Local Time-Stepping for the Shallow Water Equations using CFL Optimized Forward-Backward Runge-Kutta Schemes

TL;DR

This paper introduces a CFL-optimized local time-stepping scheme for the shallow water equations, achieving significant computational speedups while maintaining accuracy and conservation properties.

Contribution

It presents the FB-LTS scheme, a novel local time-stepping method based on forward-backward Runge-Kutta schemes, optimized for CFL conditions in shallow water simulations.

Findings

FB-LTS maintains mass and vorticity conservation.

It retains second-order temporal accuracy.

FB-LTS is up to 10 times faster than RK4 in real-world tests.

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. (2023). We show that FB-LTS maintains exact conservation of mass and absolute vorticity when applied to the TRiSK spatial discretization (Ringler et al., 2010), and provide numerical experiments showing that it retains the temporal order of the scheme on which it is based (second order). In terms of computational performance, we show that when applied to a real-world test case on a highly-variable resolution mesh, the MPAS-Ocean implementation of FB-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 (LTS3). Despite this significant increase in efficiency, the solutions produced by FB-LTS are qualitatively equivalent to those produced by both RK4 and LTS3.