Local time-stepping for the shallow water equations using CFL optimized forward-backward Runge-Kutta schemes

Abstract

The Courant–Friedrichs–Lewy (CFL) condition is a well known, necessary condition for the stability of explicit time-stepping schemes that effectively places a limit on the size of the largest admittable time-step for a given problem. We formulate and present a new local time-stepping (LTS) scheme optimized, in the CFL sense, for the shallow water equations (SWEs). This new scheme, called FB-LTS, is based on the CFL optimized forward-backward Runge-Kutta schemes from Lilly et al. [16]. We show that FB-LTS maintains exact conservation of mass and absolute vorticity when applied to the TRiSK spatial discretization [21], and provide numerical experiments showing that it retains the temporal order of the scheme on which it is based (second order). We implement FB-LTS, along with a certain operator splitting, in MPAS-Ocean to test computational performance. This scheme, SplitFB-LTS, is up to 10 times faster than the classical four-stage, fourth-order Runge-Kutta method (RK4), and 2.3 times faster than an existing strong stability preserving Runge-Kutta based LTS scheme with the same operator splitting (SplitLTS3). Despite this significant increase in efficiency, the solutions produced by SplitFB-LTS are qualitatively equivalent to those produced by both RK4 and SplitLTS3.