Convergence of a finite-volume scheme for a heat equation with a multiplicative Lipschitz noise

TL;DR

This paper proves that a finite-volume numerical scheme converges to the unique solution of a heat equation driven by multiplicative Lipschitz noise, combining probabilistic and numerical analysis techniques.

Contribution

It introduces a convergence proof for a semi-implicit finite-volume scheme applied to a stochastic heat equation with multiplicative noise, using advanced probabilistic methods.

Findings

Convergence in distribution of the scheme to a martingale solution.

Convergence in probability to the unique variational solution.

Validation of the scheme's effectiveness for stochastic heat equations.

Abstract

We study here the approximation by a finite-volume scheme of a heat equation forced by a Lipschitz continuous multiplicative noise in the sense of It\^o. More precisely, we consider a discretization which is semi-implicit in time and a two-point flux approximation scheme (TPFA) in space. We adapt the method based on the theorem of Prokhorov to obtain a convergence in distribution result, then Skorokhod's representation theorem yields the convergence of the scheme towards a martingale solution and the Gy\"{o}ngy-Krylov argument is used to prove convergence in probability of the scheme towards the unique variational solution of our parabolic problem.