Error estimates of the Godunov method for the multidimensional compressible Euler system

TL;DR

This paper establishes error estimates and convergence rates for the Godunov method applied to the multidimensional Euler system, using the relative energy principle and assuming boundedness conditions.

Contribution

It provides the first rigorous convergence rate analysis for the Godunov method in multidimensional gas dynamics under specific boundedness assumptions.

Findings

Convergence rate of 1/2 for the relative energy in L^1-norm.

First order convergence rate for the relative energy with bounded total variation.

Numerical results align with theoretical error estimates.

Abstract

We derive a priori error of the Godunov method for the multidimensional Euler system of gas dynamics. To this end we apply the relative energy principle and estimate the distance between the numerical solution and the strong solution. This yields also the estimates of the $L^2$-norm of errors in density, momentum and entropy. Under the assumption that the numerical density and energy are bounded, we obtain a convergence rate of $1/2$ for the relative energy in the $L^1$-norm. Further, under the assumption -- the total variation of numerical solution is bounded, we obtain the first order convergence rate for the relative energy in the $L^1$-norm. Consequently, numerical solutions (density, momentum and entropy) converge in the $L^2$-norm with the convergence rate of $1/2$. The numerical results presented for Riemann problems are consistent with our theoretical analysis.