IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2022i1p81-d1014661.html
   My bibliography  Save this article

Lie Symmetry Analysis of the Aw–Rascle–Zhang Model for Traffic State Estimation

Author

Listed:
  • Andronikos Paliathanasis

    (Institute of Systems Science, Durban University of Technology, P.O. Box 1334, Durban 4000, South Africa
    Departamento de Matemáticas, Universidad Católica del Norte, Avda. Angamos 0610, Casilla 1280, Antofagasta 1240000, Chile)

  • Peter G. L. Leach

    (Institute of Systems Science, Durban University of Technology, P.O. Box 1334, Durban 4000, South Africa)

Abstract

We extend our analysis on the Lie symmetries in fluid dynamics to the case of macroscopic traffic estimation models. In particular we study the Aw–Rascle–Zhang model for traffic estimation, which consists of two hyperbolic first-order partial differential equations. The Lie symmetries, the one-dimensional optimal system and the corresponding Lie invariants are determined. Specifically, we find that the admitted Lie symmetries form the four-dimensional Lie algebra A 4 , 12 . The resulting one-dimensional optimal system is consisted by seven one-dimensional Lie algebras. Finally, we apply the Lie symmetries in order to define similarity transformations and derive new analytic solutions for the traffic model. The qualitative behaviour of the solutions is discussed.

