Nonlinear analysis in the Aw-Rascle anticipation model of traffic flow

SIAM Journal on Applied Mathematics

Volumn 67, Issue 3, 2007, Pages 605-618

  Ou, Zhong Hui ^a   Dai, Shi Qiang ^a   Zhang, Peng ^a   Dong, Li Yun ^a  

^a SHANGHAI UNIVERSITY   (China)

Author keywords

Soliton; Traffic flow; WENO

Indexed keywords

COMPUTER SIMULATION; MATHEMATICAL MODELS; NONLINEAR ANALYSIS; SOLITONS; STABILITY; WAVEFRONTS;

AW-RASCLE ANTICIPATION (ARA); NONLINEAR STABILITY; WAVEFRONT EXPANSION;

TRAFFIC CONTROL;

EID: 34547372958     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/060656863     Document Type: Article

References (35)

1. G. B. WHITHAM, Linear and Nonlinear Waves, John Wiley and Sons, Inc., New York, 1974.
2. P. I. RICHARDS, Shockwaves on the highway, Oper. Res., 4 (1956), pp. 42-51.
3. M. J. LIGHTHILL AND G. B. WHITMAN, On kinematic waves: II. A theory of traffic flows on long crowded roads, in Proceedings of the Royal Society, Series A, 229 (1955), pp. 317-345.
4. H. J. PAYNE, Models of freeway traffic and control, in Mathematical Models of Public Systems Simulation Council Proc. Ser. 28, Vol. 1, Simulation Council, New York, 1971, pp. 51-61.
5. H. J. PAYNE, FREFLO: A macroscopic simulation model of freeway traffic, Transport. Res. Record, 772 (1979), pp. 68-75.
6. 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, I. Volmuller and R. Hamerslag, eds., VNU Science Press, Utecht, The Netherlands, 1984, pp. 21-42.
7. R. D. KUHNE, Freeway speed distribution and acceleration noise - Calculations from a stochastic continuum theory and comparison with measurements, in Proceedings of the 10th International Symposium on Transportation and Traffic Theory, N. H. Garter and N. H. M. Wilson, eds., Elsevier, New York, 1987, pp. 119-137.
8. B. S. KERNER AND P. KONHÄUSER, Cluster effect in initially homogeneous traffic flow, Phys. Rev. E, 48 (1993), pp. R2335-R2338.
9. B. S. KERNER AND P. KONHÄUSER, Structure and parameters of clusters in traffic flow, Phys. Rev. B, 50 (1994), pp. 54-83.
10. H. Y. LEE, H. W. LEE, AND D. KIM, Origin of synchronized traffic flow on highways and its dynamic phase transitions, Phys. Rev. Lett., 81 (1998), pp. 1130-1133.
11. H. Y. LEE, H. W. LEE, AND D. KIM, Dynamic states of a continuum traffic equation with on-ramp, Phys. Rev. E, 59 (1999), pp. 5101-5111.
12. D. HELBING, Gas-kinetic derivation of Navier-Stokes-like traffic equation, Phys. Rev. E, 53 (1996), pp. 2366-2381.
13. D. HELBING, Derivation and emprical validation of a refined traffic flow model, Phys. A, 233 (1996), pp. 253-282.
14. M. TREIBER, A. HENNECKE, AND D. HELBING, Derivation, properties, and simulation of a gas-kinetic-based, non-local traffic model, Phys. Rev. E, 59 (1999), pp. 239-253.
15. D. HELBING, Micro- and macro-simulation of freeway traffic, Math. Comput. Modelling, 35 (2002), pp. 517-547.
16. C. DAGANZO, Requiem for second-order fluid approximation to traffic flow, Trans. Res. B, 29 (1995), pp. 277-286.
17. A. AW AND M. RASCLE, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), pp. 916-938.
18. M. RASCLE, An improved macroscopic model of traffic flow: Derivation and links with the Lighthill-Whitham model, Math. Comput. Modelling, 35 (2002), pp. 581-590.
19. A. KLAR AND R. WEGENER, Kinetic derivation of macroscopic anticipation models for vehicular traffic, SIAM J. Appl. Math., 60 (2000), pp. 1749-1766.
20. R. JIANG, Q. S. WU, AND Z. J. ZHU, A new continuum model for traffic flow and numerical tests, Transport. Res. B, 36 (2002), pp. 405-419.
21. Y. XUE AND S. Q. DAI, Continuum traffic model with the consideration of two delay time scales, Phys. Rev. E, 68 (2003), pp. 066123.
22. J. G. YI, H. LIN, L. ALVAREZ, AND R. HOROWITZ, Stability of macroscopic traffic flow modeling through wavefront exapansion, Transport. Res. B, 37 (2003), pp. 661-679.
23. Z. H. OU, Equlibrium functions of traffic flow, Phys. A, 351 (2005), pp. 620-636.
24. Z. H. OU, S. Q. DAI, L. Y. DONG, Z. WU, AND M. D. TAO, New equilibrium function of traffic flow, Phys. A, 362 (2006), pp. 525-531.
25. H. M. ZHANG, Analyses of the stability and wave properties of a new continuum traffic theory, Transport. Res. B, 33 (1999), pp. 299-415.
26. H. M. ZHANG, A non-equilibrium traffic model devoid of gas-like behavior, Transport, Res. B, 36 (2002), pp. 275-290.
27. T. KOMATSU AND S. SASA, Kink soliton characterizing traffic congestion, Phys. Rev. E, 52 (1995), pp. 5574-5582.
28. M. MURAMATSU AND T. NAGATANI, Soliton and kink jams in traffic flow with open boundaries, Phys. Rev. E, 60 (1999), pp. 180-187.
29. Z. H. OU, S. Q. DAI, AND L. Y. DONG, Density waves in the full velocity difference model, J. Phys. A, 39 (2006), pp. 1251-1263.
30. D. A. KURTZE AND D. C. HONG, Traffic jams, granular flow, and soliton selection, Phys. Rev. E, 52 (1995), pp. 218-221.
31. P. BERG AND A. W. WOODS, On-ramp simulations and solitary waves of a car-following model, Phys. Rev. E, 64 (2001), pp. 035602.
32. A. AW, A. KLAR, T. MATERNE, AND M. RASCLE, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2000), pp. 259-278.
33. A. HARTEN, B. ENGQUISH, S. OSHER, AND S. CHAKRAVARTHY, Uniformly high order essentially non-oscillatory schemes, III, J. Comput. Phys., 71 (1987), pp. 231-303.
34. C.-W. SHU, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Numerical Hyperbolic Equations, B. Cockburn, C. Johnson, C.-W. Shu, and E. Tadmor, eds., Lecture Notes in Math. 1697, A. Quarteroni, ed., Springer, New York, 1998, pp. 325-432.
35. P. ZHANG, S. C. WONG, AND C.-W. SHU, A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway, J. Comput. Phys., 212 (2006), pp. 739-756.