Convergence of a stochastic collocation finite volume method for the compressible Navier-Stokes system

TL;DR

This paper introduces a stochastic collocation finite volume method for solving the compressible Navier-Stokes equations, demonstrating convergence to a statistical solution under boundedness conditions using advanced probabilistic techniques.

Contribution

It develops a novel stochastic collocation approach combined with finite volume discretization and proves its convergence to a statistical solution of the Navier-Stokes system.

Findings

Convergence of the numerical solutions to a statistical solution.

Use of stochastic compactness and Skorokhod representation in analysis.

Boundedness in probability ensures convergence.

Abstract

We propose a stochastic collocation method based on the piecewise constant interpolation on the probability space combined with a finite volume method to solve the compressible Navier-Stokes system at the nodal points. We show convergence of numerical solutions to a statistical solution of the Navier-Stokes system on condition that the numerical solutions are bounded in probability. The analysis uses the stochastic compactness method based on the Skorokhod/Jakubowski representation theorem and the criterion of convergence in probability due to Gy\"ongy and Krylov.