Discrete analyses and numerical simulation of the shallow-water system

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 inertia-gravity and Rossby waves. For most of the discretization schemes, the numerical approximation of shallow-water models is a delicate problem leading to the appearance of spurious (non physical) solutions in the representation of the fast (inertia-gravity) and slow (Rossby) waves. In order to understand these difficulties and to select appropriate spatial discretization schemes, Fourier / dispersion analyses and the study of the null space of the associated discretized problems have proven beneficial. The aim of this talk is to present such results and to propose a class of possible discretization schemes that is not affected by the spurious modes. Both discrete analyses and numerical simulations involving the eddy propagation in an oceanic context will be presented.