A local discontinuous Galerkin method for a doubly nonlinear diffusion equation arising in shallow water modeling

In this paper, we study a local discontinuous Galerkin (LDG) method to approximate solutions of a doubly nonlinear diffusion equation, known in the literature as the diffusive wave approximation of the shallow water equations (DSW). This equation arises in shallow water flow models when special assumptions are used to simplify the shallow water equations and contains as particular cases: the Porous Medium equation and the parabolic p-Laplacian. Continuous in time a priori error estimates are established between the approximate solutions obtained using the proposed LDG method and weak solutions to the DSW equation under physically consistent assumptions. The results of numerical experiments in 2D are presented to verify the numerical accuracy of the method, and to show the qualitative properties of water flow captured by the DSW equation, when used as a model to simulate an idealized dam break problem with vegetation.