Local time-stepping in Runge–Kutta discontinuous Galerkin finite element methods applied to the shallow-water equations

Abstract

Large geophysical flows often encompass both coarse and highly resolved regions. Approximating these flows using shock-capturing methods with explicit time stepping gives rise to a Courant–Friedrichs–Lewy (CFL) time step constraint. Even if the refined regions are sparse, they can restrict the global CFL condition to very small time steps, vastly increasing computational effort over the whole domain. One method to cope with this problem is to use locally varying time steps over the domain. These are also referred to as multi-rate methods in the ODE literature. Ideally, such methods must be conservative, accurate and easy to implement. In this study, we derive a second-order, local time stepping procedure within a Runge–Kutta discontinuous Galerkin (RKDG) framework to solve the shallow water equations. This procedure is based on previous first-order work of the second author and collaborator Kirby [1–3]. As we are interested in both coastal and overland flows due to, e.g., rainfall, wetting and drying is incorporated into the model. Numerical results are shown, which verify the accuracy and efficiency of the approach (compared to using a globally defined CFL time step), and the application of the method to rainfall–runoff scenarios.