Clusters in Macroscopic Traffic Flow Models


This paper concerns the traveling wave formation in macroscopic traffic flow models. The dynamics involved in this problem is described following a close analogy to compressible fluid dynamics. It is well known that vehicle clusters appear along a highway when the homogenous steady state taken as a reference is linearly unstable. The cluster properties are determined in an approximate way in terms of the parameters proper to each model and are compared between them.

Share and Cite:

R. Velasco and P. Saavedra, "Clusters in Macroscopic Traffic Flow Models," World Journal of Mechanics, Vol. 2 No. 1, 2012, pp. 51-60. doi: 10.4236/wjm.2012.21007.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] H. X. Ge, R. J. Cheng and S. Q. Dai, “KdV and Kink- Antikinksolitons in Car-Following Models,” Physica A: Statistical Mechanics and Its Applications, Vol. 357, No. 3-4, 2005, pp. 466-476. doi:10.1016/j.physa.2005.03.059
[2] H. B. Zhu and S. Q. Dai, “Numerical Simulation of Soliton and Kink Density Waves in Traffic Flow with Periodic Boundaries,” Physica A: Statistical Mechanics and Its Applications, Vol. 387, No. 16-17, 2008, pp. 4367- 4375. doi:10.1016/j.physa.2008.01.067
[3] Z. Z. Liu, X. J. Zhou, X. M. Liu and J. Luo, “Density Waves in Traffic Flow of Two Kinds of Vehicles,” Phy- sical Review E, Vol. 67, No. 1, 2003, pp. 017601-017604. doi:10.1103/PhysRevE.67.017601
[4] D. Helbing, “Traffic and Related Self-Driven Many-Particle Systems,” Reviews of Modern Physics, Vol. 73, No. 4, 2001, pp. 1067-1139. doi:10.1103/RevModPhys.73.1067
[5] M. J. Lighthill and G. B. Whitham, “On Kinematic Waves II. A Theory of Traffic Flow on Long Crowded Roads,” Proceedings of the Royal Society of London, Vol. 229, 1955, pp. 317-345.
[6] H. J. Payne, “Mathematical Models of Public Systems,” Simulation Councils, Inc., 1971.
[7] R. D. Kühne and R. Beckschulte, “Transportation and Traffic Theory,” Proceedings of 12th International Symposium on Transportation and Traffic Theory, Elsevier, Berkeley, 1993.
[8] B. S. Kerner and P. Konh?user, “Cluster Effect in Initially Homogenous Traffic Flow,” Physical Review E, Vol. 48, No. 4, 1993, R2335-R2338. doi:10.1103/PhysRevE.48.R2335
[9] B. S. Kerner and P. Konh?user, “Structure and Parameters of Clusters in Traffic Flow,” Physical Review E, Vol. 50, No. 51, 1994, pp. 54-83. doi:10.1103/PhysRevE.50.54
[10] A. Aw and M. Rascle, “Resurrection of “Second Order,” Models of Traffic Flow,” SIAM Journal of Applied Mathe- matics, Vol. 60, No. 3, 2000, pp. 916-938. doi:10.1137/S0036139997332099
[11] A. Aw, A. Klar, T. Materne and M. Rascle, “Derivation of Continuum Traffic Flow Models from Microscopic Follow the Leader Models,” SIAM Journal of Applied Mathematics, Vol. 63, No. 1, 2002, pp. 259-278. doi:10.1137/S0036139900380955
[12] W. Marques Jr. and R. M. Velasco, “An Improved Second-Order Continuum Traffic Model,” Journal of Statistical Mechanics: Theory and Experiment, Vol. 2010, 2010, P02012. doi:10.1088/1742-5468/2010/02/P02012
[13] D. Helbing, “Improved Fluid-Dynamic Model for Vehicular Traffic,” Physical Review E, Vol. 51, No. 4, 1995, pp. 3164-3169. doi:10.1103/PhysRevE.51.3164
[14] D. Helbing, “Theoretical Foundation of Macroscopic Traffic models,” Physica A: Statistical Mechanics and Its Applications, Vol. 219, No. 3-4, 1995, pp. 375-390. doi:10.1016/0378-4371(95)00174-6
[15] C. Wagner “Second-Order Continuum Traffic Flow Model,” Physical Review E, Vol. 54, No. 5, 1996, pp. 5073-5085. doi:10.1103/PhysRevE.54.5073
[16] R. M. Velasco and W. Marques Jr., “Navier-Stokes-Like Equations for Traffic Flow,” Physical Review E, Vol. 72, No. 4, 2005, pp. 046102-046110. doi:10.1103/PhysRevE.72.046102
[17] A. R. Méndez and R. M. Velasco, “An Alternative Model in Traffic Flow Equations,” Transportation Research Part B, Vol. 42, No. 9, 2008, pp. 782-797. doi:10.1016/j.trb.2008.01.003
[18] P. Berg and A. Woods, “On-Ramp Simulations and Solitary Waves of a Car-Following Model,” Physical Review E, Vol. 64, No. 3, 2001, 035602(R). doi:10.1103/PhysRevE.64.035602
[19] D. Helbing and M. Treiber, “Numerical Simulation of Macroscopic Traffic Equations,” Computing in Science & Engineering, Vol. 1, No. 5, 1999, pp. 89-99. doi:10.1109/5992.790593
[20] Y. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari, S. Tadaki and S. Yukawa, “Traffic Jams without Bottlenecks-Experimental Evidence for the Physical Mechanism of Formation of a Jam,” New Journal of Physics, Vol. 10, No. 3, 2008, 033001. doi:10.1088/1367-2630/10/3/033001
[21] B. S. Kerner, “The Physics of Traffic,” Springer, Berlin, 2005.
[22] B. S. Kerner, “Introduction of Modern Traffic Flow Theory and Control,” Springer, Berlin, 2009. doi:10.1007/978-3-642-02605-8
[23] B. R. Kerner, “Enciclopedia of Complexity and Systems Science,” Springer, Berlin, 2009, pp. 9302-9355.
[24] P. Berg, A. Mason and A. Woods, “Continuum Approach to Car-Following Models,” Physical Review E, Vol. 61, 2000, pp. 1056-1066. doi:10.1103/PhysRevE.64.035602
[25] D. A. Kürtze and D. C. Hong, “Traffic Jams, Granular Flow, and Soliton Selection,” Physical Review E, Vol. 52, No. 1, 1995, pp. 218-221. doi:10.1103/PhysRevE.52.218
[26] P. Saavedra and R. M. Velasco, “Solitons in a Macroscopic Traffic Model,” 12th IFAC Symposium on Transportation Systems, 2009, pp. 428-433.
[27] R. M. Velasco and P. Saavedra, “Clusters in the Helbing’s Improved Model,” Lecture Notes in Computational Science 6350, 2010, pp. 633-636.
[28] M. Treiber and D. Helbing, “Macroscopic Simulation of Widely Scattered Synchronized Traffic States,” Journal of Physics A: Mathematical and General, Vol. 32, No. 1, 1999, pp. L7-L23. doi:10.1088/0305-4470/32/1/003
[29] V. Shevtsov and D. Helbing, “Macroscopic Dynamics of Multilane Traffic,” Physical Review E, Vol. 59, No. 6, 1999, pp. 6328-6338. doi:10.1103/PhysRevE.59.6328
[30] P. G. Drazin and R. S. Johnson, “Solitons: An Introduction,” Cambridge University Press, Cambridge, 1990.
[31] R. S. Johnson, “Singular Perturbation Theory,” Springer, Berlin, 2004.

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.