The Riemann problem for the shallow water equations with discontinuous topography: The wet–dry case

Abstract

In this paper we consider Riemann problems for the shallow water equations with discontinuous topography whose initial conditions correspond to a wet–dry front: at time t=0 there is vacuum on the right or on the left of the step. Besides the theoretical interest of this analysis, the results may be useful to design numerical methods and/or to produce reference solutions to compare different schemes. We show that, depending on the state at the wet side, 0, 1, or 2 self-similar solutions can be constructed by composing simple waves. In problems with 0 solutions, the step acts as an obstacle for the fluid and physically meaningful solutions can be constructed by interpreting the problem as a partial Riemann problems for the homogeneous shallow water system. Some numerical results are shown where different numerical methods are compared. In particular, it is shown that, in the non-uniqueness cases, the numerical solutions can converge to one or to the other solution, what is the reason that explains the huge differences observed when different numerical methods are applied to the shallow water system with abrupt changes in the bottom.

Moreover, problems with zero solutions will be reinterpreted as Partial Riemann problems for the homogeneous system what will allow us to build a physically solution.

When one side of the step is wet and the other one is dry. We will specify the regions where we can find zero, one or two solutions, giving the form of the solution when it is possible and giving an alternative when it is not possible.