Preserving stationary discontinuities in two-layer shallow water equations with a novel well-balanced approach

Abstract

This paper proposes a novel energy-balanced numerical scheme for the two-layer shallow water equations (2LSWEs) that accurately captures internal hydraulic jumps without introducing spurious oscillations. The proposed scheme overcomes the problem of post-shock oscillations in the 2LSWE, a phenomenon commonly observed in numerical solutions of non-linear hyperbolic systems when shock-capturing schemes are used. The approach involves reconstructing the internal momentum equation of 2LSWEs using the correct Hugoniot curve via a set of shock wave fixes originally developed for single-layer shallow water equations. The scheme successfully preserves all stationary solutions, making it highly suitable for simulations of real-life scenarios involving small perturbations of these conditions.