Convergence of approximations to stochastic scalar conservation laws

TL;DR

This paper establishes a general framework demonstrating that various approximation methods for stochastic scalar conservation laws converge in law to the true solution, using kinetic and martingale techniques.

Contribution

It introduces a minimal set of conditions under which approximations to stochastic scalar conservation laws are proven to converge in law, extending previous results.

Findings

Proves convergence in law of approximation schemes

Develops a kinetic formulation for stochastic conservation laws

Applies framework to finite volume method convergence

Abstract

We develop a general framework for the analysis of approximations to stochastic scalar conservation laws. Our aim is to prove, under minimal consistency properties and bounds, that such approximations are converging to the solution to a stochastic scalar conservation law. The weak probabilistic convergence mode is convergence in law, the most natural in this context. We use also a kinetic formulation and martingale methods. Our result is applied to the convergence of the Finite Volume Method in the companion paper [15].