With cedram.org   version française
Search for an article
Table of contents for this volume | Previous article | Next article
Jyda Mint Moustapha; Benjamin Jourdain; Dimitri Daucher
A probabilistic particle approximation of the “Paveri-Fontana” kinetic model of traffic flow
SMAI-Journal of computational mathematics, 2 (2016), p. 229-253, doi: 10.5802/smai-jcm.15
Article PDF
Class. Math.: 65N35, 15A15
Keywords: Stochastic particle methods, Paveri-Fontana model, Traffic flow


This paper is devoted to the Paveri-Fontana model and its computation. The master equation of this model has no analytic solution in nonequilibrium case. We develop a stochastic approach to approximate this evolution equation. First, we give a probabilistic interpretation of the equation as a nonlinear Fokker-Planck equation. Replacing the nonlinearity by interaction, we deduce how to approximate its solution thanks to an algorithm based on a fictitious jump simulation of the interacting particle system. This algorithm is improved to obtain a linear complexity regarding the number of particles. Finally, the numerical method is illustrated on one traffic flow scenario and compared with a finite differences deterministic method.


[1] K.B. Athreya & P.E. Ney, Branching Processes, Springer-Verlag, New York, 1972  MR 373040 |  Zbl 0259.60002
[2] C. Graham & S. Méléard, “Stochastic Particle Approximations for Generalized Boltzmann Models and Convergence Estimates”, The Annals of Probability 25 (1997) no. 1, p. 115-132 Article |  MR 1428502 |  Zbl 0873.60076
[3] M. Herty, R. Illner & L. Pareschi, “Fokker-Planck Asymptotics for Traffic Flow”, Kinetic and Related Models 3 (2010), p. 165-179 Article |  MR 2580958 |  Zbl 1185.90036
[4] S.P. Hoogendoorn, Multiclass Continuum Modelling of Multilane Traffic Flow, Ph. D. Thesis, Delft University, 1999
[5] A. Klar, M. Herty & L. Pareschi, “General kinetic models for vehicular traffic and Monte Carlo methods”, Computational Methods in Applied Mathematics 5 (2005), p. 154-169  MR 2179696 |  Zbl 1114.90011
[6] B. Lapeyre, E. Pardoux & R. Sentis, Introduction to Monte-Carlo methods for transport and diffusion equations, Oxford University Press, 2003  MR 2186059 |  Zbl 1136.65133
[7] J. Mint-Moustapha, Mathematical modelling and simulation of the road traffic: statistical analysis of merging models and probabilistic simulation of a kinetic model, Ph. D. Thesis, Paris Est University, 2014
[8] S.L. Paveri-Fontana, “On Boltzmann-like treatments for traffic flow: A critical review of the basic model and an alternative proposal for dilute traffic analysis”, Transportation Research 9 (1975), p. 225-235 Article
[9] I. Prigogine & F. C. Andrews, “A Boltzmann-like Approach for Traffic Flow”, Operations Research 8 (1960), p. 789-797 Article |  MR 129011 |  Zbl 0249.90026
[10] I. Prigogine & R. Hermann, Kinetic Theory of Vehicular Traffic, American Elsevier, 1971  Zbl 0226.90011