The fully nonlinear and weakly dispersive Green-Naghdi equa- tions for shallow water waves of large amplitude is studied. An hybrid finite volume and finite difference splitting approach is proposed. Numerical validations are then performed in one hori- zontal dimension.
In the study of nearshore dynamics, the propagation and trans- formations of waves in shallow water play a key role. An accurate modelling of associated processes, such as wave-breaking and swash motions, is paramount for the study of coastal flooding due to storm waves or tsunamis, or to improve the prediction of short-term beach evolution, since they are the main source of sediment transport in the nearshore. Modelling these processes requires a phase-resolving model, able to accurately describe wave-breaking and run-up over strongly varying topographies. In an incompressible and homogeneous fluid, the propagation of surface waves is described in its full generality by the Navier-Stokes equations with nonlinear boundary conditions at the surface and at the bottom. But this problem is highly computationally demanding, and therefore is not suitable for large scale propagation applications. Therefore, more simple models have been derived to describe the behavior of the solution in some physical speci_c regimes. They are based on Nonlinear Shallow Water (NSW) or Boussinesq- type (BT) equations. The reader is referred to (Lannes and Bonneton, 2009) for a study of the various shallow-water regimes. Both nonlinear and dispersive effects can be accounted for by BT equations, with various degrees of accuracy. As the wave dynamic becomes strongly nonlinear in the final stages of shoaling and in the surf and swash zones, fully nonlinear equations, such as the Green-Naghdi (GN) equations are required (Green and Naghdi, 1976). The GN equations provide a correct description of the waves be- fore wave breaking.