Shallow Water Flows in Channels

We consider the shallow water equations for flows through channels with arbitrary cross-section. The system forms a hyperbolic set of balance laws. Exact steady-state solutions are available and are controlled by the relation between the bottom topography and the channel geometry. We use a Roetype upwind scheme for the system. Considerations of conservation, near steady-state accuracy, velocity regularization and positivity near dry states are discussed. Numerical solutions are presented illustrating the merits of the scheme for a variety of flows and demonstrating the effect of the interplay between topography and geometry on the solution.