  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.
