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

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 nonlinear 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, is to resolve accurately dispersive wave propagation on very coarse meshes is demonstrated by several of the tests.