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May  2018, 38(5): 2571-2589. doi: 10.3934/dcds.2018108

Traveling waves for a microscopic model of traffic flow

Wen Shen , and  Karim Shikh-Khalil

Mathematics Department, Pennsylvania State University, University Park, PA 16802, USA

* Corresponding author: Wen Shen

Received  September 2017 Revised  December 2017 Published  March 2018

We consider the follow-the-leader model for traffic flow. The position of each car $z_i(t)$ satisfies an ordinary differential equation, whose speed depends only on the relative position $z_{i+1}(t)$ of the car ahead. Each car perceives a local density $ρ_i(t)$. We study a discrete traveling wave profile $W(x)$ along which the trajectory $(ρ_i(t), z_i(t))$ traces such that $W(z_i(t)) = ρ_i(t)$ for all $i$ and $t>0$; see definition 2.2. We derive a delay differential equation satisfied by such profiles. Existence and uniqueness of solutions are proved, for the two-point boundary value problem where the car densities at $x\to±∞$ are given. Furthermore, we show that such profiles are locally stable, attracting nearby monotone solutions of the follow-the-leader model.

Keywords: Traffic flow, traveling waves, microscopic models, delay differential equation, local stability.
Mathematics Subject Classification: Primary: 35L02, 35L65; Secondary: 34B99, 35Q99.
Citation: Wen Shen, Karim Shikh-Khalil. Traveling waves for a microscopic model of traffic flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2571-2589. doi: 10.3934/dcds.2018108
References:
[1]

B. Argall, E. Cheleshkin, J. M. Greenberg, C. Hinde and P. Lin, A rigorous treatment of a follow-the-leader traffic model with traffic lights present, SIAM J. Appl. Math., 63 (2002), 149-168. doi: 10.1137/S0036139901391215. Google Scholar

[2]

J. Aubin, Macroscopic traffic models: Shifting from densities to "celerities", Appl. Math. Comput, 217 (2010), 963-971. doi: 10.1016/j.amc.2010.02.032. Google Scholar

[3]

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. Google Scholar

[4]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the multiscale modeling of vehicular traffic: From kinetic to hydrodynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1869-1888. doi: 10.3934/dcdsb.2014.19.1869. Google Scholar

[5]

R. M. Colombo and A. Marson, A Hölder continuous ODE related to traffic flow, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 759-772. doi: 10.1017/S0308210500002663. Google Scholar

[6]

R. M. Colombo and E. Rossi, On the micro-macro limit in traffic flow, Rend. Semin. Mat. Univ. Padova, 131 (2014), 217-235. doi: 10.4171/RSMUP/131-13. Google Scholar

[7]

E. Cristiani and S. Sahu, On the micro-to-macro limit for first-order traffic flow models on networks, Netw. Heterog. Media, 11 (2016), 395-413. doi: 10.3934/nhm.2016002. Google Scholar

[8]

M. Di Francesco and M. D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Arch. Ration. Mech. Anal., 217 (2015), 831-871. doi: 10.1007/s00205-015-0843-4. Google Scholar

[9]

R. D. Driver and M. D. Rosini, Existence and stability of solutions of a delay-differential system, Arch. Rational Mech. Anal., 10 (1962), 401-426. doi: 10.1007/BF00281203. Google Scholar

[10]

R. D. Driver, Ordinary and Delay Differential Equations, Applied Mathematical Sciences, 20, Springer-Verlag, New York-Heidelberg, 1977. Google Scholar

[11]

P. Goatin and F. Rossi, A traffic flow model with non-smooth metric interaction: well-posedness and micro-macro limit, Commun. Math. Sci., 15 (2017), 261-287. doi: 10.4310/CMS.2017.v15.n1.a12. Google Scholar

[12]

G. Guerra and W. Shen, Existence and stability of traveling waves for an integro-differential equation for slow erosion, J. Differential Equations, 256 (2014), 253-282. doi: 10.1016/j.jde.2013.09.003. Google Scholar

[13]

H. Holden and N. H. Risebro, Continuum limit of Follow-the-Leader models -a short proof, Discrete Contin. Dyn. Syst., 38 (2018), 715-722. Google Scholar

[14]

H. Holden and N. H. Risebro, Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow, preprint 2017.Google Scholar

[15]

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. doi: 10.1098/rspa.1955.0089. Google Scholar

[16]

E. Rossi, A justification of a LWR model based on a follow the leader description, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 579-591. doi: 10.3934/dcdss.2014.7.579. Google Scholar

[17]

