Convergence of a characteristic-Galerkin scheme for a shallow water problem

In this paper, we present a numerical method based on a mixed characteristic-Galerkin (or Lagrangian-Galerkin) scheme to solve a shallow water problem with dirichlet boundary conditions. In a first part, we prove an L^2-bound on the water elevation. This bound is obtained by Lions in the case of a linearized momentum equation ^[^1^] and we extend it to the nonlinear case for which the existence is shown for small data ^[^2^]. This bound allows us to construct solutions as limits of the solutions of a regularized problem and to prove the convergence of the discrete problem towards the continuous. We give a numerical criteria connecting the Lagragian discretization and the number of Galerkin eigenvectors to solve the discrete equations with a fixed-point procedure. We present a few numerical results in the case of a fixed domain showing the coherence of the scheme which seems to be adaptable to a domain depending on time.