The exact Riemann solver for the shallow water equations with a discontinuous bottom

Abstract

A new exact Riemann solver for the shallow water equations with a discontinuous bottom is proposed. The algorithm is based on the approach to overcome the non-uniqueness of the Riemann problem solution by assuming that the ‘true’ solution should have the discharge at the bottom discontinuity, which continuously depends on the initial conditions. The solver ensures the existence and uniqueness of a solution for arbitrary initial conditions. The exact Riemann solver is embedded in the Godunov scheme for numerical solution of the shallow water equations to demonstrate its advantage. It is shown that the proposed solver allows one to significantly reduce the number of computational cells for stationary flows and non-stationary ones if the mesh can resolve non-stationary features of the flow. In addition, the practical problem of rainfall-runoff is considered, and the results obtained using the exact Riemann solver show good agreement with observation data.