A finite volume scheme for the Euler system inspired by the two velocities approach

TL;DR

This paper introduces a novel finite volume scheme for the Euler system based on a two velocities approach, ensuring key physical properties and convergence to smooth solutions.

Contribution

The scheme uniquely applies numerical viscosity to the velocity field inspired by Brenner's model, preserving positivity and entropy principles.

Findings

Ensures positivity of density and pressure.

Generates dissipative measure-valued solutions.

Converges to smooth solutions when they exist.

Abstract

We propose a new finite volume scheme for the Euler system of gas dynamics motivated by the model proposed by H. Brenner. Numerical viscosity imposed through upwinding acts on the velocity field rather than on the convected quantities. The resulting numerical method enjoys the crucial properties of the Euler system, in particular positivity of the approximate density and pressure and the minimal entropy principle. In addition, the approximate solutions generate a dissipative measure-valued solutions of the limit system. In particular, the numerical solutions converge to the smooth solution of the system as long as the latter exists.