Convergence of the Finite Volume Method for Stochastic Hyperbolic Scalar Conservation Laws: A Proof by Truncation on the Sample-Time Space

Abstract

We prove the almost sure convergence of the explicit-in-time Finite Volume Method with monotone fluxes towards the unique solution of the scalar hyperbolic balance law with locally Lipschitz continuous flux and additive noise driven by a cylindrical Wiener process. We use the standard CFL condition and a martingale exponential inequality on sets whose probabilities are converging towards one. Then, with the help of stopping times on those sets, we apply theorems of convergence for approximate kinetic solutions of balance laws with stochastic forcing.