Extended water wave systems of Boussinesq equations on a finite interval: Theory and numerical analysis

TL;DR

This paper analyzes a class of Boussinesq water wave systems on finite intervals, establishing well-posedness, identifying boundary conditions, and proposing a new numerical method with convergence proofs, supported by numerical experiments.

Contribution

It provides a theoretical and numerical framework for Boussinesq systems with boundary conditions, including a new Galerkin method and analysis of wave reflection.

Findings

Analytical solution of linearized system via Fokas transform.

Identification of admissible boundary conditions for nonlinear system.

Development and convergence proof of a new Galerkin numerical scheme.

Abstract

Considered here is a class of Boussinesq systems of Nwogu type. Such systems describe propagation of nonlinear and dispersive water waves of significant interest such as solitary and tsunami waves. The initial-boundary value problem on a finite interval for this family of systems is studied both theoretically and numerically. First, the linearization of a certain generalized Nwogu system is solved analytically via the unified transform of Fokas. The corresponding analysis reveals two types of admissible boundary conditions, thereby suggesting appropriate boundary conditions for the nonlinear Nwogu system on a finite interval. Then, well-posedness is established, both in the weak and in the classical sense, for a regularized Nwogu system in the context of an initial-boundary value problem that describes the dynamics of water waves in a basin with wall-boundary conditions. In addition, a new modified Galerkin method is suggested for the numerical discretization of this regularized system in time, and its convergence is proved along with optimal error estimates. Finally, numerical experiments illustrating the effect of the boundary conditions on the reflection of solitary waves by a vertical wall are also provided.