Finite volume methods for the computation of statistical solutions of the incompressible Euler equations

TL;DR

This paper introduces a finite volume numerical scheme combined with Monte Carlo integration to efficiently compute statistical solutions of the incompressible Euler equations, offering improved convergence properties over previous spectral methods.

Contribution

The paper develops a finite volume method for statistical solutions of Euler equations, with rigorous convergence proof and practical numerical validation, surpassing spectral approaches.

Findings

Finite volume scheme converges to statistical solutions under verifiable assumptions.

The convergence is stronger than measure-valued solutions, including multi-point correlations.

Numerical experiments support the naturalness and practicality of the assumptions.

Abstract

We present an efficient numerical scheme based on Monte Carlo integration to approximate statistical solutions of the incompressible Euler equations. The scheme is based on finite volume methods, which provide a more flexible framework than previously existing spectral methods for the computation of statistical solutions for incompressible flows. This finite volume scheme is rigorously proven to, under experimentally verifiable assumptions, converge in an appropriate topology and with increasing resolution to a statistical solution. The convergence obtained is stronger than that of measure-valued solutions, as it implies convergence of multi-point correlation marginals. We present results of numerical experiments which support the claim that the aforementioned assumptions are very natural, and appear to hold in practice.