Improved error estimates for the finite volume and the MAC schemes for the compressible Navier-Stokes system

TL;DR

This paper introduces improved error estimates for finite volume and MAC schemes applied to the compressible Navier-Stokes equations, achieving better convergence rates across the full range of the adiabatic coefficient.

Contribution

It provides a refined consistency analysis and a continuous relative energy inequality to enhance error estimates for these numerical schemes.

Findings

Better convergence rates than previous results

Error estimates valid for the entire physically relevant adiabatic range

Enhanced understanding of scheme accuracy for compressible flows

Abstract

We present new error estimates for the finite volume and finite difference methods applied to the compressible Navier-Stokes equations. The main innovative ingredients of the improved error estimates are a refined consistency analysis combined with a continuous version of the relative energy inequality. Consequently, we obtain better convergence rates than those available in the literature so far. Moreover, the error estimates hold in the whole physically relevant range of the adiabatic coefficient.