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