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Jyda Mint Moustapha; Benjamin Jourdain; Dimitri Daucher A probabilistic particle approximation of the “PaveriFontana” kinetic model of traffic flow SMAIJournal of computational mathematics, 2 (2016), p. 229253, doi: 10.5802/smaijcm.15 Article PDF Class. Math.: 65N35, 15A15 Keywords: Stochastic particle methods, PaveriFontana model, Traffic flow Abstract This paper is devoted to the PaveriFontana 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 FokkerPlanck 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. Bibliography [2] C. Graham & S. Méléard, “Stochastic Particle Approximations for Generalized Boltzmann Models and Convergence Estimates”, The Annals of Probability 25 (1997) no. 1, p. 115132 Article  MR 1428502  Zbl 0873.60076 [3] M. Herty, R. Illner & L. Pareschi, “FokkerPlanck Asymptotics for Traffic Flow”, Kinetic and Related Models 3 (2010), p. 165179 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. 154169 MR 2179696  Zbl 1114.90011 [6] B. Lapeyre, E. Pardoux & R. Sentis, Introduction to MonteCarlo methods for transport and diffusion equations, Oxford University Press, 2003 MR 2186059  Zbl 1136.65133 [7] J. MintMoustapha, 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. PaveriFontana, “On Boltzmannlike treatments for traffic flow: A critical review of the basic model and an alternative proposal for dilute traffic analysis”, Transportation Research 9 (1975), p. 225235 Article [9] I. Prigogine & F. C. Andrews, “A Boltzmannlike Approach for Traffic Flow”, Operations Research 8 (1960), p. 789797 Article  MR 129011  Zbl 0249.90026 [10] I. Prigogine & R. Hermann, Kinetic Theory of Vehicular Traffic, American Elsevier, 1971 Zbl 0226.90011 
