TY - JOUR

T1 - On Convergence of Finite-Difference Shock-Capturing Schemes in Regions of Shock Waves Influence

AU - Kovyrkina, O. A.

AU - Ostapenko, V. V.

AU - Tishkin, V. F.

N1 - Funding Information: The reported study was funded in part by the Russian Foundation for Basic Research and the National Natural Science Foundation of China (project no. 21-51-53012) and by the Russian Science Foundation (project no. 21-11-00198). Publisher Copyright: © 2022, Pleiades Publishing, Ltd.

PY - 2022/6

Y1 - 2022/6

N2 - We perform a comparative accuracy study of the Rusanov, CABARETM, and WENO5 difference schemes used to compute the dam break problem for shallow water theory equations. We demonstrate that all three schemes have the first order of convergence inside the region occupied by a centered rarefaction wave, and the Rusanov scheme has the second order of convergence in the area of constant flow between the shock and the rarefaction wave, while in the CABARETM and WENO5 schemes there is no local convergence in this area. This is due to the fact that the numerical solutions obtained by the CABARETM and WENO5 schemes have undamped oscillations in the region of influence of the shock, the amplitude of which does not decrease with decreasing of the difference grid steps. As a result, taking into account the Lax-Wendroff theorem, the numerical solutions obtained by the conservative schemes CABARETM and WENO5 converge only weakly to the exact constant solution in the region of influence of the shock wave, in contrast to the Rusanov scheme, which locally converges with the second order to the exact solution in this region.

AB - We perform a comparative accuracy study of the Rusanov, CABARETM, and WENO5 difference schemes used to compute the dam break problem for shallow water theory equations. We demonstrate that all three schemes have the first order of convergence inside the region occupied by a centered rarefaction wave, and the Rusanov scheme has the second order of convergence in the area of constant flow between the shock and the rarefaction wave, while in the CABARETM and WENO5 schemes there is no local convergence in this area. This is due to the fact that the numerical solutions obtained by the CABARETM and WENO5 schemes have undamped oscillations in the region of influence of the shock, the amplitude of which does not decrease with decreasing of the difference grid steps. As a result, taking into account the Lax-Wendroff theorem, the numerical solutions obtained by the conservative schemes CABARETM and WENO5 converge only weakly to the exact constant solution in the region of influence of the shock wave, in contrast to the Rusanov scheme, which locally converges with the second order to the exact solution in this region.

KW - CABARET scheme

KW - local convergence of difference solution

KW - Rusanov scheme

KW - shock

KW - WENO5 scheme

UR - http://www.scopus.com/inward/record.url?scp=85135458183&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/689afb7a-f415-3d90-9939-ee6a5bf1f850/

U2 - 10.1134/S1064562422030048

DO - 10.1134/S1064562422030048

M3 - Article

AN - SCOPUS:85135458183

VL - 105

SP - 171

EP - 174

JO - Doklady Mathematics

JF - Doklady Mathematics

SN - 1064-5624

IS - 3

ER -