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

0557842 - MÚ 2023 RIV US eng J - Journal Article
Lukáčová-Medviďová, M. - She, Bangwei - Yuan, Y.
Error estimates of the Godunov method for the multidimensional compressible Euler system.
Journal of Scientific Computing. Roč. 91, č. 3 (2022), č. článku 71. ISSN 0885-7474. E-ISSN 1573-7691
R&D Projects: GA ČR(CZ) GA21-02411S
Institutional support: RVO:67985840
Keywords : compressible Euler system * consistency formulation * error estimates * Godunov method
OECD category: Pure mathematics
Impact factor: 2.5, year: 2022
Method of publishing: Open access
https://doi.org/10.1007/s10915-022-01843-6

We derive a priori error estimates of the Godunov method for the multidimensional compressible 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-norms of the errors in density, momentum and entropy. Under the assumption, that the numerical density is uniformly bounded from below by a positive constant and that the energy is uniformly bounded from above and stays positive, we obtain a convergence rate of 1/2 for the relative energy in the L1-norm, that is to say, a convergence rate of 1/4 for the L2-error of the numerical solution. Further, under the assumption—the total variation of the numerical solution is uniformly bounded, we obtain the first order convergence rate for the relative energy in the L1-norm, consequently, the numerical solution converges in the L2-norm with the convergence rate of 1/2. The numerical results presented are consistent with our theoretical analysis.
Permanent Link: http://hdl.handle.net/11104/0331687