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

  • * Corresponding author: Gabriella Puppo

    * Corresponding author: Gabriella Puppo 
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  • 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.


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  • 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
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    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 $[0,{V_{\max }}]$ $T=\frac{{V_{\max }}}{\Delta v}$
     | Show Table
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  •   A. Aw , A. Klar , T. 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.
      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.
      R. Borsche , M. 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.
      R.M. Colombo , Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002) , 708-721 (electronic).  doi: 10.1137/S0036139901393184.
      V. Coscia , M. 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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      A. Klar  and  R. Wegener , A kinetic model for vehicular traffic derived from a stochastic microscopic model, Transport. Theor. Stat., 25 (1996) , 785-798. 
      A. Klar  and  R. Wegener , Enskog-like kinetic models for vehicular traffic, J. Statist. Phys., 87 (1997) , 91-114.  doi: 10.1007/BF02181481.
      J.P. Lebacque , Two-phase bounded-acceleration traffic flow model: Analytical solutions and applications, Transport. Res. Record, 1852 (2003) , 220-230. 
      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. 
      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. 
      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.
      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. 
      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.
      I. Prigogine, A Boltzmann-like approach to the statistical theory of traffic flow, in Theory of traffic flow, Elsevier, Amsterdam, 1961,158–164.
      I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic, American Elsevier Publishing Co., New York, 1971.
      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.
      G. Puppo , M. Semplice , A. 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.
      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.
      B. Seibold , M.R. Flynn , A.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.
      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.
      H.M. Zhang  and  T. Kim , A car-following theory for multiphase vehicular traffic flow, Transport. Res. B-Meth., 39 (2005) , 385-399. 
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