  We propose a variational framework for the resolution of a non-hydrostatic
Saint-Venant type model with bottom topography. This model is a shallow water
type approximation of the incompressible Euler system with free surface and
slightly differs from the Green-Nagdhi model, see [13] for more details about
the model derivation. The numerical approximation relies on a
prediction-correction type scheme initially introduced by Chorin-Temam [17] to
treat the incompressibility in the Navier-Stokes equations. The hyperbolic part
of the system is approximated using a kinetic finite volume solver and the
correction step implies to solve a mixed problem where the velocity and the
pressure are defined in compatible finite element spaces. The resolution of the
incompressibility constraint leads to an elliptic problem involving the
non-hydrostatic part of the pressure. This step uses a variational formulation
of a shallow water version of the incompressibility condition. Several
numerical experiments are performed to confirm the relevance of our approach.
