Berthon, Christophe; Loubère, Raphaël; Michel-Dansac, Victor 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]. Cited in 1 Document 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 Keywords:shallow-water equations; Godunov-type schemes; fully well-balanced schemes; second-order accuracy; moving steady states Citations:Zbl 1359.76206 PDFBibTeX XMLCite \textit{C. Berthon} et al., Springer Proc. Math. Stat. 236, 165--177 (2018; Zbl 1405.76032) Full Text: DOI HAL