Finite element solution of the shallow water equations by a quasi-direct decomposition procedure

We describe in this paper some new methods for solving the shallow water flow problem; this problem is composed of two coupled non-linear equations. First a fractional step method is used to decouple the difficulties due to the convection and the propagation. The diffusion propagation step amounts to solving two coupled linear equations on the water depth and the fluxes. We used then some quasi-direct decomposition techniques to decouple the unknowns, leading to symmetric systems only. In addition some non-trivial boundary conditions on the flux are examined. Particularly we are interested to get decoupled systems on the two components of the flux. Numerical tests are presented to enlighten the behaviour and the capabilities of the method. Finally an industrial example is treated on the Dunkirk harbour.