A convergent finite volume scheme for the stochastic barotropic compressible Euler equations

TL;DR

This paper introduces a finite volume scheme for the stochastic 3D barotropic compressible Euler equations, proving convergence of numerical solutions to the true solutions under stochastic influences.

Contribution

It is the first to establish convergence of numerical approximations for stochastic compressible Euler equations, including a measure-valued to strong solution convergence.

Findings

Convergence of Young measures to dissipative martingale solutions.

Strong convergence of numerical solutions to regular solutions.

First proof of numerical approximation convergence for this stochastic system.

Abstract

In this paper, we analyze a semi-discrete finite volume scheme for the three-dimensional barotropic compressible Euler equations driven by a multiplicative Brownian noise. We derive necessary a priori estimates for numerical approximations, and show that the Young measure generated by the numerical approximations converge to a dissipative measure--valued martingale solution to the stochastic compressible Euler system. These solutions are probabilistically weak in the sense that the driving noise and associated filtration are integral part of the solution. Moreover, we demonstrate strong convergence of numerical solutions to the regular solution of the limit systems at least on the lifespan of the latter, thanks to the weak (measure-valued)--strong uniqueness principle for the underlying system. To the best of our knowledge, this is the first attempt to prove the convergence of numerical approximations for the underlying system.