# American Institute of Mathematical Sciences

September  2017, 10(3): 823-854. doi: 10.3934/krm.2017033

## Kinetic models for traffic flow resulting in a reduced space of microscopic velocities

 1 Universitá degli Studi dell'Insubria, Dipartimento di Scienza ed Alta Tecnologia, Via Valleggio, 11 -22070, Como, Italy 2 Universitá degli Studi di Torino, Dipartimento di Matematica "G. Peano", Via Carlo Alberto, 8 -10123 Torino, Italy 3 Consiglio Nazionale delle Ricerche, Istituto per le Applicazioni del Calcolo "M. Picone", Via dei Taurini, 19 -00185 Roma, Italy

* Corresponding author: Gabriella Puppo

Received  August 2015 Revised  April 2016 Published  December 2016

The purpose of this paper is to study the properties of kinetic models for traffic flow described by a Boltzmann-type approach and based on a continuous space of microscopic velocities. In our models, the particular structure of the collision kernel allows one to find the analytical expression of a class of steady-state distributions, which are characterized by being supported on a quantized space of microscopic speeds. The number of these velocities is determined by a physical parameter describing the typical acceleration of a vehicle and the uniqueness of this class of solutions is supported by numerical investigations. This shows that it is possible to have the full richness of a kinetic approach with the simplicity of a space of microscopic velocities characterized by a small number of modes. Moreover, the explicit expression of the asymptotic distribution paves the way to deriving new macroscopic equations using the closure provided by the kinetic model.

Citation: Gabriella Puppo, Matteo Semplice, Andrea Tosin, Giuseppe Visconti. Kinetic models for traffic flow resulting in a reduced space of microscopic velocities. Kinetic & Related Models, 2017, 10 (3) : 823-854. doi: 10.3934/krm.2017033
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##### References:
Connection between the $\delta$ model and a discrete-velocity model, having the same steady-state distribution
Structure of the probability matrices of the $\delta$ model, with $\Delta v={V_{\max }}/2$. The shaded areas correspond to the non-zero elements of the matrices. For the meaning of the different hatchings, please see Table Table 1
Structure of the probability matrices of the $\delta$ model with $\delta v$ integer sub-multiple of $\Delta v$. The shaded areas correspond to the non-zero elements of the matrices. For the meaning of the different hatchings, please see Table 1
Structure of the probability matrices of the $\chi$ model with $\delta v$ integer sub-multiple of $\Delta v$
Approximation of the asymptotic kinetic distribution function obtained with two acceleration terms $\Delta v=1/T$, $T=3$ (top), $T=5$ (bottom), and $N=rT+1$ velocity cells, with $r\in\left\{1,4,8\right\}$; $\rho=0.3$ (left) and $\rho=0.6$ (right) are the initial densities. We mark with red circles on the x-axes the center of the $T+1$ cells obtained with $r=1$
Evolution towards equilibrium of the discretized model (13) with $N=4$ (green), $N=7$ (blue) and $N=10$ (red) grid points. The acceleration parameter $\Delta v$ is taken as ${V_{\max }}/3$ and the density is $\rho=0.6$. Black circles indicate the equilibrium values
Cumulative density at equilibrium for several values of $\delta v\to 0$. The density is $\rho=0.6$ and $\Delta v$ is chosen as $1/3$ (left), $1/5$ (right)
Evolution towards equilibrium, $\rho=0.7$, $T=4$, $N=17$. Left: $f_j(t=0)\equiv \rho/N$. Middle: $f_j(t=0)=0, j=1,2,3$, $f_j(t=0)\equiv(\rho/(N-3)), j>3$. Right: $f_1=\epsilon=10^{-6}, f_2=f_3=0$ and $f_j(t=0)\equiv((\rho-\epsilon)/(N-3))$. The thick lines highlight the components $f_j$ and the blue ones are for those that appear in stable equilibria, i.e. with $j=kr+1$ for $k=0,\ldots,T$
Speed of convergence towards the stable equilibria of the $\delta$ model. The initial condition is a small random perturbation of the steady-states
Evolution of the macroscopic velocity in time. Left: comparison of different values of $T$ and $\delta v$. The dot-dashed lines without markers correspond to the $\chi$ model. Right: relaxation to steady state for different combinations of $\eta$ and $T$
Evolution of the macroscopic velocity in time, for different values of $T$ and $\eta$. Left: $\rho=0.65$. Right: $\rho=0.9$
Fundamental diagrams resulting from the $\delta$ model (blue *-symbols) and from the $\chi$ model with acceleration parameter $\Delta v_{\delta}=\frac12\Delta v_{\chi}$ (red circles). The dashed line is the flux of the $\delta$ model in the limit $r\to\infty$
Fundamental diagrams resulting from the $\delta$ model with acceleration parameter $\Delta v_{\delta}=\frac14$. The probability $P$ is taken as in (5) with $\gamma=1$ (blue data), $\gamma=3/4$ (green data) and $\gamma=1/4$ (cyan data). The dashed lines are the fluxes in the limit $r\to\infty$
Top: fundamental diagrams provided by the $\chi$ model with $N=4$ (left) and $N=61$ (right) velocities. Bottom: equilibria of the function $f_1$ (blue solid line), $f_{N-1}$ (green dashed) and $f_N$ (red dot-dashed) for any density in $\left[0,1\right]$
Comparison between experimental data and the diagram resulting from the $\delta$ model, with $\Delta v=1/3$, $P=1-\rho^{3/4}$ (left panel) and $\Delta v=1/4$, $P=1-\rho^{3/4}$ (right panel). The experimental diagram is reproduced by kind permission of Seibold et al. [27]
Table describing the patterns in the matrices of Figure 2, 3 and 9.
 Pattern Entries of the matrices Proportional to P ▨ 1 − P ▩ P ■ 1
 Pattern Entries of the matrices Proportional to P ▨ 1 − P ▩ P ■ 1
Table of the numerical parameters.
 Parameter Description Definition N number of discrete speeds δv cell amplitude $\delta v=\frac{{V_{\max }}}{N-1}$ r ratio between the speed jump $\Delta v$ and the cell size $\delta v$ $r=\frac{\Delta v}{\delta v}$ T number of speed jumps $\Delta v$ contained in $[0,{V_{\max }}]$ $T=\frac{{V_{\max }}}{\Delta v}$
 Parameter Description Definition N number of discrete speeds δv cell amplitude $\delta v=\frac{{V_{\max }}}{N-1}$ r ratio between the speed jump $\Delta v$ and the cell size $\delta v$ $r=\frac{\Delta v}{\delta v}$ T number of speed jumps $\Delta v$ contained in $[0,{V_{\max }}]$ $T=\frac{{V_{\max }}}{\Delta v}$
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