Convergence and error analysis of compressible fluid flows with random data: Monte Carlo method

TL;DR

This paper analyzes the convergence and error estimates of the Monte Carlo method combined with finite volume discretization for solving the stochastic compressible Navier-Stokes equations, providing theoretical and numerical validation.

Contribution

It introduces a novel convergence analysis for Monte Carlo finite volume methods applied to stochastic compressible Navier-Stokes equations, addressing non-uniqueness issues.

Findings

Monte Carlo finite volume method converges to a statistical strong solution.

Established convergence rates for the method.

Numerical experiments confirm theoretical predictions.

Abstract

The goal of this paper is to study convergence and error estimates of the Monte Carlo method for the Navier-Stokes equations with random data. To discretize in space and time, the Monte Carlo method is combined with a suitable deterministic discretization scheme, such as a finite volume method. We assume that the initial data, force and the viscosity coefficients are random variables and study both, the statistical convergence rates as well as the approximation errors. Since the compressible Navier-Stokes equations are not known to be uniquely solvable in the class of global weak solutions, we cannot apply pathwise arguments to analyze the random Navier-Stokes equations. Instead we have to apply intrinsic stochastic compactness arguments via the Skorokhod representation theorem and the Gy\"{o}ngy-Krylov method. Assuming that the numerical solutions are bounded in probability, we prove that the Monte Carlo finite volume method converges to a statistical strong solution. The convergence rates are discussed as well. Numerical experiments illustrate theoretical results.