Discretization of a new model of dispersive waves with improved dispersive properties and exact conservation of energy

In this work, we derive a hyperbolic system of dispersive equations for the numerical simulation of coastal waves with improved dispersive properties and admitting an exact energy conservation equation. This system is derived with the assumption of a moderate non-linearity and of a correction coefficient close to 1. This system contains the same non-linear terms as the Serre-Green-Naghdi equations, which are obtained in the limit where the Mach number tends to zero. The assumptions are only used to neglect non-linear terms related to the improvement of dispersive properties. The bathymetry can be included with a mild-slope hypothesis. On this basis, we propose an energy-stable numerical scheme relying on a splitting between the hyperbolic and dispersive parts of the model. The stability of the method is achieved through the discrete dissipation of the energy balance specific to each step. We also establish the existence of soliton solutions for this model. Numerical simulations are proposed to highlight the dispersive properties of the model, as well as the dissipative character of the scheme.