Structure-preserving approximations of the Serre-Green-Naghdi equations in standard and hyperbolic form

TL;DR

This paper introduces structure-preserving numerical methods for the Serre-Green-Naghdi equations, ensuring conservation of mass, energy, and momentum, and applicable to various discretization schemes for accurate dispersive wave simulations.

Contribution

The paper presents novel, energy-preserving numerical schemes for the Serre-Green-Naghdi equations, including hyperbolic approximations enabling explicit time stepping and broad discretization compatibility.

Findings

Methods conserve total water mass and energy.

Schemes are well-balanced for lake-at-rest state.

Structure preservation improves accuracy on coarse meshes.

Abstract

We develop structure-preserving numerical methods for the Serre-Green-Naghdi equations, a model for weakly dispersive free-surface waves. We consider both the classical form, requiring the inversion of a non-linear elliptic operator, and a hyperbolic approximation of the equations, allowing fully explicit time stepping. Systems for both flat and variable topography are studied. Our novel numerical methods conserve both the total water mass and the total energy. In addition, the methods for the original Serre-Green-Naghdi equations conserve the total momentum for flat bathymetry. For variable topography, all the methods proposed are well-balanced for the lake-at-rest state. We provide a theoretical setting allowing us to construct schemes of any kind (finite difference, finite element, discontinuous Galerkin, spectral, etc.) as long as summation-by-parts operators are available in the chosen setting. Energy-stable variants are proposed by adding a consistent high-order artificial viscosity term. The proposed methods are validated through a large set of benchmarks to verify all the theoretical properties. Whenever possible, comparisons with exact, reference numerical, or experimental data are carried out. The impressive advantage of structure preservation, and in particular energy preservation, to resolve accurately dispersive wave propagation on very coarse meshes is demonstrated by several of the tests.