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  4. Central finite volume schemes for non-local traffic flow models with Arrhenius-type look-ahead rules

Central finite volume schemes for non-local traffic flow models with Arrhenius-type look-ahead rules

MMC.
2023;
Volume 10, Number 4
: pp. 1100–1108
https://doi.org/10.23939/mmc2023.04.1100
Received: August 20, 2023
Revised: October 28, 2023
Accepted: October 30, 2023

Mathematical Modeling and Computing, Vol. 10, No. 4, pp. 1100–1108 (2023)

Authors:
  1. S. Belkadi
  2. M. Atounti
1
Mohammed First University, Multidisciplinary Faculty of Nador
2
Mohammed First University, Multidisciplinary Faculty of Nador

We present a central finite volume method and apply it to a new class of nonlocal traffic flow models with an Arrhenius-type look-ahead interaction.  These models can be stated as scalar conservation laws with nonlocal fluxes.  The suggested scheme is a development of the Nessyah–Tadmor non-oscillatory central scheme.  We conduct several numerical experiments in which we carry out the following actions: i) we show the robustness and high resolution of the suggested method;  ii) we compare the equations' solutions with local and nonlocal fluxes;  iii) we examine how the look-ahead distance affects the numerical solution.

finite volume methods
conservation laws
nonlocal flux
traffic flow modeling
limiters
  1. Lighthill M., Whitham G. B.  On kinematic waves. II. A theory of traffic flow on long crowded roads.  Proceedings of the Royal Society A.  229 (1178), 317–345 (1995).
  2. Kuhne R. D., Gartner N. H.  75 Years of the Fundamental Diagram for Traffic Flow Theory: Greenshields Symposium.  Transportation Research Board E-Circular (2011).
  3. Sopasakis A., Katsoulakis M. A.  Stochastic modeling and simulation of traffic flow: asymmetric single exclusion process with Arrhenius look-ahead dynamics. SIAM Journal on Applied Mathematics.  66 (3), 921–944 (2006).
  4. Kurganov A., Polizzi A.  Non-oscillatory central schemes for a traffic flow model with Arrhenius look-ahead dynamics.  Networks and Heterogeneous Media.  4 (3), 431–451 (2009).
  5. Lee Y.  Thresholds for shock formation in traffic flow models with nonlocal-concave-convex flux.  Journal of Differential Equations.  266 (1), 580–599 (2019).
  6. Eymard R., Gallouët T., Herbin R.  Finite Volume Method. Handbook of Numerical Analysis. Lions, Janvier (2013).
  7. Godlewski E., Raviart P. A.  Hyperbolic Systems of Conservation Laws.  Ellipses (1991).
  8. Helbing D., Treiber M.  Gas-kinetic-based traffic model explaining observed hysteretic phase transition.  Physical Review Letters.  81 (14), 3042–3045 (1998).
  9. Chiarello F. A., Goatin P.  Global entropy weak solutions for general non-local traffic flow models with the anisotropic kernel.  ESAIM: M2AN.  52 (1), 163–180 (2018).
  10. Belkadi S., Atounti M.  Non-oscillatory central schemes for general non-local traffic flow models.  International Journal of Applied Mathematics.  35 (4), 515–528 (2022).
  11. Nessyahu N., Tadmor E.  Non-oscillatory central differencing for hyperbolic conservation laws.  Journal of Computational Physics.  87 (2), 408–463 (1990).
  12. Sweby P. K.  High-resolution schemes using flux limiters for hyperbolic conservation laws.  SIAM Journal on Numerical Analysis.  21 (5), 995–1011 (1984).
  13. Blandin S., Goatin P.  Well-posedness of a conservation law with non-local flux arising in traffic flow modeling.  Numerische Mathematik.  132 (2), 217–241 (2017).
  14. Goatin P., Scialanga S.  Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity.  Networks and Heterogeneous Media.  11 (1), 107–121 (2016).