We are interested in characterizing the resonant behavior that occurs in a shallow water system with an arbitrary bottom topography. The inhomogeneity introduced by the bottom topography elevation term in the system of equations is modeled by a linearly degenerate field. Strict hyperbolicity fails when one of the different nonlinear waves has a zero speed. A proliferation of reflected waves can occur by interactions and then resonance effects are important for a transonic flow. Classical nonlinear waves interact with standing waves and exhibit overcompressive, undercompressive, and marginal compressive waves.
Analytical understanding is used to introduce a new kind of numerical processing of the source term.
The numerical flux and source terms are simultaneously solved in both space and time to overcome resonance. 65M12 PII .