Exact solution of the Riemann problem for the shallow water equations with discontinuous bottom geometry

In this paper we present the exact solution of the Riemann Problem for the non-linear shallow water equations with a step-like bottom. The solution has been obtained by solving an enlarged system obtained by adding an additional equation for the bottom geometry and then using the principles of conservation of mass and momentum across the step. The resulting solution is unique and satis es the principle of dissipation of energy across the shock wave. We provide a few examples of possible wave patterns. The proposed exact Riemann problem solution is validated against numerical solutions by the rst-order centred Lax-Friedrichs scheme. A practical implementation of the proposed exact Riemann solver in the framework of a second-order upwind TVD method is also illustrated.