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Flux-gradient and source-term balancing for certain high resolution shock-capturing schemes. (English) Zbl 1237.76100
Summary: We present an extension of Marquina’s flux formula, as introduced in R.P. Fedkiw et al. The penultimate scheme for systems of conservation laws: finite difference ENO with Marquina’s flux splitting. In: M. Hafez, (edit.) Progress in numerical solutions of partial differential equations, Arcachon, France, (1998), for the shallow water system. We show that the use of two different Jacobians at cell interfaces prevents the scheme from satisfying the exact C-property [A. Bermúdez , M.E. Vázquez. Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23, No. 8, 1049–1071 (1994; Zbl 0816.76052)]. While the approximate C-property is satisfied for higher order versions of the scheme. The use of a single Jacobian in Marquina’s flux splitting formula leads to a numerical scheme satisfying the exact C-property, hence we propose a combined technique that uses Marquina’s two sided decomposition when the two adjacent states are not close and a single decomposition otherwise. Finally, we propose a special treatment at wet/dry fronts and situations of dry bed generation.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
76M12 Finite volume methods applied to problems in fluid mechanics
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