Solving Shallow Water flows in 2D with FreeFem++ on structured mesh

Abstract- FreeFem++ is an open source platform to solve partial differential equations numerically, based on finite element methods. What we will present in this work is inspired from the finite volume method for hyperbolic problem in order to solve it with finite element method in FreeFem++. In particular we will see how it can be possible, with an approach of finite element and finite volume method, to solve the Shallow Water equations (Saint- Venant system) in 2D with topographic source term. More precisely, we will define the quantity types of finite volume such as up-wind scheme, HLL flux, well balanced scheme with hydrostatic reconstruction that are used in the numerical resolution of the variational problem. To this end we will consider a particular rectangular structured isotropic domain using P1 finite element space. We note that in other situations, for example with unstructured anisotropic mesh or with other type of finite element space, an approach of Discontinuous Galerkin method could be used as in [11]. We note also that, there is a FreeVol project in order to introduce Finite Volume technic in FreeFem++ for hyperbolic PDEs by Frédéric Hecht et al. which is an ongoing work