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 Roe-type 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 the topography and the geometry on the solution.

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