American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Dynamics for the diffusive Leslie-Gower model with double free boundaries
  • DCDS Home
  • This Issue
  • Next Article
    Global existence in the critical space for the Thirring and Gross-Neveu models coupled with the electromagnetic field
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
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Numerical simulations of stationary profiles $W(x)$ with various values of $(\rho_-, \rho_+)$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Johanna Ridder, Wen Shen. Traveling waves for nonlocal models of traffic flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4001-4040. doi: 10.3934/dcds.2019161
[2]

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
[3]

Wen Shen. Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads. Networks & Heterogeneous Media, 2019, 14 (4) : 709-732. doi: 10.3934/nhm.2019028
[4]

Felisia Angela Chiarello, Paola Goatin. Non-local multi-class traffic flow models. Networks & Heterogeneous Media, 2019, 14 (2) : 371-387. doi: 10.3934/nhm.2019015
[5]

Tong Li, Jeungeun Park. Stability of traveling waves of models for image processing with non-convex nonlinearity. Communications on Pure & Applied Analysis, 2018, 17 (3) : 959-985. doi: 10.3934/cpaa.2018047
[6]

Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445
[7]

Jan Friedrich, Oliver Kolb, Simone Göttlich. A Godunov type scheme for a class of LWR traffic flow models with non-local flux. Networks & Heterogeneous Media, 2018, 13 (4) : 531-547. doi: 10.3934/nhm.2018024
[8]

John M. Hong, Cheng-Hsiung Hsu, Bo-Chih Huang, Tzi-Sheng Yang. Geometric singular perturbation approach to the existence and instability of stationary waves for viscous traffic flow models. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1501-1526. doi: 10.3934/cpaa.2013.12.1501
[9]

Shi-Liang Wu, Wan-Tong Li, San-Yang Liu. Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 347-366. doi: 10.3934/dcdsb.2012.17.347
[10]

Zhao-Xing Yang, Guo-Bao Zhang, Ge Tian, Zhaosheng Feng. Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 581-603. doi: 10.3934/dcdss.2017029
[11]

Abraham Solar. Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5799-5823. doi: 10.3934/dcds.2019255
[12]

Alberto Bressan, Khai T. Nguyen. Conservation law models for traffic flow on a network of roads. Networks & Heterogeneous Media, 2015, 10 (2) : 255-293. doi: 10.3934/nhm.2015.10.255
[13]

Tong Li. Qualitative analysis of some PDE models of traffic flow. Networks & Heterogeneous Media, 2013, 8 (3) : 773-781. doi: 10.3934/nhm.2013.8.773
[14]

Paola Goatin. Traffic flow models with phase transitions on road networks. Networks & Heterogeneous Media, 2009, 4 (2) : 287-301. doi: 10.3934/nhm.2009.4.287
[15]

Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic & Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165
[16]

Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689
[17]

Grégory Faye. Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2473-2496. doi: 10.3934/dcds.2016.36.2473
[18]

Hua Chen, Ling-Jun Wang. A perturbation approach for the transverse spectral stability of small periodic traveling waves of the ZK equation. Kinetic & Related Models, 2012, 5 (2) : 261-281. doi: 10.3934/krm.2012.5.261
[19]

Aslihan Demirkaya, Milena Stanislavova. Numerical results on existence and stability of standing and traveling waves for the fourth order beam equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 197-209. doi: 10.3934/dcdsb.2018097
[20]

Luyi Ma, Hong-Tao Niu, Zhi-Cheng Wang. Global asymptotic stability of traveling waves to the Allen-Cahn equation with a fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2457-2472. doi: 10.3934/cpaa.2019111

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (73)
  • HTML views (183)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences