Computation of measure-valued solutions for the incompressible Euler equations

TL;DR

This paper introduces a spectral viscosity and ensemble averaging algorithm to compute measure-valued solutions for the incompressible Euler equations, demonstrating convergence, robustness, and insights into non-uniqueness of weak solutions.

Contribution

It presents a novel numerical method combining spectral viscosity and ensemble averaging to approximate measure-valued solutions for Euler equations, with proven convergence and extensive computational analysis.

Findings

Algorithm converges to measure-valued solutions with increasing resolution

Numerical experiments show robustness and efficiency of the method

Indications of non-atomic solutions suggest non-uniqueness of weak solutions

Abstract

We combine the spectral (viscosity) method and ensemble averaging to propose an algorithm that computes admissible measure valued solutions of the incompressible Euler equations. The resulting approximate young measures are proved to converge (with increasing numerical resolution) to a measure valued solution. We present numerical experiments demonstrating the robustness and efficiency of the proposed algorithm, as well as the appropriateness of measure valued solutions as a solution framework for the Euler equations. Furthermore, we report an extensive computational study of the two dimensional vortex sheet, which indicates that the computed measure valued solution is non-atomic and implies possible non-uniqueness of weak solutions constructed by Delort.