Some considerations on numerical schemes for treating hyperbolicity issues in two-layer models

Multi-layer depth-averaged models are widely employed in various hydraulic engineering applications. Yet, such models are not strictly hyperbolic. Their equation systems typically lose hyperbolicity when the relative velocities between layers become too large, which is associated with Kelvin–Helmholtz instabilities involving turbulent momentum exchanges between the layers. Focusing on the two-layer case, we present a numerical improvement that locally avoids the loss of hyperbolicity. The proposed modification introduces an additional momentum exchange between layers, whose value is iteratively calculated to be strictly sufficient to keep the system hyperbolic. The approach can be easily implemented in any finite volume scheme and there is no limitation concerning the density ratio between layers. Numerical examples, employing both HLL-type and Roe-type approximate Riemann solvers, are reported to validate the method and its key features.