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A nonconservative macroscopic traffic flow model in a two-dimensional urban-porous city

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  • García-Chan, Néstor
  • Alvarez-Vázquez, Lino J.
  • Martínez, Aurea
  • Vázquez-Méndez, Miguel E.

Abstract

In this paper we propose a novel traffic flow model based on understanding the city as a porous media, this is, streets and building-blocks characterizing the urban landscape are seen now as the fluid-phase and the solid-phase of a porous media, respectively. Moreover, based in the interchange of mass in the porous media models, we can model the interchange of cars between streets and off-street parking-spaces. Therefore, our model is not a standard conservation law, being formulated as the coupling of a non-stationary convection–diffusion–reaction PDE with a Darcy–Brinkman–Forchheimer PDE system. To solve this model, the classical Galerkin P1 finite element method combined with an explicit time marching scheme of strong stability preserving type was enough to stabilize our numerical solutions. Numerical experiences on an urban-porous domain inspired by the city of Guadalajara (Mexico) allow us to simulate the influence of the porosity terms on the traffic speed, the traffic flow at rush-valley hours, and traffic congestion due to the lack of parking spaces.

Suggested Citation

  • García-Chan, Néstor & Alvarez-Vázquez, Lino J. & Martínez, Aurea & Vázquez-Méndez, Miguel E., 2025. "A nonconservative macroscopic traffic flow model in a two-dimensional urban-porous city," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 233(C), pages 60-74.
    • Handle: RePEc:eee:matcom:v:233:y:2025:i:c:p:60-74
    DOI: 10.1016/j.matcom.2025.01.016
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    References listed on IDEAS

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    1. Jiang, Yan-Qun & Zhou, Shu-Guang & Tian, Fang-Bao, 2015. "A higher-order macroscopic model for bi-direction pedestrian flow," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 425(C), pages 69-78.
    2. Néstor García-Chan & Juan A. Licea-Salazar & Luis G. Gutierrez-Ibarra, 2023. "Urban Heat Island Dynamics in an Urban–Rural Domain with Variable Porosity: Numerical Methodology and Simulation," Mathematics, MDPI, vol. 11(5), pages 1-18, February.
    3. Leclercq, Ludovic & Sénécat, Alméria & Mariotte, Guilhem, 2017. "Dynamic macroscopic simulation of on-street parking search: A trip-based approach," Transportation Research Part B: Methodological, Elsevier, vol. 101(C), pages 268-282.
    4. Daganzo, Carlos F., 1995. "Requiem for second-order fluid approximations of traffic flow," Transportation Research Part B: Methodological, Elsevier, vol. 29(4), pages 277-286, August.
    5. Tingzhen Ming & Shengnan Lian & Yongjia Wu & Tianhao Shi & Chong Peng & Yueping Fang & Renaud de Richter & Nyuk Hien Wong, 2021. "Numerical Investigation on the Urban Heat Island Effect by Using a Porous Media Model," Energies, MDPI, vol. 14(15), pages 1-23, August.
    6. Robert Herman & Tenny Lam & Ilya Prigogine, 1972. "Kinetic Theory of Vehicular Traffic: Comparison with Data," Transportation Science, INFORMS, vol. 6(4), pages 440-452, November.
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