The Aw-Rascle and Zhang's model: Vacuum problems, existence and regularity of the solutions of the Riemann problem

In this paper we will discuss some aspects of the recent macroscopic models of the second-order proposed by [Aw, A., Rascle, M., 2000. Resurrection of second order models of traffic flow. SIAM Journal of Applied Mathematics 60 (3), 916-938] and [Zhang, H.M., 2002. A non-equilibrium traffic model devoid of gas-like behavior. Transportation Research Part B 36, 275-290]. These models were suggested after the publication of an article written by [Daganzo, C.F., 1995. Requiem for second-order fluid approximations of traffic flow. Transportation Research Part B 29, 277-286] showing that some classical second-order models can exhibit non-physical solutions. It is shown in this note that the ARZ (Aw-Rascle-Zhang) model respects the anisotropic character of traffic flow, that it yields physical solutions, and that vacuum problems can be solved satisfactorily, provided that the fundamental diagram (equilibrium speed-density relationship) is extended in a suitable fashion. It follows that the Riemann problem for the ARZ model with extended fundamental diagram always admits a solution, and that this solution depends continuously on the initial conditions.