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

2021/11/09 by Mária Lukáčová-Medviďová, Lukáčová-Medviďová, Mária, Bangwei She +3 · 1 citation
Earth and Planetary Sciences · Engineering · Mathematics · #Aquatic and Environmental Studies #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2111.05009

openalex publication_date 2021/11/09 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

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 L²-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 L¹-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 L¹-norm. Consequently, numerical solutions (density, momentum and entropy) converge in the L²-norm with the convergence rate of 1/2. The numerical results presented for Riemann problems are consistent with our theoretical analysis.

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