Riemann Problem with Nonlinear Resonance Effects and Well-Balanced Godunov Scheme for Shallow Fluid Flow Past an Obstacle

Abstract

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 Riemann problem "with nonlinear resonance" is solved. The so-called well-balanced Godunov scheme is constructed, which yields more accurate asymptotic states and preserves the balance between the source terms and flux terms. The numerical flux and source terms are simultaneously solved in both space and time to overcome resonance.