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.
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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© 2024 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- The Burr distribution as an asymptotic law for extreme order statistics and its application to the analysis of statistical regularities in the interplanetary magnetic field
- Hidden attractors and nonlocal oscillations in gene networks models
- Computation and analysis of optimal disturbances of stationary solutions of the hepatitis B dynamics model
- On the accuracy of shock-capturing schemes when calculating Cauchy problems with periodic discontinuous initial data
- Constructing low-rank Tucker tensor approximations using generalized completion
- The Editorial Board of the Journal with deep regret is announcing the death of Prof. Yuri Alekseevich KUZNETSOV aged 79, on January 30, 2024
Artikel in diesem Heft
- Frontmatter
- The Burr distribution as an asymptotic law for extreme order statistics and its application to the analysis of statistical regularities in the interplanetary magnetic field
- Hidden attractors and nonlocal oscillations in gene networks models
- Computation and analysis of optimal disturbances of stationary solutions of the hepatitis B dynamics model
- On the accuracy of shock-capturing schemes when calculating Cauchy problems with periodic discontinuous initial data
- Constructing low-rank Tucker tensor approximations using generalized completion
- The Editorial Board of the Journal with deep regret is announcing the death of Prof. Yuri Alekseevich KUZNETSOV aged 79, on January 30, 2024