Convergence of a finite volume method to weak solutions for the compressible Navier-Stokes-Fourier system

TL;DR

This paper proves that an upwind finite volume method converges strongly to a weak solution of the compressible Navier-Stokes-Fourier system, ensuring the method's reliability for simulating complex fluid flows.

Contribution

It establishes the convergence of a finite volume scheme to weak solutions for the compressible Navier-Stokes-Fourier system, including entropy and energy inequalities, using novel consistency and oscillation analysis.

Findings

Strong convergence of the finite volume method to weak solutions.

Validation of the method's adherence to entropy and energy inequalities.

Application of renormalisation techniques to control oscillations.

Abstract

We prove strong convergence of an upwind-type finite volume method to a weak solution of the Navier-Stokes-Fourier system with the Dirichlet boundary conditions. The limit solution satisfies a weak form of the mass and momentum equations, together with a weak form of the entropy and ballistic energy inequalities, and complies with the weak-strong uniqueness principle. The finite volume method uses piecewise-constant spatial approximations. The convergence proof is based on a combination of delicate consistency estimates with a careful analysis of the oscillations of numerical densities via renormalisation of the continuity equation.