Boussinesq Systems of Bona-Smith Type on Plane Domains: Theory and Numerical Analysis

TL;DR

This paper analyzes Boussinesq systems of Bona-Smith type for surface wave modeling, establishing well-posedness, developing a convergent numerical method, and demonstrating simulations in complex domains.

Contribution

It provides the first well-posedness results for these systems on bounded domains and introduces a convergent Galerkin method for their numerical approximation.

Findings

Proved local well-posedness for initial-boundary-value problems.

Developed a Galerkin method with proven $L^2$ convergence.

Performed numerical simulations comparing Bona-Smith and BBM-BBM systems.

Abstract

We consider a class of Boussinesq systems of Bona-Smith type in two space dimensions approximating surface wave flows modelled by the three-dimensional Euler equations. We show that various initial-boundary-value problems for these systems, posed on a bounded plane domain are well posed locally in time. In the case of reflective boundary conditions, the systems are discretized by a modified Galerkin method which is proved to converge in $L^2$ at an optimal rate. Numerical experiments are presented with the aim of simulating two-dimensional surface waves in complex plane domains with a variety of initial and boundary conditions, and comparing numerical solutions of Bona-Smith systems with analogous solutions of the BBM-BBM system.