The layerwise Green-Naghdi model: derivation and numerical resolution

This work focuses on solving the incompressible free surface Euler equations also called water wave equations.
The objective is to provide a robust numerical strategy that is sufficiently effective to be applied at the scale of applications: coastal areas or rivers.
Reduced models based on vertical integration are widely used, particularly the shallow water equations.
This model is based on two main assumptions: the homogeneity of the horizontal velocity over the water column and the hydrostatic pressure.
However, this model failed to recover some of the essential properties of the flow, i.e. the dispersion relation essential to wave propagation and the vertical vorticity essential to estimate the energy lost in a flow.
More sophisticated models have been derived and no longer require these assumptions taken separately, namely the Green-Naghdi model and the shallow water layered model (other strategies exist).
Following this previous works, we propose a derivation of a layerwise version of the Green-Naghdi model, which could be understood as a vertical semi-discretization of the water wave equations (without assumption except regularity).
Finally, we propose a fully-discretization of the layerwise Green-Naghdi model that satisfies the dissipation of mechanical energy (acting as mathematical entropy).
A few numerical simulations will illustrate the robustness of the strategy and the consistency with the analytical solutions.