W. Shen, http://www.personal.psu.edu/wxs27/SIM/TrafficODE/, Scilab code used to generate the approximate solutions in this paper, 2017.Google Scholar

[18]

W. Shen, Traveling wave profiles for a Follow-the-Leader model for traffic flow with rough road condition, preprint, 2017. To appear in Netw. Heterog. Media, 2018.Google Scholar

show all references

References:
[1]

B. Argall, E. Cheleshkin, J. M. Greenberg, C. Hinde and P. Lin, A rigorous treatment of a follow-the-leader traffic model with traffic lights present, SIAM J. Appl. Math., 63 (2002), 149-168. doi: 10.1137/S0036139901391215. Google Scholar

[2]

J. Aubin, Macroscopic traffic models: Shifting from densities to "celerities", Appl. Math. Comput, 217 (2010), 963-971. doi: 10.1016/j.amc.2010.02.032. Google Scholar

[3]

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. Google Scholar

[4]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the multiscale modeling of vehicular traffic: From kinetic to hydrodynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1869-1888. doi: 10.3934/dcdsb.2014.19.1869. Google Scholar

[5]

R. M. Colombo and A. Marson, A Hölder continuous ODE related to traffic flow, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 759-772. doi: 10.1017/S0308210500002663. Google Scholar

[6]

R. M. Colombo and E. Rossi, On the micro-macro limit in traffic flow, Rend. Semin. Mat. Univ. Padova, 131 (2014), 217-235. doi: 10.4171/RSMUP/131-13. Google Scholar

[7]

E. Cristiani and S. Sahu, On the micro-to-macro limit for first-order traffic flow models on networks, Netw. Heterog. Media, 11 (2016), 395-413. doi: 10.3934/nhm.2016002. Google Scholar

[8]

M. Di Francesco and M. D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Arch. Ration. Mech. Anal., 217 (2015), 831-871. doi: 10.1007/s00205-015-0843-4. Google Scholar

[9]

R. D. Driver and M. D. Rosini, Existence and stability of solutions of a delay-differential system, Arch. Rational Mech. Anal., 10 (1962), 401-426. doi: 10.1007/BF00281203. Google Scholar

[10]

R. D. Driver, Ordinary and Delay Differential Equations, Applied Mathematical Sciences, 20, Springer-Verlag, New York-Heidelberg, 1977. Google Scholar

[11]

P. Goatin and F. Rossi, A traffic flow model with non-smooth metric interaction: well-posedness and micro-macro limit, Commun. Math. Sci., 15 (2017), 261-287. doi: 10.4310/CMS.2017.v15.n1.a12. Google Scholar

[12]

G. Guerra and W. Shen, Existence and stability of traveling waves for an integro-differential equation for slow erosion, J. Differential Equations, 256 (2014), 253-282. doi: 10.1016/j.jde.2013.09.003. Google Scholar

[13]

H. Holden and N. H. Risebro, Continuum limit of Follow-the-Leader models -a short proof, Discrete Contin. Dyn. Syst., 38 (2018), 715-722. Google Scholar

[14]

H. Holden and N. H. Risebro, Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow, preprint 2017.Google Scholar

[15]

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. doi: 10.1098/rspa.1955.0089. Google Scholar

[16]

E. Rossi, A justification of a LWR model based on a follow the leader description, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 579-591. doi: 10.3934/dcdss.2014.7.579. Google Scholar

[17]

W. Shen, http://www.personal.psu.edu/wxs27/SIM/TrafficODE/, Scilab code used to generate the approximate solutions in this paper, 2017.Google Scholar

[18]

W. Shen, Traveling wave profiles for a Follow-the-Leader model for traffic flow with rough road condition, preprint, 2017. To appear in Netw. Heterog. Media, 2018.Google Scholar

Figure 1.  Typical graphs of $G(\lambda)$ and location of the zeros
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Figure 2.  Numerical simulations for the approximate sequence $W_n(x)$ for various values of $\hat x_n$. We use $\rho_- = 0.3$, $\rho_+ = 0.7$, $\ell = 0.5$, $V = 1$, and $\phi(\rho) = 1-\rho$. The solid curve is the graph of $\psi(x) = \rho_+-0.2 e^{-\lambda_+x}$, plotted on the interval $0\le x\le 2$. The dotted curves are plots for $W_n(x)$ with $\hat x_n = 0, 0.1, 0.25, 0.5, 1$
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Figure 3.  Numerical simulations of stationary profiles $W(x)$ with various values of $(\rho_-, \rho_+)$
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