Skip to main content
SpringerLink
Log in

Solitary wave solutions to higher-order traffic flow model with large diffusion

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

This paper uses the Taylor expansion to seek an approximate Kortewegde Vries equation (KdV) solution to a higher-order traffic flow model with sufficiently large diffusion. It demonstrates the validity of the approximate KdV solution considering all the related parameters to ensure the physical boundedness and the stability of the solution. Moreover, when the viscosity coefficient depends on the density and velocity of the flow, the wave speed of the KdV solution is naturally related to either the first or the second characteristic field. The finite element method is extended to solve the model and examine the stability and accuracy of the approximate KdV solution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Payne, H. J. Models of freeway traffic and control. Mathematical Models of Public Systems, Simulation Council Proceedings (ed. Bekey, G. A.), Series 1, Vol. 1, Chapter 6, Simulation Council, La Jolla, California, 51–61 (1971)

    Google Scholar 

  2. Whitham, G. B. Linear and Nonlinear Waves, John Wiely and Sons, Inc., New York (1974)

    MATH  Google Scholar 

  3. Kerner, B. S. and Konhäuser, P. Cluster effect in initially homogeneous traffic flow. Phys. Rev. E, 48, R2335–R2338 (1993)

    Article  Google Scholar 

  4. Aw, A. and Rascle, M. Resurrection of “second order” models of traffic flow. SIAM J. Appl. Math., 60, 916–938 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Berthelin, F., Degond, P., Delitala, M., and Rascle, M. A model for the formation and evolution of traffic jams. Arch. Ration. Mech. An., 187, 185–220 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Jiang, R., Wu, Q. S., and Zhu, Z. J. A new continuum model for traffic flow and numerical tests. Transport. Res. B, 36, 405–419 (2002)

    Article  Google Scholar 

  7. Siebel, F. and Mauser, W. On the fundamental diagram of traffic flow. SIAM J. Appl. Math., 66, 1150–1162 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Zhang, P., Wong, S. C., and Dai, S. Q. A conserved higher-order anisotropic traffic flow model: description of equilibrium and non-equilibrium flows. Transport. Res. B, 43, 562–574 (2009)

    Article  Google Scholar 

  9. Greenberg, J. M. Congestion redux. SIAM J. Appl. Math., 64, 1175–1185 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Zhang, P. and Wong, S. C. Essence of conservation forms in the traveling wave solutions of higherorder traffic flow models. Phys. Rev. E, 74, 026109 (2006)

    Article  MathSciNet  Google Scholar 

  11. Kerner, B. S., Klenov, S. L., and Konhäuser, P. Asymptotic theory of traffic jams. Phys. Rev. E, 56, 4200–4216 (1997)

    Article  Google Scholar 

  12. Wu, C. X., Zhang, P., Dai, S. Q., and Wong, S. C. Asymptotic solution of a wide cluster in Kühne’s higher-order traffic flow model. Proceedings of the Fifth International Conference on Nonlinear Mechanics (eds. Chien, W. Z., Dai, S. Q., Zhong, W. Z., Cheng, Q. J.), Shanghai University Press, Shanghai, 1132–1136 (2007)

    Google Scholar 

  13. Kurtze, D. A. and Hong, D. C. Traffic jams, granular flow, and soliton selection. Phys. Rev. E, 52, 218–221 (1995)

    Article  MathSciNet  Google Scholar 

  14. Wada, S. and Hayakawa, H. Kink solution in a fluid model of traffic flow. J. Phys. Soc. Jpn., 67, 763–766 (1998)

    Article  Google Scholar 

  15. Nagatani, T. Density waves in traffic flow. Phys. Rev. E, 61, 3564–3570 (2000)

    Article  Google Scholar 

  16. Tang, T. Q., Huang, H. J., and Shang, H. Y. A new macro model for traffic flow with the consideration of the driver’s forecast effect. Phys. Lett. A, 374, 1668–1672 (2010)

    Article  MATH  Google Scholar 

  17. Yu, L., Li, T., and Shi, Z. K. Density waves in a traffic flow model with reaction-time delay. Physica A, 389, 2607–2616 (2010)

    Article  MathSciNet  Google Scholar 

  18. Tang, T. Q., Li, C. Y., Huang, H. J., and Shang, H. Y. Macro modeling and analysis of traffic flow with road width. J. Cent. South Univ. Technol., 18, 1757–1764 (2011)

    Article  Google Scholar 

  19. Tang, T. Q., Li, P., Wu, Y. H., and Huang, H. J. A macro model for traffic flow with consideration of static bottleneck. Commun. Theor. Phys., 58, 300–306 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Berg, P. and Woods, A. On-ramp simulations and solitary waves of a car-following model. Phys. Rev. E, 64, 035602 (2001)

    Article  Google Scholar 

  21. Tang, T. Q., Huang, H. J., and Xu, G. A new macro model with consideration of the traffic interruption probability. Physica A, 387, 6845–6856 (2008)

    Article  Google Scholar 

  22. Ge, H. X. and Lo, S. M. The KdV-Burgers equation in speed gradient viscous continuum model. Physica A, 391, 1652–1656 (2012)

    Article  Google Scholar 

  23. Wu, C. X., Zhang, P., Wong, S. C., Qiao, D. L., and Dai, S. Q. Solitary wave solution to Aw-Rascle viscous model of traffic flow. Appl. Math. Mech. -Engl. Ed., 34, 523–528 (2013) DOI 10.1007/s10483-013-1687-9

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, 200072, P. R. China

    Xiao-xia Jian  (菅肖霞), Peng Zhang  (张 鹏) & Dian-liang Qiao  (乔殿梁)

  2. Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, P. R. China

    S. C. Wong

  3. Department of Transportation Engineering, TOD-based Sustainable Urban Transportation Center, Ajou University, Suwon, 443-749, Korea

    Kee-choo Choi  (崔岐柱)

  4. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai, 200072, P. R. China

    Xiao-xia Jian  (菅肖霞), Peng Zhang  (张 鹏) & Dian-liang Qiao  (乔殿梁)

Authors
  1. Xiao-xia Jian  (菅肖霞)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peng Zhang  (张 鹏)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. C. Wong
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Dian-liang Qiao  (乔殿梁)
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Kee-choo Choi  (崔岐柱)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peng Zhang  (张 鹏).

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 11072141 and 11272199), the National Basic Research Program of China (No. 2012CB725404), the Shanghai Program for Innovative Research Team in Universities, the Research Grants Council of the Hong Kong Special Administrative Region, China (No.HKU7184/10E), and the National Research Foundation of Korea (MEST)(No.NRF-2010-0029446)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jian, Xx., Zhang, P., Wong, S.C. et al. Solitary wave solutions to higher-order traffic flow model with large diffusion. Appl. Math. Mech.-Engl. Ed. 35, 167–176 (2014). https://doi.org/10.1007/s10483-014-1781-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-014-1781-x

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation