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On the convergence of semi-discrete high resolution schemes with superbee flux limiter for conservation laws. (English) Zbl 1286.65110

Li, Tatsien (ed.) et al., Hyperbolic problems. Theory, numerics and applications. Vol. 2. Proceedings of the 13th international conference on hyperbolic problems, HYP 2010, Beijing, China, June 15–19, 2010. Hackensack, NJ: World Scientific; Beijing: Higher Education Press (ISBN 978-981-4417-08-2/v.2; 978-981-4417-06-8/set). Series in Contemporary Applied Mathematics CAM 18, 431-438 (2012).
Summary: A class of high resolution schemes, using flux limiters for hyperbolic conservation laws, was introduced by P. K. Sweby [SIAM J. Numer. Anal. 21, 995–1011 (1984; Zbl 0565.65048)] in the 1980s. In the semi-discrete case, for the convex conservation laws with or without a source term, N. Yang and H. Jiang [Methods Appl. Anal. 12, No. 1, 89–101 (2005; Zbl 1128.65071)] have shown the entropy consistence of the schemes of this class based on minmod limiter when the building block of the schemes is an arbitrary \(E\)-scheme, and based on Chakravarthy-Osher or van Leer’s limiter when the building block of the scheme is Godunov, the Engquist-Osher, or the Lax-Friedrichs. However, the convergence problems related to many other flux limiters, such as Roe’s superbee, have been open.
In this paper, we use the convergence criteria, established by H. Yang and N. Jiang [ibid. 10, No. 4, 487–512 (2003; Zbl 1077.65099)] to show the convergence of the schemes with superbee limiter.
For the entire collection see [Zbl 1255.35003].

MSC:

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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