Limiting behavior of non-self-similar Riemann solutions for the Umami Chaplygin Aw-Rascle model with friction

The Riemann problem for the Umami Chaplygin Aw-Rascle model of traffic flow with Coulomb-like friction is solved by introducing the reasonable flux-function. Three kinds of non-self-similar Riemann solutions involving delta-shock are given exactly by the means of variable substitution method. The serious traffic congestion may be explained by the delta-shock, for which, the generalized Rankine–Hugoniot relation and entropy condition are clarified. Since the friction term is involved, all waves containing the rarefaction wave, contact discontinuity, shock wave, and delta-shock are bent into parabolic curves. By analyzing the limiting behavior of the solutions, it is discovered that as the Umami Chaplygin gas pressure and flux-function vanish simultaneously, the Riemann solution containing the rarefaction wave and the contact discontinuity tends to the vacuum solution of the zero-pressure Euler equations with friction, and the solution including the shock wave and the contact discontinuity tends to the delta-shock solution of the zero-pressure Euler equations with friction.