundary conditions
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