Solution of 2D Boussinesq systems with FreeFem++: The flat bottom case

TL;DR

This paper develops a FreeFem++ numerical method for solving 2D Boussinesq systems modeling surface wave propagation over flat bottoms, incorporating mesh adaptation and algorithm optimization, with results consistent with existing literature.

Contribution

It introduces a detailed FreeFem++ implementation for 2D Boussinesq systems with mesh adaptation and compares different system families.

Findings

Solutions agree with literature results

Optimized algorithm improves computational efficiency

Comparison of Boussinesq families highlights differences

Abstract

We consider here different family of Boussinesq systems in two space dimensions. These systems approximate the three-dimensional Euler equations and consist of three coupled nonlinear dispersive wave equations that describe propagation of long surface waves of small amplitude in ideal fluids over a horizontal bottom. We present here a FreeFem++ code aimed at solving numerically these systems where a discretization using P1 finite element for these systems was taken in space and a second order Runge-Kutta scheme in time. We give the detail of our code where we use a mesh adaptation technique. An optimization of the used algorithm is done and a comparison of the solution for different Boussinesq family is done too. The results we obtained agree with those of the literature.