On the Galerkin / finite-element method for the Serre equations

TL;DR

This paper introduces a highly accurate Galerkin finite-element scheme with explicit Runge-Kutta time integration for the Serre equations, effectively modeling dispersive shallow water waves and their complex interactions.

Contribution

It presents a novel fully-discrete numerical scheme combining smooth spline-based Galerkin methods with explicit Runge-Kutta, achieving optimal accuracy and stability for the Serre system.

Findings

Scheme achieves optimal spatial and temporal accuracy.

Stability does not require restrictive time step conditions.

Effective modeling of solitary, cnoidal, and dispersive shock waves.

Abstract

A highly accurate numerical scheme is presented for the Serre system of partial differential equations, which models the propagation of dispersive shallow water waves in the fully-nonlinear regime. The fully-discrete scheme utilizes the Galerkin / finite-element method based on smooth periodic splines in space, and an explicit fourth-order Runge-Kutta method in time. Computations compared with exact solitary and cnoidal wave solutions show that the scheme achieves the optimal orders of accuracy in space and time. These computations also show that the stability of this scheme does not impose restrictive conditions on the temporal step size. In addition, solitary, cnoidal, and dispersive shock waves are studied in detail using this numerical scheme for the Serre system and compared with the 'classical' Boussinesq system for small-amplitude shallow water waves. The results show that the interaction of solitary waves in the Serre system is more inelastic. The efficacy of the numerical scheme for modeling dispersive shocks is shown by comparison with asymptotic results. These results have application to the modeling of shallow water waves of intermediate or large amplitude, such as occurs in the nearshore zone.