Numerical discretizations for shallow water equations with source terms on unstructured meshes

In the following lines we introduce two frictional schemes for the discretization of the 2D Shallow Water system, on unstructured meshes. The starting point consists in writing both of them as convex combinations of 1D schemes. Then, we propose to include the resistance effects proceeding to a slight adaptation of the gathered convex components, using the frictional approach recently developed in [2]. This method turns out to provide an excellent behavior for vanishing water heights, and does not require a modification of the CFL. Numerical experiments will be performed in order to assess the capacity of the two schemes in dealing with wetting and drying, complex geometry and topography. Introduction. In this work we consider discretizations of the 2D Non linear Shallow Water equations (NSW). As the name suggests, the NSW model consists in an hyperbolic set of non linear equations, and is used to describe the motion of shallow flows, as flood waves, flooding and drying, dam breaks, or more generally any kind of hydrodynamic processes near coasts or in bed rivers. Denoting h the water height, q = (qx, qy) the discharge and z a parametrization of the bed slope, we write the set of NSW equations under its conservative form : wt +∇.H(w) = −B(w, z)−F(w) , (1) with w =  h qx qy  H(w) =  qx qy q x h + 12gh qxqy h qxqy h q y h + 1 2gh 2  , (2) and the following expressions for bathymetry and friction : B(w, z) =  0 ghzx ghzy  F(w) =  0 κ hγ qx κ hγ qy  . (3) 2000 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35.