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
References:
[1]

A. AwA. KlarT. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278. doi: 10.1137/S0036139900380955.

[2]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938 (electronic). doi: 10.1137/S0036139997332099.

[3]

R. BorscheM. Kimathi and A. Klar, A class of multi-phase traffic theories for microscopic, kinetic and continuum traffic models, Comput. Math. Appl., 64 (2012), 2939-2953. doi: 10.1016/j.camwa.2012.08.013.

[4]

R.M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721 (electronic). doi: 10.1137/S0036139901393184.

[5]

V. CosciaM. Delitala and P. Frasca, On the mathematical theory of vehicular traffic flow. Ⅱ. Discrete velocity kinetic models, Internat. J. Non-Linear Mech., 42 (2007), 411-421. doi: 10.1016/j.ijnonlinmec.2006.02.008.

[6]

M. Delitala and A. Tosin, Mathematical modeling of vehicular traffic: A discrete kinetic theory approach, Math. Models Methods Appl. Sci., 17 (2007), 901-932. doi: 10.1142/S0218202507002157.

[7]

L. Fermo and A. Tosin, A fully-discrete-state kinetic theory approach to modeling vehicular traffic, SIAM J. Appl. Math., 73 (2013), 1533-1556. doi: 10.1137/120897110.

[8]

L. Fermo and A. Tosin, Fundamental diagrams for kinetic equations of traffic flow, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 449-462. doi: 10.3934/dcdss.2014.7.449.

[9]

P. Freguglia and A. Tosin, Proposal of a Risk Model for Vehicular Traffic: A Boltzmann-type Kinetic Approach, Commun. Math. Sci., Accepted, arXiv: 1506.05422.

[10]

M. Herty and R. Illner, On stop-and-go waves in dense traffic, Kinet. Relat. Models, 1 (2008), 437-452. doi: 10.3934/krm.2008.1.437.

[11]

M. Herty and R. Illner, Analytical and numerical investigations of refined macroscopic traffic flow models, Kinet. Relat. Models, 3 (2010), 311-333. doi: 10.3934/krm.2010.3.311.

[12]

M. Herty and L. Pareschi, Fokker-Planck asymptotics for traffic flow models, Kinet. Relat. Models, 3 (2010), 165-179. doi: 10.3934/krm.2010.3.165.

[13]

R. Illner, A. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow, Commun. Math. Sci. , 1 (2003), 1–12, URL http://projecteuclid.org/euclid.cms/1118150395.

[14]

A. Klar and R. Wegener, A kinetic model for vehicular traffic derived from a stochastic microscopic model, Transport. Theor. Stat., 25 (1996), 785-798.

[15]

A. Klar and R. Wegener, Enskog-like kinetic models for vehicular traffic, J. Statist. Phys., 87 (1997), 91-114. doi: 10.1007/BF02181481.

[16]

J.P. Lebacque, Two-phase bounded-acceleration traffic flow model: Analytical solutions and applications, Transport. Res. Record, 1852 (2003), 220-230.

[17]

J.P. Lebacque and M.M. Khoshyaran, A variational formulation for higher order macroscopic traffic flow models of the GSOM family, Procedia -Social and Behavioral Sciences, 80 (2013), 370-394.

[18]

M.J. Lighthill and G.B. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345.

[19]

A. R. Méndez and R. M. Velasco, Kerner's free-synchronized phase transition in a macroscopic traffic flow model with two classes of drivers, J. Phys. A, 46 (2013), 462001, 9PP. doi: 10.1088/1751-8113/46/46/462001.

[20]

S.L. Paveri-Fontana, On Boltzmann-like treatments for traffic flow: A critical review of the basic model and an alternative proposal for dilute traffic analysis, Transport. Res., 9 (1975), 225-235.

[21]

B. Piccoli and A. Tosin, Vehicular traffic: a review of continuum mathematical models, in Mathematics of complexity and dynamical systems. Vols. 1-3, Springer, New York, 2012,1748-1770. doi: 10.1007/978-1-4614-1806-1_112.

[22]

I. Prigogine, A Boltzmann-like approach to the statistical theory of traffic flow, in Theory of traffic flow, Elsevier, Amsterdam, 1961,158–164.

[23]

I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic, American Elsevier Publishing Co., New York, 1971.

[24]

G. Puppo, M. Semplice, A. Tosin and G. Visconti, Analysis of a multi-population kinetic model for traffic flow, Commun. Math. Sci. , Accepted. arXiv: 1511.06395v2.

[25]

G. PuppoM. SempliceA. Tosin and G. Visconti, Fundamental diagrams in traffic flow: The case of heterogeneous kinetic models, Commun. Math. Sci., 14 (2016), 643-669. doi: 10.4310/CMS.2016.v14.n3.a3.

[26]

M. D. Rosini, Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Understanding Complex Systems, Springer, Heidelberg, 2013. doi: 10.1007/978-3-319-00155-5.

[27]

B. SeiboldM.R. FlynnA.R. Kasimov and R.R. Rosales, Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models, Netw. Heterog. Media, 8 (2013), 745-772. doi: 10.3934/nhm.2013.8.745.

[28]

G. Visconti, M. Herty, G. Puppo and A. Tosin, Multivalued fundamental diagrams of traffic flow in the kinetic Fokker-Planck limit, Multiscale Model. Simul. Accepted. arXiv: 1607.08530.

