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A higher-order macroscopic model for pedestrian flows

Author

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  • Jiang, Yan-qun
  • Zhang, Peng
  • Wong, S.C.
  • Liu, Ru-xun

Abstract

This paper develops a higher-order macroscopic model of pedestrian crowd dynamics derived from fluid dynamics that consists of two-dimensional Euler equations with relaxation. The desired directional motion of pedestrians is determined by an Eikonal-type equation, which describes a problem that minimizes the instantaneous total walking cost from origin to destination. A linear stability analysis of the model demonstrates its ability to describe traffic instability in crowd flows. The algorithm to solve the macroscopic model is composed of a splitting technique introduced to treat the relaxation terms, a second-order positivity-preserving central-upwind scheme for hyperbolic conservation laws, and a fast-sweeping method for the Eikonal-type equation on unstructured meshes. To test the applicability of the model, we study a challenging pedestrian crowd flow problem of the presence of an obstruction in a two-dimensional continuous walking facility. The numerical results indicate the rationality of the model and the effectiveness of the computational algorithm in predicting the flux or density distribution and the macroscopic behavior of the pedestrian crowd flow. The simulation results are compared with those obtained by the two-dimensional Lighthill–Whitham–Richards pedestrian flow model with various model parameters, which further shows that the macroscopic model is able to correctly describe complex phenomena such as “stop-and-go waves” observed in empirical pedestrian flows.

Suggested Citation

  • Jiang, Yan-qun & Zhang, Peng & Wong, S.C. & Liu, Ru-xun, 2010. "A higher-order macroscopic model for pedestrian flows," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(21), pages 4623-4635.
    • Handle: RePEc:eee:phsmap:v:389:y:2010:i:21:p:4623-4635
    DOI: 10.1016/j.physa.2010.05.003
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    Citations

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    Cited by:

    1. Maity, Somnath & Sundar, S., 2022. "A coupled model for macroscopic behavior of crowd in flood induced evacuation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 607(C).
    2. Aghamohammadi, Rafegh & Laval, Jorge A., 2020. "A continuum model for cities based on the macroscopic fundamental diagram: A semi-Lagrangian solution method," Transportation Research Part B: Methodological, Elsevier, vol. 132(C), pages 101-116.
    3. He, Mengchen & Wang, Qiao & Chen, Juan & Xu, Shiwei & Ma, Jian, 2023. "Modeling pedestrian walking behavior in the flow field with moving walkways," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 619(C).
    4. Haghani, Milad, 2021. "The knowledge domain of crowd dynamics: Anatomy of the field, pioneering studies, temporal trends, influential entities and outside-domain impact," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 580(C).
    5. Shunqiang Ye & Lu Wang & Kang Hao Cheong & Nenggang Xie, 2017. "Pedestrian Group-Crossing Behavior Modeling and Simulation Based on Multidimensional Dirty Faces Game," Complexity, Hindawi, vol. 2017, pages 1-12, December.
    6. 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.
    7. Liang, Haoyang & Du, Jie & Wong, S.C., 2021. "A Continuum model for pedestrian flow with explicit consideration of crowd force and panic effects," Transportation Research Part B: Methodological, Elsevier, vol. 149(C), pages 100-117.
    8. Mollier, Stéphane & Delle Monache, Maria Laura & Canudas-de-Wit, Carlos & Seibold, Benjamin, 2019. "Two-dimensional macroscopic model for large scale traffic networks," Transportation Research Part B: Methodological, Elsevier, vol. 122(C), pages 309-326.
    9. Goldsztein, Guillermo H., 2017. "Crowd of individuals walking in opposite directions. A toy model to study the segregation of the group into lanes of individuals moving in the same direction," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 479(C), pages 162-173.
    10. Aghamohammadi, Rafegh & Laval, Jorge A., 2020. "Dynamic traffic assignment using the macroscopic fundamental diagram: A Review of vehicular and pedestrian flow models," Transportation Research Part B: Methodological, Elsevier, vol. 137(C), pages 99-118.
    11. Jiang, Yan-Qun & Hu, Ying-Gang & Huang, Xiaoqian, 2022. "Modeling pedestrian flow through a bottleneck based on a second-order continuum model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 608(P1).
    12. Li, Xiao-Yang & Lin, Zhi-Yang & Zhang, Peng & Zhang, Xiao-Ning, 2023. "Reconstruction of density and cost potential field of Eikonal equation: Applications to discrete pedestrian flow models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 629(C).
    13. Jiang, Yan-Qun & Zhou, Shu-Guang & Duan, Ya-Li & Huang, Xiao-Qian, 2023. "A viscous continuum model with smoke effect for pedestrian evacuation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 621(C).
    14. Jiang, Yan-Qun & Zhang, Wei & Zhou, Shu-Guang, 2016. "Comparison study of the reactive and predictive dynamic models for pedestrian flow," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 441(C), pages 51-61.
    15. Petr Kubera & Jiří Felcman, 2021. "On the Verification of the Pedestrian Evacuation Model," Mathematics, MDPI, vol. 9(13), pages 1-23, June.

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