A regularized shallow-water waves system with slip-wall boundary conditions in a basin: Theory and numerical analysis

TL;DR

This paper introduces a new well-posed Boussinesq system for simulating long, nonlinear dispersive waves in bounded basins with slip-wall boundary conditions, combining theoretical analysis and numerical validation.

Contribution

A novel Boussinesq system is derived and proven well-posed for slip-wall boundary conditions, with a finite-element numerical method and convergence analysis.

Findings

The new system is mathematically well-posed.

Numerical method shows optimal convergence.

Simulations agree well with experimental data.

Abstract

The simulation of long, nonlinear dispersive waves in bounded domains usually requires the use of slip-wall boundary conditions. Boussinesq systems appearing in the literature are generally not well-posed when such boundary conditions are imposed, or if they are well-posed it is very cumbersome to implement the boundary conditions in numerical approximations. In the present paper a new Boussinesq system is proposed for the study of long waves of small amplitude in a basin when slip-wall boundary conditions are required. The new system is derived using asymptotic techniques under the assumption of small bathymetric variations, and a mathematical proof of well-posedness for the new system is developed. The new system is also solved numerically using a Galerkin finite-element method, where the boundary conditions are imposed with the help of Nitsche's method. Convergence of the numerical method is analyzed, and precise error estimates are provided. The method is then implemented, and the convergence is verified using numerical experiments. Numerical simulations for solitary waves shoaling on a plane slope are also presented. The results are compared to experimental data, and excellent agreement is found.