Convergence of finite volume schemes for the Euler equations via dissipative measure--valued solutions

TL;DR

This paper proves that entropy stable finite volume schemes for the Euler equations generate dissipative measure-valued solutions and converge to classical solutions when they exist, addressing issues of ill-posedness and oscillations.

Contribution

It introduces a framework linking finite volume schemes with dissipative measure-valued solutions and proves convergence to classical solutions using the weak-strong uniqueness principle.

Findings

Finite volume schemes generate dissipative measure-valued solutions.

Numerical solutions converge to classical solutions when they exist.

The approach handles oscillations and ill-posedness in Euler equations.

Abstract

The Cauchy problem for the complete Euler system is in general ill posed in the class of admissible (entropy producing) weak solutions. This suggests there might be sequences of approximate solutions that develop fine scale oscillations. Accordingly, the concept of measure--valued solution that capture possible oscillations is more suitable for analysis. We study the convergence of a class of entropy stable finite volume schemes for the barotropic and complete compressible Euler equations in the multidimensional case. We establish suitable stability and consistency estimates and show that the Young measure generated by numerical solutions represents a dissipative measure--valued solution of the Euler system. Here dissipative means that a suitable form of the Second law of thermodynamics is incorporated in the definition of the measure--valued solutions. In particular, using the recently established weak-strong uniqueness principle, we show that the numerical solutions converge pointwise to the regular solution of the limit systems at least on the lifespan of the latter.