Semi-discrete entropy-preserving surface reconstruction schemes for the shallow water equations: Analysis of physical structures

Abstract

We aim to introduce a new variant of hydrostatic reconstruction scheme, which is physical-structure-preserving and semi-discrete entropy-preserving for shallow water equations with a discontinuous bottom topography, based on novel surface reconstructions (NSR). The NSR is used to define approximate Riemann states with respect to the water depth and the velocity to compute consistent numerical fluxes across the cell interface and the upwind interface source term. This work is motivated by path-conservative surface reconstructions (Dong (2023) [19]), in which the conditions to satisfy the entropy inequality are complicated. We propose the NSR scheme to satisfy the entropy inequality without the corresponding complicated conditions. A key advantage of the NSR scheme is to preserve the physical structure of the water surface in the sense that when the water surface is continuous, then the defined approximate Riemann state should inherit this property. The physical-structure-preserving property is indispensable for obtaining a more accurate discretized source term and avoiding artificial “waterfall” effects. We prove that the NSR scheme satisfies a semi-discrete entropy condition and can guarantee the water depth to be nonnegative, and maintain the stationary state. We show several computed results to compare the NSR scheme with the hydrostatic reconstruction scheme based on subcell reconstructions and the path-conservative surface reconstruction scheme. We also extend the NSR scheme for pollutant transport in shallow water based on adaptive moving triangular meshes. Finally, several numerical results obtained by the adaptive NSR scheme for pollutant transport in shallow water are shown.