CFL Optimized Forward-Backward Runge-Kutta Schemes for the Shallow Water Equations

TL;DR

This paper introduces an optimized Runge-Kutta scheme, FB-RK(3,2), that significantly increases the maximum stable time-step for shallow water equations, enhancing computational efficiency while maintaining accuracy.

Contribution

The paper develops and optimizes a new Runge-Kutta-type scheme using weighted averaging and von Neumann analysis, achieving larger stable time-steps for shallow water simulations.

Findings

FB-RK(3,2) allows time-steps up to 2.8 times larger than traditional methods.

The scheme maintains second-order accuracy and low dispersion/dissipation errors.

In nonlinear tests, it doubles the efficiency with minimal impact on solution quality.

Abstract

We present the formulation and optimization of a Runge-Kutta-type time-stepping scheme for solving the shallow water equations, aimed at substantially increasing the effective allowable time-step over that of comparable methods. This scheme, called FB-RK(3,2), uses weighted forward-backward averaging of thickness data to advance the momentum equation. The weights for this averaging are chosen with an optimization process that employs a von Neumann-type analysis, ensuring that the weights maximize the admittable Courant number. Through a simplified local truncation error analysis and numerical experiments, we show that the method is at least second order in time for any choice of weights and exhibits low dispersion and dissipation errors for well-resolved waves. Further, we show that an optimized FB-RK(3,2) can take time-steps up to 2.8 times as large as a popular three-stage, third-order strong stability preserving Runge-Kutta method in a quasi-linear test case. In fully nonlinear shallow water test cases relevant to oceanic and atmospheric flows, FB-RK(3,2) outperforms SSPRK3 in admittable time-step by factors between roughly between 1.6 and 2.2, making the scheme approximately twice as computationally efficient with little to no effect on solution quality.