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

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  • 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
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    References listed on IDEAS

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