Communications in Mathematical Sciences

Volume 15 (2017)

Number 2

Analysis of a multi-population kinetic model for traffic flow

Pages: 379 – 412



Gabriella Puppo (Dipartimento di Scienza ed Alta Tecnologia, Università dell’Insubria, Como, Italy)

Matteo Semplice (Dipartimento di Matematica, Università di Torino, Italy)

Andrea Tosin (Istituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche, Roma, Italy)

Giuseppe Visconti (Dipartimento di Scienza ed Alta Tecnologia, Università dell’Insubria, Como, Italy)


In this work we extend a recent kinetic traffic model [G. Puppo, M. Semplice, A. Tosin, and G. Visconti, Kinet. Relat. Models, in press, 2016] to the case of more than one class of vehicles, each of which is characterized by few different microscopic features. We consider a Boltzmann-like framework with only binary interactions, which take place among vehicles belonging to the various classes. Our approach differs from the multi-population kinetic model proposed in [G. Puppo, M. Semplice, A. Tosin, and G. Visconti, Commun. Math. Sci., 14:643–669, 2016] because here we assume continuous velocity spaces and we introduce a parameter describing the physical velocity jump performed by a vehicle that increases its speed after an interaction. The model is discretized in order to investigate numerically the structure of the resulting fundamental diagrams and the system of equations is analyzed by studying well posedness. Moreover, we compute the equilibria of the discretized model and we show that the exact asymptotic kinetic distributions can be obtained with a small number of velocities in the grid. Finally, we introduce a new probability law in order to attenuate the sharp capacity drop occurring in the diagrams of traffic.


multispecies models, Boltzmann-like kinetic models, discrete velocity models, multi-valued diagrams, two-phases diagrams

2010 Mathematics Subject Classification

35Q20, 65Z05, 90B20

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Published 21 February 2017