Boussinesq-Peregrine water wave models and their numerical approximation

TL;DR

This paper investigates the numerical approximation of Boussinesq-Peregrine water wave models using Galerkin finite element methods, introduces a new BBM-BBM type system with improved convergence, and demonstrates their effectiveness through numerical experiments.

Contribution

It introduces a new BBM-BBM type Boussinesq system with better convergence properties and analyzes the Galerkin finite element method's effectiveness for these models.

Findings

The new BBM-BBM system exhibits improved convergence.

Both systems accurately model small amplitude long waves.

Galerkin finite element method is more effective for BBM-BBM systems.

Abstract

In this paper we consider the numerical solution of Boussinesq-Peregrine type systems by the application of the Galerkin finite element method. The structure of the Boussinesq systems is explained and certain alternative nonlinear and dispersive terms are compared. A detailed study of the convergence properties of the standard Galerkin method, using various finite element spaces on unstructured triangular grids, is presented. Along with the study of the Peregrine system, a new Boussinesq system of BBM-BBM type is derived. The new system has the same structure in its momentum equation but differs slightly in the mass conservation equation compared to the Peregrine system. Further, the finite element method applied to the new system has better convergence properties, when used for its numerical approximation. Due to the lack of analytical formulas for solitary wave solutions for the systems under consideration, a Galerkin finite element method combined with the Petviashvili iteration is proposed for the numerical generation of accurate approximations of line solitary waves. Various numerical experiments related to the propagation of solitary and periodic waves over variable bottom topography and their interaction with the boundaries of the domains are presented. We conclude that both systems have similar accuracy when approximate long waves of small amplitude while the Galerkin finite element method is more effective when applied to BBM-BBM type systems.