Convergence of first-order Finite Volume Method based on Exact Riemann Solver for the Complete Compressible Euler Equations

TL;DR

This paper proves the convergence of a finite volume numerical method based on the exact Riemann solver for the complete compressible Euler equations, using dissipative measure-valued solutions to handle oscillations and singularities.

Contribution

It establishes entropy inequalities, consistency, and both weak and strong convergence of the method, linking numerical solutions to measure-valued solutions for complex flows.

Findings

Numerical results align with theoretical convergence analysis.

Method effectively captures oscillations and singularities in compressible flows.

Convergence proven for complex flow problems like Kelvin-Helmholtz instability.

Abstract

Recently developed concept of dissipative measure-valued solution for compressible flows is a suitable tool to describe oscillations and singularities possibly developed in solutions of multidimensional Euler equations. In this paper we study the convergence of the first-order finite volume method based on the exact Riemann solver for the complete compressible Euler equations. Specifically, we derive entropy inequality and prove the consistency of numerical method. Passing to the limit, we show the weak and strong convergence of numerical solutions and identify their limit. The numerical results presented for the spiral, Kelvin-Helmholtz and the Richtmyer-Meshkov problem are consistent with our theoretical analysis.