[29]

H.M. Zhang and T. Kim, A car-following theory for multiphase vehicular traffic flow, Transport. Res. B-Meth., 39 (2005), 385-399.

show all references

References:
[1]

A. AwA. KlarT. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278. doi: 10.1137/S0036139900380955.

[2]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938 (electronic). doi: 10.1137/S0036139997332099.

[3]

R. BorscheM. Kimathi and A. Klar, A class of multi-phase traffic theories for microscopic, kinetic and continuum traffic models, Comput. Math. Appl., 64 (2012), 2939-2953. doi: 10.1016/j.camwa.2012.08.013.

[4]

R.M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721 (electronic). doi: 10.1137/S0036139901393184.

[5]

V. CosciaM. Delitala and P. Frasca, On the mathematical theory of vehicular traffic flow. Ⅱ. Discrete velocity kinetic models, Internat. J. Non-Linear Mech., 42 (2007), 411-421. doi: 10.1016/j.ijnonlinmec.2006.02.008.

[6]

M. Delitala and A. Tosin, Mathematical modeling of vehicular traffic: A discrete kinetic theory approach, Math. Models Methods Appl. Sci., 17 (2007), 901-932. doi: 10.1142/S0218202507002157.

[7]

L. Fermo and A. Tosin, A fully-discrete-state kinetic theory approach to modeling vehicular traffic, SIAM J. Appl. Math., 73 (2013), 1533-1556. doi: 10.1137/120897110.

[8]

L. Fermo and A. Tosin, Fundamental diagrams for kinetic equations of traffic flow, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 449-462. doi: 10.3934/dcdss.2014.7.449.

[9]

P. Freguglia and A. Tosin, Proposal of a Risk Model for Vehicular Traffic: A Boltzmann-type Kinetic Approach, Commun. Math. Sci., Accepted, arXiv: 1506.05422.

[10]

M. Herty and R. Illner, On stop-and-go waves in dense traffic, Kinet. Relat. Models, 1 (2008), 437-452. doi: 10.3934/krm.2008.1.437.

[11]

M. Herty and R. Illner, Analytical and numerical investigations of refined macroscopic traffic flow models, Kinet. Relat. Models, 3 (2010), 311-333. doi: 10.3934/krm.2010.3.311.

[12]

M. Herty and L. Pareschi, Fokker-Planck asymptotics for traffic flow models, Kinet. Relat. Models, 3 (2010), 165-179. doi: 10.3934/krm.2010.3.165.

[13]

R. Illner, A. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow, Commun. Math. Sci. , 1 (2003), 1–12, URL http://projecteuclid.org/euclid.cms/1118150395.

[14]

A. Klar and R. Wegener, A kinetic model for vehicular traffic derived from a stochastic microscopic model, Transport. Theor. Stat., 25 (1996), 785-798.

[15]

A. Klar and R. Wegener, Enskog-like kinetic models for vehicular traffic, J. Statist. Phys., 87 (1997), 91-114. doi: 10.1007/BF02181481.

[16]

J.P. Lebacque, Two-phase bounded-acceleration traffic flow model: Analytical solutions and applications, Transport. Res. Record, 1852 (2003), 220-230.

[17]

J.P. Lebacque and M.M. Khoshyaran, A variational formulation for higher order macroscopic traffic flow models of the GSOM family, Procedia -Social and Behavioral Sciences, 80 (2013), 370-394.

[18]

M.J. Lighthill and G.B. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345.

[19]

A. R. Méndez and R. M. Velasco, Kerner's free-synchronized phase transition in a macroscopic traffic flow model with two classes of drivers, J. Phys. A, 46 (2013), 462001, 9PP. doi: 10.1088/1751-8113/46/46/462001.

[20]

S.L. Paveri-Fontana, On Boltzmann-like treatments for traffic flow: A critical review of the basic model and an alternative proposal for dilute traffic analysis, Transport. Res., 9 (1975), 225-235.

[21]

B. Piccoli and A. Tosin, Vehicular traffic: a review of continuum mathematical models, in Mathematics of complexity and dynamical systems. Vols. 1-3, Springer, New York, 2012,1748-1770. doi: 10.1007/978-1-4614-1806-1_112.

[22]

I. Prigogine, A Boltzmann-like approach to the statistical theory of traffic flow, in Theory of traffic flow, Elsevier, Amsterdam, 1961,158–164.

[23]

I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic, American Elsevier Publishing Co., New York, 1971.

[24]

G. Puppo, M. Semplice, A. Tosin and G. Visconti, Analysis of a multi-population kinetic model for traffic flow, Commun. Math. Sci. , Accepted. arXiv: 1511.06395v2.

[25]

G. PuppoM. SempliceA. Tosin and G. Visconti, Fundamental diagrams in traffic flow: The case of heterogeneous kinetic models, Commun. Math. Sci., 14 (2016), 643-669. doi: 10.4310/CMS.2016.v14.n3.a3.

[26]

M. D. Rosini, Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Understanding Complex Systems, Springer, Heidelberg, 2013. doi: 10.1007/978-3-319-00155-5.

[27]

B. SeiboldM.R. FlynnA.R. Kasimov and R.R. Rosales, Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models, Netw. Heterog. Media, 8 (2013), 745-772. doi: 10.3934/nhm.2013.8.745.

[28]

G. Visconti, M. Herty, G. Puppo and A. Tosin, Multivalued fundamental diagrams of traffic flow in the kinetic Fokker-Planck limit, Multiscale Model. Simul. Accepted. arXiv: 1607.08530.

[29]

H.M. Zhang and T. Kim, A car-following theory for multiphase vehicular traffic flow, Transport. Res. B-Meth., 39 (2005), 385-399.

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