Surface reconstruction schemes for shallow water equations with a nonconservative product source term

Abstract

We aim to introduce surface reconstruction (SR) schemes for shallow water equations with a nonconservative product source term. The SR scheme is used to define the intermediate water depth and the bottom topography on the cell boundaries. The key ingredient of the SR scheme is to smooth the water surface level or the bottom topography while maintaining their the monotone property. The discretization of the integral of the nonconservative product term is established with the aid of a family of pathes chosen in the phase space. The discretized source term is closer to the exact one. For obtaining a second-order accuracy, we introduce a non-oscillatory monotone-preserving reconstruction method. We establish the conditions of the SR scheme for obtaining the semi-discrete and fully-discrete entropy inequalities and prove that the introduced SR scheme can maintain the stationary solutions and guarantee the water depth to be nonnegative. Several numerical results of the one- and two-dimensional shallow water equations confirm that the SR scheme can preserve the stationary state, the positivity of the water depth, and is efficient when computing the shallow water flows over a step.