JOURNAL OF COMPUTATIONAL MATHEMATICS

Browse issues
A probabilistic particle approximation of the “Paveri-Fontana” kinetic model of traffic flow
Jyda Mint Moustapha1; Benjamin Jourdain2; Dimitri Daucher1
1 Université Paris-Est, LEPSIS - IFSTTAR, France
2 Université Paris-Est, CERMICS - ENPC, France
The SMAI Journal of computational mathematics, Volume 2 (2016), pp. 229-253.

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.

Published online:
DOI: 10.5802/smai-jcm.15
Classification: 65N35, 15A15
Keywords: Stochastic particle methods, Paveri-Fontana model, Traffic flow

Jyda Mint Moustapha 1; Benjamin Jourdain 2; Dimitri Daucher 1

1 Université Paris-Est, LEPSIS - IFSTTAR, France
2 Université Paris-Est, CERMICS - ENPC, France 
@article{SMAI-JCM_2016__2__229_0,
     author = {Jyda Mint Moustapha and Benjamin Jourdain and Dimitri Daucher},
     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},
     zbl = {1416.65038},
     mrnumber = {3633551},
     language = {en},
     url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.15/}
}
TY  - JOUR
AU  - Jyda Mint Moustapha
AU  - Benjamin Jourdain
AU  - Dimitri Daucher
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  - https://smai-jcm.centre-mersenne.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 Jyda Mint Moustapha
%A Benjamin Jourdain
%A Dimitri Daucher
%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 https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.15/
%R 10.5802/smai-jcm.15
%G en
%F SMAI-JCM_2016__2__229_0
Jyda Mint Moustapha; Benjamin Jourdain; Dimitri Daucher. A probabilistic particle approximation of the “Paveri-Fontana” kinetic model of traffic flow. The SMAI Journal of computational mathematics, Volume 2 (2016), pp. 229-253. doi : 10.5802/smai-jcm.15. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.15/

[1] K.B. Athreya; P.E. Ney Branching Processes, Springer-Verlag, New York, 1972

[2] C. Graham; S. Méléard 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] M. Herty; R. Illner; L. Pareschi Fokker-Planck Asymptotics for Traffic Flow, Kinetic and Related Models, Volume 3 (2010), pp. 165-179 | DOI | MR | Zbl

[4] S.P. Hoogendoorn Multiclass Continuum Modelling of Multilane Traffic Flow, Delft University (1999) (Ph. D. Thesis)

[5] A. Klar; M. Herty; L. Pareschi General kinetic models for vehicular traffic and Monte Carlo methods, Computational Methods in Applied Mathematics, Volume 5 (2005), pp. 154-169 | MR | Zbl

[6] B. Lapeyre; E. Pardoux; R. Sentis Introduction to Monte-Carlo methods for transport and diffusion equations, Oxford University Press, 2003 | Zbl

[7] J. Mint-Moustapha 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] 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, Volume 9 (1975), pp. 225-235 | DOI

[9] I. Prigogine; F. C. Andrews A Boltzmann-like Approach for Traffic Flow, Operations Research, Volume 8 (1960), pp. 789-797 | DOI | MR | Zbl

[10] I. Prigogine; R. Hermann Kinetic Theory of Vehicular Traffic, American Elsevier, 1971

Cited by Sources: