Analysis of a multi-population kinetic model for traffic flow

Abstract

In this work we extend a recent kinetic traffic model [G. Puppo, M. Semplice, A. Tosin, 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, 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

Hybrid Stochastic Kinetic Description of Two-Dimensional Traffic Dynamics

“…The value ρ = ρ c such that P (ρ c ) = 1 2 is called the critical density. For the function (2.2) it results, for instance, ρ c = (1/2) 1/δ , which is consistent with the values found in [41,42,43] for different kinetic models of traffic flow.…”

The BGK approximation of kinetic models for traffic

“…This means that the model cannot reproduce and explain the scattering behavior of the experimental measurements, at least not at equilibrium. Mathematical models being able to describe multivalued fundamental diagrams were also studied and they link the observed scattered data to several reasons: for instance the presence of heterogeneous components, such as different drivers [31,33] or populations of vehicles [42], or the deviation of microscopic speeds at equilibrium [13], or the presence of non-equilibrium phenomena such as stop and go waves [25,46,49]. In this work we will focus on the last motivation and consider non-equilibrium phenomena that might lead to complex traffic phenomena as e.g.…”

Multivalued Fundamental Diagrams of Traffic Flow in the Kinetic Fokker--Planck Limit

“…Although several laws of the probability function P (ρ) are taken into account in order to better fix experimental data, see e.g. [30,31,32], we will consider the simplest choice…”

“…The parameter ∆v would be a finite parameter which models the physical velocity jump performed in acceleration. Clearly, it may depend on the mechanical microscopic characteristics of vehicles, see [32], but here we will assume that it is constant. With this choice and evaluating explicitly the integrals in (21), the asymptotic distribution f ∞ writes as…”

“…In other words, we can think that r synthesizes the different properties of the flow (as the composition of the traffic flow or the structural characteristics of the road) which induce different macroscopic dynamics for a given value of the density ρ. For instance, comparing this result with our multi-population models [32,34], we can suppose that r takes into account the heterogeneous composition of the vehicles on the road.…”