Discrete energy estimates for a hyperbolic model of dispersive equations

Arnaud Duran

We propose a numerical approach for a hyperbolic system of dispersive equations for coastal wave simulation with improved dispersive properties and admitting an exact energy equation. This system is derived with an assumption of mean non-linearity and a correction coefficient close to 1. This system contains the same non-linear terms as the Serre-Green-Naghdi equations in the limit where the Mach number tends to zero. The proposed scheme is based on a splitting between the hyperbolic part and the relaxed part containing the dispersive terms. The stability of the scheme is ensured by proposing an energy-dissipating approach at each of the two stages of the splitting. This is a collaboration with Gaël L. Richard (INRAE Université Grenoble Alpes).