A P-adaptive Discontinuous Galerkin Method Using Local Time-stepping Strategy Applied to the Shallow Water Equations

摘要

This article describes the application of a p-adaptive Discontinuous Galerkin (DG) method, which is based on a Taylor expansion in both space and time and local time-stepping strategy, to simulate shallow water flows without viscosity. Approximating these flows with explicit time stepping gives rise to a CFL time step constraint. This can lead to an unreasonable high CPU effort when the smallest time steps resulted from the global CFL condition are used or high order approximations are applied. One method to deal with this inefficiency is to use locally varying time steps or some adaptive methods, e.g. h-, p-, or hp-adaptive algorithm. In this study, we combine a p-adaptive algorithm with an explicit discontinuous Galerkin scheme using local time-stepping strategy to solve the Shallow Water Equations (SWE). A slope limiter for DG using local time steps is also proposed to avoid numerical oscillations. Numerical results are presented, which verify the accuracy and efficiency of the approach (compared to using a globally defined CFL time step and fixed high order approximations).