  We propose an efficient numerical scheme for the resolution of a
non-hydrostatic Saint-Venant type model. The model is a shallow water type
approximation of the incompressbile Euler system with free surface and slightly
differs from the Green-Naghdi model. The numerical approximation relies on a
kinetic interpretation of the model and a projection-correction type scheme.
The hyperbolic part of the system is approximated using a kinetic based finite
volume solver and the correction step implies to solve an elliptic problem
involving the non-hydrostatic part of the pressure. We prove the numerical
scheme satisfies properties such as positivity, well-balancing and a fully
discrete entropy inequality. The numerical scheme is confronted with various
time-dependent analytical solutions. Notice that the numerical procedure
remains stable when the water depth tends to zero.
