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On the accuracy of shock-capturing schemes when calculating Cauchy problems with periodic discontinuous initial data

  • Olyana A. Kovyrkina EMAIL logo and Vladimir V. Ostapenko
Published/Copyright: April 8, 2024

Abstract

We study the accuracy of shock-capturing schemes for the shallow water Cauchy problems with piecewise smooth discontinuous initial data. We consider the second order balance-characteristic (CABARETM) scheme, the third order finite-difference Rusanov–Burstein–Mirin (RBM) scheme and the fifth order in space, the third order in time weighted essentially non-oscillatory (WENO5) scheme. We have shown that the maximum loss of accuracy occurs in the centered rarefaction waves of the exact solutions, where all these schemes have the first order of convergence and fairly close values of the numerical disbalances (errors), regardless of their formal approximation order on the smooth solutions. In the same time, inside the shock influence areas the considered schemes can have different convergence orders and, as a result, significantly different accuracy. In particular, when solving the Cauchy problem with periodic initial data, when the exact solution has no centered rarefaction waves, the RBM scheme has a significantly higher accuracy inside the shock influence areas, compared to the CABARETM and WENO5 schemes. It means that the combined scheme, in which the RBM scheme is a basic scheme and the CABARETM scheme is an internal one, can be effectively used to compute weak solutions of such type Cauchy problems.

MSC 2010: 65M06; 65M12

Funding statement: The reported study was funded in part by the Russian Science Foundation, project No. 22-11-00060 (Sections 2 and 3).

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Received: 2023-12-27
Accepted: 2024-02-06
Published Online: 2024-04-08
Published in Print: 2024-04-25

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 3.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/rnam-2024-0009/pdf
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