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Maximum principle of central schemes with \(k\)-monotone fluxes. (English) Zbl 1284.65106

Li, Tatsien (ed.) et al., Hyperbolic problems. Theory, numerics and applications. Vol. 1. 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 978-981-4417-07-5/v.1; 978-981-4417-06-8/set). Series in Contemporary Applied Mathematics CAM 17, 227-237 (2012).
Summary: The Nessyahu-Tadmor [H. Nessyahu and E. Tadmor, J. Comput. Phys. 87, No. 2, 408–463 (1990; Zbl 0697.65068)] scheme is a simple yet robust second-order non-oscillatory scheme, which relies on a piecewise linear reconstruction. A typical reconstruction choice is based on the standard minmod limiter. This choice gives a maximum principle for the scheme. However, it reduces the reconstruction to first order at local extrema. In this paper, we show that a maximum principle is still valid for second-order schemes when a new MAPR-like limiter is used, see [I. Christov and B. Popov, ibid. 227, No. 11, 5736–5757 (2008; Zbl 1151.65068)]. To prove this result, we require that the flux is \(k\)-monotone. We also show that the maximum principle implies the usual total variation diminishing bound.
For the entire collection see [Zbl 1255.35002].

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
35B50 Maximum principles in context of PDEs
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