Error estimates of a finite volume method for the compressible Navier--Stokes--Fourier system

TL;DR

This paper analyzes the convergence rate of a finite volume method applied to the compressible Navier--Stokes--Fourier system, establishing error estimates based on the relative energy method under boundedness assumptions.

Contribution

It provides the first rigorous convergence analysis of a finite volume scheme for the compressible Navier--Stokes--Fourier system with error bounds.

Findings

Established local existence of a strong solution.

Derived a priori error estimates using relative energy.

Assumed uniform boundedness of density and temperature.

Abstract

In this paper we study the convergence rate of a finite volume approximation of the compressible Navier--Stokes--Fourier system. To this end we first show the local existence of a highly regular unique strong solution and analyse its global extension in time as far as the density and temperature remain bounded. We make a physically reasonable assumption that the numerical density and temperature are uniformly bounded from above and below. The relative energy provides us an elegant way to derive a priori error estimates between finite volume solutions and the strong solution.