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A second-order well-balanced scheme for the shallow water equations with topography. (English) Zbl 1405.76032

Klingenberg, Christian (ed.) et al., Theory, numerics and applications of hyperbolic problems I, Aachen, Germany, August 2016. Cham: Springer (ISBN 978-3-319-91544-9/hbk; 978-3-319-91545-6/ebook). Springer Proceedings in Mathematics & Statistics 236, 165-177 (2018).
Summary: We consider the well-balanced numerical scheme for the shallow water equations with topography introduced in [V. Michel-Dansac et al., Comput. Math. Appl. 72, No. 3, 568–593 (2016; Zbl 1359.76206)] and its second-order well-balanced extension, which requires two heuristic parameters. The goal of the present contribution is to derive a parameter-free second-order well-balanced scheme. To that end, we consider a convex combination between the well-balanced scheme and a second-order scheme. We then prove that a relevant choice of the parameter of this convex combination ensures that the resulting scheme is both second-order accurate and well-balanced. Afterward, we perform several numerical experiments, in order to illustrate both the second-order accuracy and the well-balanced property of this numerical scheme. Finally, we outline some perspectives in a short conclusion.
For the entire collection see [Zbl 1398.65011].

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
86A05 Hydrology, hydrography, oceanography

Citations:

Zbl 1359.76206
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