Article Contents
Article Contents

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

• * Corresponding author: Gabriella Puppo
• 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.

Mathematics Subject Classification: Primary: 35Q20, 90B20; Secondary: 65Z05.

 Citation:

• Figure 1.  Connection between the $\delta$ model and a discrete-velocity model, having the same steady-state distribution

Figure 2.  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

Figure 3.  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

Figure 9.  Structure of the probability matrices of the $\chi$ model with $\delta v$ integer sub-multiple of $\Delta v$

Figure 4.  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 $r=1$

Figure 5.  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

Figure 6.  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)

Figure 7.  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 $k=0,\ldots,T$

Figure 8.  Speed of convergence towards the stable equilibria of the $\delta$ model. The initial condition is a small random perturbation of the steady-states

Figure 10.  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 $T$

Figure 11.  Evolution of the macroscopic velocity in time, for different values of $T$ and $\eta$. Left: $\rho=0.65$. Right: $\rho=0.9$

Figure 12.  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 $r\to\infty$

Figure 13.  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 $r\to\infty$

Figure 14.  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 $\left[0,1\right]$

Figure 15.  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 1.  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

Table 2.  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 $[0,{V_{\max }}]$ $T=\frac{{V_{\max }}}{\Delta v}$
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