A steady-state-preserving scheme for shallow water flows in channels

Abstract

A steady-state-preserving numerical model is developed for the shallow water equations in channels in the current study. The proposed numerical model is based on the finite volume method and capable to preserve both the moving-water and still-water steady states. In order to preserve the general steady states, the source terms in the momentum equation are incorporated into a global flux term; and the new momentum equation with global flux term is then relaxed in order to avoid the non-linearity in the computation. One can thus develop an upwind/modified-HLL hybrid Riemann solver to estimate the fluxes in different flow regimes. The developed numerical model can preserve the general steady states, and one avoids: (1) non-trivial root-finding for point values of wetted cross-sectional area A at cell interfaces; (2) complex discretization of geometric source terms incorporating artificial conservation corrections. The developed numerical model has second order accuracy in both space and time. Numerical solutions are presented to demonstrate that the proposed numerical model is capable to exactly preserve both moving-water and still-water steady states.