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.
DOI : 10.5802/smai-jcm.15
Mots-clés : Stochastic particle methods, Paveri-Fontana model, Traffic flow
@article{SMAI-JCM_2016__2__229_0,
author = {Mint Moustapha, Jyda and Jourdain, Benjamin and Daucher, Dimitri},
title = {A probabilistic particle approximation of the {{\textquotedblleft}Paveri-Fontana{\textquotedblright}} kinetic model of traffic flow},
journal = {The SMAI Journal of computational mathematics},
pages = {229--253},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {2},
year = {2016},
doi = {10.5802/smai-jcm.15},
mrnumber = {3633551},
zbl = {1416.65038},
language = {en},
url = {http://www.numdam.org/articles/10.5802/smai-jcm.15/}
}
TY - JOUR AU - Mint Moustapha, Jyda AU - Jourdain, Benjamin AU - Daucher, Dimitri TI - A probabilistic particle approximation of the “Paveri-Fontana” kinetic model of traffic flow JO - The SMAI Journal of computational mathematics PY - 2016 SP - 229 EP - 253 VL - 2 PB - Société de Mathématiques Appliquées et Industrielles UR - http://www.numdam.org/articles/10.5802/smai-jcm.15/ DO - 10.5802/smai-jcm.15 LA - en ID - SMAI-JCM_2016__2__229_0 ER -
%0 Journal Article %A Mint Moustapha, Jyda %A Jourdain, Benjamin %A Daucher, Dimitri %T A probabilistic particle approximation of the “Paveri-Fontana” kinetic model of traffic flow %J The SMAI Journal of computational mathematics %D 2016 %P 229-253 %V 2 %I Société de Mathématiques Appliquées et Industrielles %U http://www.numdam.org/articles/10.5802/smai-jcm.15/ %R 10.5802/smai-jcm.15 %G en %F SMAI-JCM_2016__2__229_0
Mint Moustapha, Jyda; Jourdain, Benjamin; Daucher, Dimitri. A probabilistic particle approximation of the “Paveri-Fontana” kinetic model of traffic flow. The SMAI Journal of computational mathematics, Tome 2 (2016), pp. 229-253. doi : 10.5802/smai-jcm.15. http://www.numdam.org/articles/10.5802/smai-jcm.15/
[1] Branching Processes, Springer-Verlag, New York, 1972
[2] Stochastic Particle Approximations for Generalized Boltzmann Models and Convergence Estimates, The Annals of Probability, Volume 25 (1997) no. 1, pp. 115-132 | DOI | MR | Zbl
[3] Fokker-Planck Asymptotics for Traffic Flow, Kinetic and Related Models, Volume 3 (2010), pp. 165-179 | DOI | MR | Zbl
[4] Multiclass Continuum Modelling of Multilane Traffic Flow, Delft University (1999) (Ph. D. Thesis)
[5] General kinetic models for vehicular traffic and Monte Carlo methods, Computational Methods in Applied Mathematics, Volume 5 (2005), pp. 154-169 | MR | Zbl
[6] Introduction to Monte-Carlo methods for transport and diffusion equations, Oxford University Press, 2003 | Zbl
[7] Mathematical modelling and simulation of the road traffic: statistical analysis of merging models and probabilistic simulation of a kinetic model, Paris Est University (2014) (Ph. D. Thesis)
[8] On Boltzmann-like treatments for traffic flow: A critical review of the basic model and an alternative proposal for dilute traffic analysis, Transportation Research, Volume 9 (1975), pp. 225-235 | DOI
[9] A Boltzmann-like Approach for Traffic Flow, Operations Research, Volume 8 (1960), pp. 789-797 | DOI | MR | Zbl
[10] Kinetic Theory of Vehicular Traffic, American Elsevier, 1971
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