On Energy Stable Runge-Kutta Methods for the Water Wave Equation and Its Simplified Non-Local Hyperbolic Model (Communications in Computational Physics, 2022)
Although interest in numerical approximations of the water wave equationgrows in recent years, the lack of rigorous analysis of its time discretization inhibits thedesign of more efficient algorithms. In practice of water wave simulations, the trade-off between efficiency and stability has been a challenging problem. Thus to shed lighton the stability condition for simulations of water waves, we focus on a model simplified from the water wave equation of infinite depth. This model preserves two mainproperties of the water wave equation: non-locality and hyperbolicity. For the constantcoefficient case, we conduct systematic stability studies of the fully discrete approximation of such systems with the Fourier spectral approximation in space and generalRunge-Kutta methods in time. As a result, an optimal time discretization strategy isprovided in the form of a modified CFL condition, i.e. $∆t = \mathcal{O}(\sqrt{∆x}).$ Meanwhile,the energy stable property is established for certain explicit Runge-Kutta methods.This CFL condition solves the problem of efficiency and stability: it allows numericalschemes to stay stable while resolves oscillations at the lowest requirement, which onlyproduces acceptable computational load. In the variable coefficient case, the convergence of the semi-discrete approximation of it is presented, which naturally connectsto the water wave equation. Analogue of these results for the water wave equationof finite depth is also discussed. To validate these theoretic observation, extensive numerical tests have been performed to verify the stability conditions. Simulations of thesimplified hyperbolic model in the high frequency regime and the water wave equation are also provided.