Asymptotic preserving finite volume method for the compressible Euler equations: analysis via dissipative measure-valued solutions

TL;DR

This paper introduces and analyzes an asymptotic preserving finite volume scheme for the compressible Euler equations, demonstrating its stability, consistency, and convergence to classical solutions in the low Mach number limit using dissipative measure-valued solutions.

Contribution

The paper develops a new AP finite volume scheme for multidimensional Euler equations and rigorously proves its convergence properties via dissipative measure-valued solutions.

Findings

The scheme is energy stable and consistent.

Numerical solutions converge to DMV solutions as mesh refines.

The scheme accurately captures low Mach number limits.

Abstract

We propose and analyze a new asymptotic preserving (AP) finite volume scheme for the multidimensional compressible barotropic Euler equations to simulate low Mach number flows. The proposed scheme uses a stabilized upwind numerical flux, with the stabilization term being proportional to the stiff pressure gradient, and we prove its conditional energy stability and consistency. Utilizing the concept of dissipative measure-valued (DMV) solutions, we rigorously illustrate the AP properties of the scheme for well-prepared initial data. In particular, we prove that the numerical solutions will converge weakly to a DMV solution of the compressible Euler equations as the mesh parameter vanishes, while the Mach number is fixed. The DMV solutions then converge to a classical solution of the incompressible Euler system as the Mach number goes to zero. Conversely, we show that if the mesh parameter is kept fixed, we obtain an energy stable and consistent finite-volume scheme approximating the incompressible Euler equations as the Mach number goes to zero. The numerical solutions generated by this scheme then converge weakly to a DMV solution of the incompressible Euler system as the mesh parameter vanishes. Invoking the weak-strong uniqueness principle, we conclude that the DMV solution and classical solution of the incompressible Euler system coincide, proving the AP property of the scheme. We also present an extensive numerical case study in order to illustrate the theoretical convergences, wherein we utilize the techniques of K-convergence.