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

TL;DR

This paper analyzes the stability and convergence of Runge-Kutta methods for a simplified nonlocal hyperbolic model related to the water wave equation, providing optimal time step constraints and numerical validation.

Contribution

It offers a systematic stability analysis and optimal CFL-like conditions for Runge-Kutta methods applied to a simplified water wave model, bridging theory and numerical experiments.

Findings

Optimal time step constraints identified for explicit Runge-Kutta methods.

Stability conditions verified through extensive numerical tests.

Convergence of semi-discrete approximation demonstrated for variable coefficients.

Abstract

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.