Delta-shocks and vacuums as flux-approximation limit of solutions to the Aw-Rascle model with time-dependent source

Abstract

The flux-approximation limit of solutions to the Aw-Rascle (AR) traffic flow model with time-dependent source is considered. First, the Riemann problem containing flux perturbation and generalized Chaplygin gas (GCG) pressure is solved. Then, it is proved that, as the flux perturbation and pressure decrease to some certain critical value, any Riemann solution containing a shock wave and a contact discontinuity will approach a delta shock wave of the perturbed AR model itself. Furthermore, as both the flux perturbation and pressure vanish, this delta shock wave with critical value eventually tends to the delta-shock solution of the pressureless gas dynamics (PGD) model with time-dependent source. By contrast, any solution containing a rarefaction wave and a contact discontinuity converges to a two-contact-discontinuity solution of the PGD model with time-dependent source and the nonvacuum intermediate state in between tends to a vacuum state. Finally, some representatively numerical results are presented.