Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations

Shallow water equations with a non-flat bottom topography have been widely used to model flows in rivers and coastal areas. An important difficulty arising in these simulations is the appearance of dry areas, as standard numerical methods may fail in the presence of these areas. These equations also have steady state solutions in which the flux gradients are nonzero but exactly balanced by the source term. In this paper we propose a high order discontinuous Galerkin method which can maintain the steady state exactly, and at the same time preserves the non-negativity of the water height without loss of mass conservation. A simple positivity preserving limiter, valid under suitable CFL condition, will be introduced in one dimension and then extended to two dimensions with rectangular meshes. Numerical tests are performed to verify the positivity preserving property, well balanced property, high order accuracy, and good resolution for smooth and discontinuous solutions.