A hyperbolic reformulation of the Serre-Green-Naghdi model for general bottom topographies

We present a novel hyperbolic reformulation of the Serre-Green-Naghdi (SGN) model for the description of dispersive water waves. Contrarily to the classical Boussinesq-type models, it contains only first order derivatives, thus allowing to overcome the numerical difficulties arising from higher order derivative terms, especially in the context of high order discontinuous Galerkin finite element schemes. The proposed model reduces to the original SGN model when an artificial sound speed tends to infinity. Moreover, it is endowed with an extra conservation law from which the energy-type conservation law associated with the original SGN model is retrieved when the artificial sound speed goes to infinity. In order to provide a theoretical basis for the proposed model, a derivation from the vertical average of the compressible Euler equations has been proposed. The governing partial differential equations are then solved at the aid of high order ADER discontinuous Galerkin finite element schemes. The new model has been successfully validated against numerical and experimental results, for both flat and non-flat bottom. For bottom topographies with large variations, the new model proposed in this paper provides more accurate results with respect to the hyperbolic reformulation of the SGN model with the mild bottom approximation.