A.I. Aleksyuk, V.V. Belikov. The uniqueness of the exact solution of the Riemann problem for the shallow water equations with discontinuous bottom // Journal of Computational Physics, vol. 390, 2019, pp. 232-248

The Riemann problem for the shallow water equations with discontinuous topography is considered. In a general case the exact solution of this problem is not unique, which complicates the application of an exact Riemann solver in numerical methods, since it is not clear which solution should be chosen. In the present work it is shown that involving an additional physical assumption makes it possible to prove the existence and uniqueness of the solution. The assumption is that the discharge at the bottom discontinuity should continuously depend on the initial conditions. The proven uniqueness opens up a possibility to use an exact Riemann solver for a numerical solution of the shallow water equations with complex discontinuous topography.