A conservative fully-discrete numerical method for the regularised shallow water wave equations

TL;DR

This paper introduces a new conservative fully-discrete numerical scheme combining finite element and Runge-Kutta methods for simulating long water waves, ensuring energy preservation for accurate long-time wave propagation modeling.

Contribution

The paper presents a novel conservative fully-discrete scheme for the regularised shallow water equations, integrating Galerkin finite elements with explicit Runge-Kutta time stepping.

Findings

Demonstrates improved accuracy and convergence properties

Ensures energy conservation in long-time wave simulations

Reveals a new convergence pattern for the numerical method

Abstract

The paper proposes a new, conservative fully-discrete scheme for the numerical solution of the regularised shallow water Boussinesq system of equations in the cases of periodic and reflective boundary conditions. The particular system is one of a class of equations derived recently and can be used in practical simulations to describe the propagation of weakly nonlinear and weakly dispersive long water waves, such as tsunamis. Studies of small-amplitude long waves usually require long-time simulations in order to investigate scenarios such as the overtaking collision of two solitary waves or the propagation of transoceanic tsunamis. For long-time simulations of non-dissipative waves such as solitary waves, the preservation of the total energy by the numerical method can be crucial in the quality of the approximation. The new conservative fully-discrete method consists of a Galerkin finite element method for spatial semidiscretisation and an explicit relaxation Runge--Kutta scheme for integration in time. The Galerkin method is expressed and implemented in the framework of mixed finite element methods. The paper provides an extended experimental study of the accuracy and convergence properties of the new numerical method. The experiments reveal a new convergence pattern compared to standard Galerkin methods.