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May
2018, 38(5): 2571-2589.
doi: 10.3934/dcds.2018108
Traveling waves for a microscopic model of traffic flow
Mathematics Department, Pennsylvania State University, University Park, PA 16802, USA |
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 and Continuous Dynamical Systems,
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [10] |
R. D. Driver, Ordinary and Delay Differential Equations, Applied Mathematical Sciences, 20, Springer-Verlag, New York-Heidelberg, 1977. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [17] |
W. Shen, http://www.personal.psu.edu/wxs27/SIM/TrafficODE/, Scilab code used to generate the approximate solutions in this paper, 2017. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [10] |
R. D. Driver, Ordinary and Delay Differential Equations, Applied Mathematical Sciences, 20, Springer-Verlag, New York-Heidelberg, 1977. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [17] |
W. Shen, http://www.personal.psu.edu/wxs27/SIM/TrafficODE/, Scilab code used to generate the approximate solutions in this paper, 2017. |
| [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. |
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 $\hat x_n = 0, 0.1, 0.25, 0.5, 1$
Figure 3.
Numerical simulations of stationary profiles $W(x)$ with various values of $(\rho_-, \rho_+)$
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