×

A traffic-flow model with constraints for the modeling of traffic jams. (English) Zbl 1197.35159

Recently, Berthelin et al. introduced a traffic-flow model describing the formation and the dynamics of traffic jams. This model, which consists of a constrained pressureless gas dynamics system, assumes that the maximal density constraint is independent of the velocity. However, in practice, the distribution of vehicles on a highway depends on their velocity. In this paper, a more realistic model, namely the Second-Order Model with Constraints (in short SOMC), is proposed, derived from the Aw and Rascle model, which takes into account this feature. Moreover, when the maximal density constraint is saturated, the SOMC model ‘relaxes’ to the Lighthill and Whitham model. An existence result of weak solutions for this model by means of cluster dynamics is proved in order to construct a sequence of approximations, and the associated Riemann problem is solved completely.

MSC:

35L60 First-order nonlinear hyperbolic equations
90B20 Traffic problems in operations research
35D99 Generalized solutions to partial differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1137/S0036139997332099 · Zbl 0957.35086 · doi:10.1137/S0036139997332099
[2] DOI: 10.1103/PhysRevE.51.1035 · doi:10.1103/PhysRevE.51.1035
[3] DOI: 10.1142/S0218202502001635 · Zbl 1027.35079 · doi:10.1142/S0218202502001635
[4] DOI: 10.1007/s00205-007-0061-9 · Zbl 1153.90003 · doi:10.1007/s00205-007-0061-9
[5] DOI: 10.1137/S0036142997317353 · Zbl 0924.35080 · doi:10.1137/S0036142997317353
[6] DOI: 10.1016/S0895-7177(02)80029-2 · Zbl 0994.90021 · doi:10.1016/S0895-7177(02)80029-2
[7] DOI: 10.1007/978-3-662-22019-1 · doi:10.1007/978-3-662-22019-1
[8] DOI: 10.1016/0191-2615(95)00007-Z · doi:10.1016/0191-2615(95)00007-Z
[9] DOI: 10.1142/S0218202507002157 · Zbl 1117.35320 · doi:10.1142/S0218202507002157
[10] DOI: 10.1287/opre.9.4.545 · Zbl 0096.14205 · doi:10.1287/opre.9.4.545
[11] DOI: 10.1137/S0036139900378657 · Zbl 1006.35064 · doi:10.1137/S0036139900378657
[12] DOI: 10.1103/PhysRevE.51.3164 · doi:10.1103/PhysRevE.51.3164
[13] Herman R., Kinetic Theory of Vehicular Traffic (1971) · Zbl 0226.90011
[14] DOI: 10.4310/CMS.2003.v1.n1.a1 · Zbl 1153.82335 · doi:10.4310/CMS.2003.v1.n1.a1
[15] DOI: 10.1007/978-3-540-40986-1 · doi:10.1007/978-3-540-40986-1
[16] Klar A., Surv. Math. Ind. 6 pp 215–
[17] DOI: 10.1007/BF02181481 · Zbl 0917.90124 · doi:10.1007/BF02181481
[18] DOI: 10.1137/1.9781611970562 · doi:10.1137/1.9781611970562
[19] Lighthill M. J., Proc. Roy. Soc. A 229 pp 1749–
[20] Nagel K., J. Phys. 2 pp 2221–
[21] DOI: 10.1080/00411459508205136 · Zbl 0817.90028 · doi:10.1080/00411459508205136
[22] DOI: 10.1016/0041-1647(75)90063-5 · doi:10.1016/0041-1647(75)90063-5
[23] Payne H. J., Trans. Res. Rec. 722 pp 68–
[24] DOI: 10.1287/opre.4.1.42 · doi:10.1287/opre.4.1.42
[25] DOI: 10.1090/conm/017/16 · Zbl 0538.35050 · doi:10.1090/conm/017/16
[26] DOI: 10.1016/S0191-2615(00)00050-3 · doi:10.1016/S0191-2615(00)00050-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.