Discontinous Galerkin Methods for the Sahllow-Water Equations with bathymetric terms and dry areas

In this paper, we develop Discontinuous Galerkin Methods to deal with the Shallow-Water Equations involving bathymetric temrms and dry areas. Our major goals are on the one hand to preserve quiescent flows and, more generally, steady-states and on the other hand to ensure stability when the flow includes flooding and drying. Drawing on earlier ideas for Finite Volume Methods [Aud,Ber], we design two Discontinuous Galerkin Methods that preserve flows at rest. We investigate their acuracy on various numerical tests including subcritical and trancritical flows. Moreover, for one of the methods, we analyze its behaviour for some test cases including flooding and drying. [Aud] : E. Audusse, F. Bouchut, M.O. Bristeau, R. Klein, B. Perthame (2004) : "A fast and well-balanced scheme with hydrostatic reconstruction fo the shallow water flows", SIAM J. Sci. Comput., Vol 25, n°6, p. 2050-2065. [Ber] : A. Bermudez, M.E. Vazquez (1994) : "Upwind methods for hyperbolic conservation laws with source terms", Computers and Fluids, 23, p.1049-1071.