Multilevel finite volume methods and boundary conditions for geophysical flows

This review article concerns multi-level methods for finite volume discretizations. They are presented in the context of the non-viscous shallow water equations in space dimension one and two, although the ideas are of more general validity. The second, partly related topic, addressed in this article is that of boundary conditions also presented in the context of the inviscid shallow water equations. Beside algorithmic presentation, the article contains the discussion of some numerical stability issues and the results of some numerical simulations. The multi-level method is applied numerically to the equations of shallow water using a central-upwind scheme for finite volumes and we prove the stability for the linear shallow water equations in different situations. The question of the boundary conditions is illustrated for the two-dimensional equations in the reference case of the so-called Rossby soliton.