Discrete analyses and numerical simulation of typical waves for environmental problems

The shallow-water system, obtained from the Navier-Stokes
equations by vertical averaging, is extensively used in
environmental studies (ocean, atmosphere, rivers) to simulate
the propagation of typical waves (inertia-gravity, Rossby, ...). 
For most of the discretization schemes, the numerical 
approximation of shallow-water models leads to the
appearance of spurious (non physical) solutions in the
representation of the waves. In order to understand these 
difficulties and to select appropriate spatial discretization 
schemes, discrete analyses may be performed. The aim of this
talk is to present such results in the context of continuous 
and mostly discontinuous Galerkin methods, and to propose a 
class of possible discretization schemes that is not affected 
by the spurious modes. Both discrete analyses and numerical
simulations involving wave propagation for environmental
problems will be presented.