A robust second-order surface reconstruction for shallow water flows with a discontinuous topography and a Manning friction

Abstract

A second-order surface reconstruction (SR) method for the shallow water equations with a discontinuous bottom topography and a Manning friction source term is presented. We redefine the water surface level at the cell interface by using the minimum difference between the bottom level and the original water surface level. The reconstructed water surface level is used to define the intermediate bottom level and the intermediate water height at the cell interface. We propose an explicit-implicit method to address the friction source term. The new second-order SR scheme together with the explicit-implicit method can preserve a special steady-state solution of the system and can maintain the positivity of the water depth. We also extend the new scheme to two-dimensional shallow water flows. To demonstrate the robustness and effectiveness of the new scheme, we use several classical numerical experiments for the shallow water flows over a complex bottom topography.