Admissibility of a wide cluster solution in "anisotropic" higher-order traffic flow models

Abstract

We analytically investigate a wide cluster solution and show that it is not admitted in some of the traffic flow models in the literature. For those traffic flow models that admit the wide cluster solution, the relationship between two important control parameters and the critical densities that divide an equilibrium solution into stable and unstable regions is thoroughly discussed in detail. We find that such wide clusters exist with a free traffic density in an unstable region, and with one or three critical densities. These results are different from the cases in the well-known higher-order traffic flow models of Payne and Whitham [H. J. Payne, "Models of freeway traffic and control," in Mathematical Models of Public Systems, A. G. Bekey, ed., Simulation Council Proc. Ser. 1, La Jolla, CA, 1971, pp. 51-61], [G. B. Whitham, Linear and Nonlinear Waves, John Wiley and Sons, New York, 1974], Kühne [R. D. Kühne, "Macroscopic freeway model for dense traffic-stop-start waves and incident detection," in Proceedings of the 9th International Symposium on Transportation and Traffic Theory, J. Volmuller and R. Hamerslag, eds., VNU Science Press, Utrecht, 1984, pp. 21-42], and Kerner and Konhäuser [B. S. Kerner and P. Konhäuser, Phys. Rev. E (3), 50 (1994), pp. 54-83]. © 2007 Society for Industrial and Applied Mathematics.