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We present the formulation and optimization of a Runge-Kutta-typetime-stepping scheme for solving the shallow water equations, aimed atsubstantially increasing the effective allowable time-step over that ofcomparable methods. This scheme, called FB-RK(3,2), uses weightedforward-backward averaging of thickness data to advance the momentum equation.The weights for this averaging are chosen with an optimization process thatemploys a von Neumann-type analysis, ensuring that the weights maximize theadmittable Courant number. Through a simplified local truncation error analysisand numerical experiments, we show that the method is at least second order intime for any choice of weights and exhibits low dispersion and dissipationerrors 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-lineartest case. In fully nonlinear shallow water test cases relevant to oceanic andatmospheric flows, FB-RK(3,2) outperforms SSPRK3 in admittable time-step byfactors between roughly between 1.6 and 2.2, making the scheme approximatelytwice as computationally efficient with little to no effect on solutionquality.

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