Maximal energy dissipation principle and the Riemann problem for shallow water flow: Application to the dam-break problem

Abstract

The shallow water system with a single discontinuity of the bottom topography is a simple model of a fluid flow with a sudden change in bottom elevation. The discontinuity is modeled by an increasing step function for simplicity and further application to a specific problem. Using an piecewise continuous approximation with the parameter ε ≪ 1 of the step function at $x=0$, the Riemann problem for the system is transferred after letting ε → 0 into the minimization of energy problem with the constraints: The approximated solution has to satisfy the Dafermos’s maximal dissipation condition around the discontinuity, while shock and rarefaction waves must have non-positive velocity for x < 0 and non-negative one for x > 0. The application of the procedure is made for the dam-break problem with the additional assumptions that fluid is moving from left to right for x < 0 and the bed is dry for x > 0. The existence of an admissible solution corresponds to the dam failure. A shadow wave is a perturbation that consists of two shock waves with an infinitesimally small area between. It is used to model the fluid behavior near the discontinuity.