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There is a growing interest in investigating numerical approximations of thewater wave equation in recent years, whereas the lack of rigorous analysis ofits time discretization inhibits the design of more efficient algorithms. Inthis work, we focus on a nonlocal hyperbolic model, which essentially inheritsthe features of the water wave equation, and is simplified from the latter. Forthe constant coefficient case, we carry out systematical stability studies ofthe fully discrete approximation of such systems with the Fourier spectralapproximation in space and general Runge-Kutta method in time. In particular,we discover the optimal time step constraints, in the form of a modified CFLcondition, when certain explicit Runge-Kutta method are applied. Besides, theconvergence of the semi-discrete approximation of variable coefficient case isshown, which naturally connects to the water wave equation. Extensive numericaltests have been performed to verify the stability conditions and simulations ofthe simplified hyperbolic model in the high frequency regime and the water waveequation are also provided.

On Runge-Kutta methods for the water wave equation and its simplified nonlocal hyperbolic model