A conservative high-order discontinuous Galerkin method for the shallow water equations with arbitrary topography

摘要

A conservative high-order Godunov-type scheme is presented for solving the balance laws of the 1D shallow water equations (SWE). The scheme adopts a finite element Runge-Kutta (RK) discontinuous Galerkin (DG) framework. Based on an overall third-order accurate formulation, the model is referred to as RKDG3. Treatment of topographic source term is built in the DG approximation. Simplified formulae for initializing bed data at a discrete level are derived by assuming a local linear bed function to ease practical flow simulations. Owing to the adverse effects caused by using an uncontrolled global limiting process in an RKDG3 scheme (RKDG3-GL), a new conservative RKDG3 scheme with user-parameter-free local limiting method (RKDG3-LL) is designed to gain better accuracy and conservativeness. The advantages of the new RKDG3-LL model are demonstrated by applying to several steady and transient benchmark flow tests with irregular (either differentiable or non-differentiable) topography.