We investigate the Riemann problem for the shallow water equations withvariable and (possibly) discontinuous topography and provide a completedescription of the properties of its solutions: existence; uniqueness in thenon-resonant regime; multiple solutions in the resonant regime. This analysisleads us to a numerical algorithm that provides one with a Riemann solver.Next, we introduce a Godunov-type scheme based on this Riemann solver, which iswell-balanced and of quasi-conservative form. Finally, we present numericalexperiments which demonstrate the convergence of the proposed scheme even inthe resonance regime, except in the limiting situation when Riemann dataprecisely belong to the resonance hypersurface.
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