A well-balanced Runge-Kutta discontinuous Galerkin method for the shallow-water equations with flooding and drying

We build and analyze a Runge-Kutta discontinuous Galerkin method to approximate the one- and two-dimensional shallow-water equations. We introduce a flux modification technique to derive a well-balanced scheme preserving steady states at rest with variable ground elevation and a slope modification technique to deal satisfactorily with flooding and drying. Numerical results illustrating the performance of the proposed scheme are presented.