Bona-Smith-type systems in bounded domains with slip-wall boundary conditions: Theoretical justification and a conservative numerical scheme

TL;DR

This paper establishes the mathematical well-posedness and conservation properties of Bona-Smith-type water wave systems in bounded domains with slip-wall boundaries, and introduces a conservative numerical scheme validated by numerical experiments.

Contribution

It provides the first theoretical proof of existence and uniqueness for these systems with slip-wall conditions and develops a conservative finite element scheme for their numerical simulation.

Findings

Proved local-in-time existence and uniqueness of solutions.

Demonstrated conservation of mass, vorticity, and energy in the numerical scheme.

Validated the numerical method through challenging experiments.

Abstract

Considered herein is a class of Boussinesq systems of Bona-Smith type that describe water waves in bounded two-dimensional domains with slip-wall boundary conditions and variable bottom topography. Such boundary conditions are necessary in situations involving water waves in channels, ports, and generally in basins with solid boundaries. We prove that, given appropriate initial conditions, the corresponding initial-boundary value problems have unique solutions locally in time, which is a fundamental property of deterministic mathematical modeling. Moreover, we demonstrate that the systems under consideration adhere to three basic conservation laws for water waves: mass, vorticity, and energy conservation. The theoretical analysis of these specific Boussinesq systems leads to a conservative mixed finite element formulation. Using explicit, relaxation Runge-Kutta methods for the discretization in time, we devise a fully discrete scheme for the numerical solution of initial-boundary value problems with slip-wall conditions, preserving mass, vorticity, and energy. Finally, we present a series of challenging numerical experiments to assess the applicability of the new numerical model.