Hamiltonian regularisation of shallow water equations with uneven bottom

TL;DR

This paper extends Hamiltonian regularisation to shallow water equations over uneven, possibly time-dependent bottoms, preserving energy without artificial dissipation and validating the approach through numerical simulations.

Contribution

It generalizes Hamiltonian regularisation to variable bathymetry in shallow water models, maintaining energy conservation without artificial dissipation.

Findings

Regularised solutions preserve energy and lack dispersive effects.

Numerical results show consistency with classical models over uneven bottoms.

Method effectively handles bathymetry variations without introducing dissipation.

Abstract

The regularisation of nonlinear hyperbolic conservation laws has been a problem of great importance for achieving uniqueness of weak solutions and also for accurate numerical simulations. In a recent work, the first two authors proposed a so-called Hamiltonian regularisation for nonlinear shallow water and isentropic Euler equations. The characteristic property of this method is that the regularisation of solutions is achieved without adding any artificial dissipation or ispersion. The regularised system possesses a Hamiltonian structure and, thus, formally preserves the corresponding energy functional. In the present article we generalise this approach to shallow water waves over general, possibly time-dependent, bottoms. The proposed system is solved numerically with continuous Galerkin method and its solutions are compared with the analogous solutions of the classical shallow water and dispersive Serre-Green-Naghdi equations. The numerical results confirm the absence of dispersive and dissipative effects in presence of bathymetry variations.