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 that includes 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 satisfies the principle of dissipation of energy across the shock wave. We provide examples of possible wave patterns. Numerical solution of a first-order dissipative scheme as well as an implementation of our Riemann solver in the second-order upwind method are compared with the proposed exact Riemann problem solution. A practical implementation of the proposed exact Riemann solver in the framework of a second-order upwind TVD method is also illustrated.