Flux-gradient and source-term balancing for certain high resolution shock-capturing schemes

Abstract

We present an extension of Marquina’s flux formula, as introduced in Fedkiw et al. [Fedkiw RP, Merriman B, Donat R, Osher S. The penultimate scheme for systems of conservation laws: finite difference ENO with Marquina’s flux splitting. In: Hafez M, editor. Progress in numerical solutions of partial differential equations, Arcachon, France; July 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 [Bermúdez A, Vázquez ME. Upwind methods for hyperbolic conservation laws with source terms. Comput Fluids 1994;23(8):1049–71] 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.