Maximal dissipation and shallow water flow -- the dam-break problem

The shallow water system with a bed jump describes a fluid flow over a dam, represented by a simple step function with the Riemann initial data is such that the vacuum is on the right-hand side. The main question is whether the fluid will stay on the left-hand side of the dam. After approximating the step function space derivative with a fixed shadow wave, solutions to that problem stay bounded. A proper connection between waves on both sides of the dam, and a unique solution afterward is found by using the maximal dissipation principle.