A novel energy-bounded Boussinesq model and a well balanced and stable numerical discretisation

TL;DR

This paper introduces a new energy-bounded Boussinesq model tailored for problems with complex bathymetry, featuring a stable numerical scheme and tunable dispersive parameters for accurate wave simulation.

Contribution

A novel nonlinear energy-stable Boussinesq system with tunable dispersive parameters and a robust finite-volume scheme for one- and two-dimensional wave modeling.

Findings

System is nonlinearly entropy-stable and flexible.

Finite-volume scheme demonstrates robustness and accuracy.

Model effectively captures dispersive relations up to high wavenumbers.

Abstract

In this work, a novel Boussinesq system is put forward. The system is naturally nonlinearly entropy/energy-stable, and is designed for problems with sharply varying bathymetric features. The system is flexible and allows tuning of the dispersive parameters to the relevant wavenumber range of the problem at hand. We present a few such parameter sets, including one that tracks the dispersive relation of the underlying Euler equations up to a nondimensional wavenumber of about $30$. In the one-dimensional case, we design a stable finite-volume scheme and demonstrate its robustness and accuracy in a suite of test problems including Dingemans's wave experiment. We generalise the system to the two-dimensional case and sketch how the numerical scheme can be straightforwardly generalised.