Weakly Compressible Two-Layer Shallow-Water Flows Along Channels

In this paper, we formulate a model for weakly compressible two-layer shallow water flows with friction in general channels. The formulated model is non-conservative, and in contrast to the incompressible limit, our system is strictly hyperbolic. The generalized Rankine–Hugoniot conditions are provided for the present system with non-conservative products to define weak solutions. We write the Riemann invariants along each characteristic field for channels with constant width in an appendix. A robust well-balanced path-conservative semi-discrete central-upwind scheme is proposed and implemented to validate exact solutions to the Riemann problem. We also present numerical tests in general channels to show the merits of the scheme.