Suggested Citation

  • Andronikos Paliathanasis & Peter G. L. Leach, 2022. "Lie Symmetry Analysis of the Aw–Rascle–Zhang Model for Traffic State Estimation," Mathematics, MDPI, vol. 11(1), pages 1-11, December.
    • Handle: RePEc:gam:jmathe:v:11:y:2022:i:1:p:81-:d:1014661
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/1/81/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/1/81/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Paul I. Richards, 1956. "Shock Waves on the Highway," Operations Research, INFORMS, vol. 4(1), pages 42-51, February.
    2. Daganzo, Carlos F., 1994. "The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory," Transportation Research Part B: Methodological, Elsevier, vol. 28(4), pages 269-287, August.
    3. Gipps, P.G., 1981. "A behavioural car-following model for computer simulation," Transportation Research Part B: Methodological, Elsevier, vol. 15(2), pages 105-111, April.
    4. Newell, G. F., 1993. "A simplified theory of kinematic waves in highway traffic, part II: Queueing at freeway bottlenecks," Transportation Research Part B: Methodological, Elsevier, vol. 27(4), pages 289-303, August.
    5. Mohamed R. Ali, 2019. "A Truncation Method for Solving the Time-Fractional Benjamin-Ono Equation," Journal of Applied Mathematics, Hindawi, vol. 2019, pages 1-7, May.
    6. Zhang, H. M., 2002. "A non-equilibrium traffic model devoid of gas-like behavior," Transportation Research Part B: Methodological, Elsevier, vol. 36(3), pages 275-290, March.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Blandin, Sébastien & Argote, Juan & Bayen, Alexandre M. & Work, Daniel B., 2013. "Phase transition model of non-stationary traffic flow: Definition, properties and solution method," Transportation Research Part B: Methodological, Elsevier, vol. 52(C), pages 31-55.
    2. Zhou, Fang & Li, Xiaopeng & Ma, Jiaqi, 2017. "Parsimonious shooting heuristic for trajectory design of connected automated traffic part I: Theoretical analysis with generalized time geography," Transportation Research Part B: Methodological, Elsevier, vol. 95(C), pages 394-420.
    3. Li, Xiaopeng & Ouyang, Yanfeng, 2011. "Characterization of traffic oscillation propagation under nonlinear car-following laws," Transportation Research Part B: Methodological, Elsevier, vol. 45(9), pages 1346-1361.
    4. Li, Jia & Zhang, H.M., 2013. "The variational formulation of a non-equilibrium traffic flow model: Theory and implications," Transportation Research Part B: Methodological, Elsevier, vol. 57(C), pages 314-325.
    5. Seo, Toru & Kawasaki, Yutaka & Kusakabe, Takahiko & Asakura, Yasuo, 2019. "Fundamental diagram estimation by using trajectories of probe vehicles," Transportation Research Part B: Methodological, Elsevier, vol. 122(C), pages 40-56.
    6. McCrea, Jennifer & Moutari, Salissou, 2010. "A hybrid macroscopic-based model for traffic flow in road networks," European Journal of Operational Research, Elsevier, vol. 207(2), pages 676-684, December.
    7. Huanping Li & Jian Wang & Guopeng Bai & Xiaowei Hu, 2021. "Exploring the Distribution of Traffic Flow for Shared Human and Autonomous Vehicle Roads," Energies, MDPI, vol. 14(12), pages 1-21, June.
    8. Ma, Tao & Zhou, Zhou & Antoniou, Constantinos, 2018. "Dynamic factor model for network traffic state forecast," Transportation Research Part B: Methodological, Elsevier, vol. 118(C), pages 281-317.
    9. Pedro Cesar Lopes Gerum & Andrew Reed Benton & Melike Baykal-Gürsoy, 2019. "Traffic density on corridors subject to incidents: models for long-term congestion management," EURO Journal on Transportation and Logistics, Springer;EURO - The Association of European Operational Research Societies, vol. 8(5), pages 795-831, December.
    10. Pereira, Mike & Boyraz Baykas, Pinar & Kulcsár, Balázs & Lang, Annika, 2022. "Parameter and density estimation from real-world traffic data: A kinetic compartmental approach," Transportation Research Part B: Methodological, Elsevier, vol. 155(C), pages 210-239.
    11. Rehborn, Hubert & Klenov, Sergey L. & Palmer, Jochen, 2011. "An empirical study of common traffic congestion features based on traffic data measured in the USA, the UK, and Germany," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(23), pages 4466-4485.
    12. Kontorinaki, Maria & Spiliopoulou, Anastasia & Roncoli, Claudio & Papageorgiou, Markos, 2017. "First-order traffic flow models incorporating capacity drop: Overview and real-data validation," Transportation Research Part B: Methodological, Elsevier, vol. 106(C), pages 52-75.
    13. Flötteröd, G. & Osorio, C., 2017. "Stochastic network link transmission model," Transportation Research Part B: Methodological, Elsevier, vol. 102(C), pages 180-209.
    14. Canepa, Edward S. & Claudel, Christian G., 2017. "Networked traffic state estimation involving mixed fixed-mobile sensor data using Hamilton-Jacobi equations," Transportation Research Part B: Methodological, Elsevier, vol. 104(C), pages 686-709.
    15. Jiang, Chenming & Bhat, Chandra R. & Lam, William H.K., 2020. "A bibliometric overview of Transportation Research Part B: Methodological in the past forty years (1979–2019)," Transportation Research Part B: Methodological, Elsevier, vol. 138(C), pages 268-291.
    16. Kai Nagel & Peter Wagner & Richard Woesler, 2003. "Still Flowing: Approaches to Traffic Flow and Traffic Jam Modeling," Operations Research, INFORMS, vol. 51(5), pages 681-710, October.
    17. Tian, Junfang & Treiber, Martin & Ma, Shoufeng & Jia, Bin & Zhang, Wenyi, 2015. "Microscopic driving theory with oscillatory congested states: Model and empirical verification," Transportation Research Part B: Methodological, Elsevier, vol. 71(C), pages 138-157.
    18. Ngoduy, D. & Hoang, N.H. & Vu, H.L. & Watling, D., 2016. "Optimal queue placement in dynamic system optimum solutions for single origin-destination traffic networks," Transportation Research Part B: Methodological, Elsevier, vol. 92(PB), pages 148-169.
    19. Costeseque, Guillaume & Lebacque, Jean-Patrick, 2014. "A variational formulation for higher order macroscopic traffic flow models: Numerical investigation," Transportation Research Part B: Methodological, Elsevier, vol. 70(C), pages 112-133.
    20. Wong, S. C. & Wong, G. C. K., 2002. "An analytical shock-fitting algorithm for LWR kinematic wave model embedded with linear speed-density relationship," Transportation Research Part B: Methodological, Elsevier, vol. 36(8), pages 683-706, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2022:i:1:p:81-:d:1014661. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.