Discontinuous Galerkin Methods for the Shallow-Water Equations with bathymetric terms and dry areas

Shallow-Water Equations are encountered in many applications related to hydraulics, flood propagation or dam break problems. This set of nonlinear hyperbolic equations models a fluid flow in a domain where the depth of the fluid is much lower than its horizontal dimensions. Usually, these equations are treated using Finite Difference, Finite Volume or Continuous Finite Element methods. In this paper, we develop a Runge-Kutta Discontinuous Galerkin Finite Element Method to deal with the one-dimensional Shallow-Water Equations. Discontinuous Galerkin Methods are currently experiencing a vigorous development. Advantages of such methods include their high order of accuracy and their ease to handle complicated geometries. For Shallow-Water Equations, the goals of this work are on the one hand to preserve quiescent flows (and, more generally, steady-states) in the presence of bathymetric terms and, on the other hand, to handle accurately flooding and drying. Drawing on earlier ideas for Finite Volume Methods [1], we design a Discontinuous Galerkin Method that preserves flows at rest. We assess the scheme on various one-dimensional problems including subcritical and trancritical flows as well as cases with flooding and drying [2]. These tests confirm the stability of the scheme and its high accuracy. Moreover we present a generalization of the method to the two-dimensional case that we illustrate with some numerical results