American Institute of Mathematical Sciences

Advanced
September  2018, 13(3): 449-478. doi: 10.3934/nhm.2018020

Traveling wave profiles for a Follow-the-Leader model for traffic flow with rough road condition

Wen Shen ,

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

* Corresponding author: Wen Shen

Received  November 2017 Revised  January 2018 Published  July 2018

We study a Follow-the-Leader (FtL) ODE model for traffic flow with rough road condition, and analyze stationary traveling wave profiles where the solutions of the FtL model trace along, near the jump in the road condition. We derive a discontinuous delay differential equation (DDDE) for these profiles. For various cases, we obtain results on existence, uniqueness and local stability of the profiles. The results here offer an alternative approximation, possibly more realistic than the classical vanishing viscosity approach, to the conservation law with discontinuous flux for traffic flow.

Keywords: Traffic flow, rough coefficient, traveling waves, microscopic models, delay differential equation, local stability.
Mathematics Subject Classification: Primary: 65M20, 35L02, 35L65; Secondary: 34B99, 35Q99.
Citation: Wen Shen. Traveling wave profiles for a Follow-the-Leader model for traffic flow with rough road condition. Networks & Heterogeneous Media, 2018, 13 (3) : 449-478. doi: 10.3934/nhm.2018020
References:
[1]

B. Andreianov, New approaches to describing admissibility of solutions of scalar conservation laws with discontinuous flux, CANUM 2014—42e Congrès National d'Analyse Numérique, ESAIM Proc. Surveys, EDP Sci., Les Ulis, 50 (2015), 40–65. doi: 10.1051/proc/201550003.  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]

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

[4]

A. Bressan, Unique solutions for a class of discontinuous differential equations, Proc. Amer. Math. Soc., 104 (1988), 772-778.  doi: 10.1090/S0002-9939-1988-0964856-0.  Google Scholar

[5]

A. Bressan and W. Shen, Uniqueness for discontinuous ODE and conservation laws, Nonlinear Anal, 34 (1998), 637-652.  doi: 10.1016/S0362-546X(97)00590-7.  Google Scholar

[6]

A. Bressan and W. Shen, Unique solutions of discontinuous O.D.E.'s in Banach spaces, Anal. Appl. (Singap.), 4 (2006), 247-262.  doi: 10.1142/S0219530506000772.  Google Scholar

[7]

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

[8]

A. Corli, L. di Ruvo, L. Malaguti and M. D. Rosini, Traveling waves for degenerate diffusive equations on networks, Netw. Heterog. Media, 12 (2017), 339-370.  doi: 10.3934/nhm.2017015.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications (Soviet Series), 18, Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[14]

T. Gimse and N. H. Risebro, Riemann problems with a discontinuous flux function, Third International Conference on Hyperbolic Problems, Vol. I, II, 488–502, Studentlitteratur, Lund, 1990.  Google Scholar

[15]

G. Guerra and W. Shen, Vanishing viscosity solutions of riemann problems for models in polymer flooding, Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis, (2017), 261-285.  doi: 10.4171/186-1/12.  Google Scholar

[16]

H. Holden and N. H. Risebro, Continuum limit of Follow-the-Leader models - a short proof, To appear in DCDS, 38 (2018), 715-722.  doi: 10.3934/dcds.2018031.  Google Scholar

[17]

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, Networks & Heterogeneous Media, 13 (2018). Google Scholar

[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.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[19]

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

[20]

W. Shen, On the uniqueness of vanishing viscosity solutions for Riemann problems for polymer flooding, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 37, 25pp. doi: 10.1007/s00030-017-0461-y.  Google Scholar

[21]

W. Shen, Scilab codes for simulations and plots used in this paper www.personal.psu.edu/wxs27/SIM/Traffic-DDDE, 2017. Google Scholar

[22]

W. Shen and K. Shikh-Khalil, Traveling waves for a microscopic model of traffic flow, Discrete and Continuous Dynamical Systems, 38 (2018), 2571-2589.  doi: 10.3934/dcds.2018108.  Google Scholar

show all references

References:
[1]

B. Andreianov, New approaches to describing admissibility of solutions of scalar conservation laws with discontinuous flux, CANUM 2014—42e Congrès National d'Analyse Numérique, ESAIM Proc. Surveys, EDP Sci., Les Ulis, 50 (2015), 40–65. doi: 10.1051/proc/201550003.  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]

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

[4]

A. Bressan, Unique solutions for a class of discontinuous differential equations, Proc. Amer. Math. Soc., 104 (1988), 772-778.  doi: 10.1090/S0002-9939-1988-0964856-0.  Google Scholar

[5]

A. Bressan and W. Shen, Uniqueness for discontinuous ODE and conservation laws, Nonlinear Anal, 34 (1998), 637-652.  doi: 10.1016/S0362-546X(97)00590-7.  Google Scholar

[6]

A. Bressan and W. Shen, Unique solutions of discontinuous O.D.E.'s in Banach spaces, Anal. Appl. (Singap.), 4 (2006), 247-262.  doi: 10.1142/S0219530506000772.  Google Scholar

[7]

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

[8]

A. Corli, L. di Ruvo, L. Malaguti and M. D. Rosini, Traveling waves for degenerate diffusive equations on networks, Netw. Heterog. Media, 12 (2017), 339-370.  doi: 10.3934/nhm.2017015.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications (Soviet Series), 18, Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[14]

T. Gimse and N. H. Risebro, Riemann problems with a discontinuous flux function, Third International Conference on Hyperbolic Problems, Vol. I, II, 488–502, Studentlitteratur, Lund, 1990.  Google Scholar

[15]

G. Guerra and W. Shen, Vanishing viscosity solutions of riemann problems for models in polymer flooding, Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis, (2017), 261-285.  doi: 10.4171/186-1/12.  Google Scholar

[16]

H. Holden and N. H. Risebro, Continuum limit of Follow-the-Leader models - a short proof, To appear in DCDS, 38 (2018), 715-722.  doi: 10.3934/dcds.2018031.  Google Scholar

[17]

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, Networks & Heterogeneous Media, 13 (2018). Google Scholar

[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.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[19]

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

[20]

W. Shen, On the uniqueness of vanishing viscosity solutions for Riemann problems for polymer flooding, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 37, 25pp. doi: 10.1007/s00030-017-0461-y.  Google Scholar

[21]

W. Shen, Scilab codes for simulations and plots used in this paper www.personal.psu.edu/wxs27/SIM/Traffic-DDDE, 2017. Google Scholar

[22]

W. Shen and K. Shikh-Khalil, Traveling waves for a microscopic model of traffic flow, Discrete and Continuous Dynamical Systems, 38 (2018), 2571-2589.  doi: 10.3934/dcds.2018108.  Google Scholar

Figure 1.  Graph of $Q(x)$ on the interval $[z_k, z_{k+2}]$. Illustration of the locations for $y_k, y^\sharp_k$ and $y'_{k+1}$, used in the proof of Lemma 2.8
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Flux functions $f_-, f_+$, and the locations of $\rho^-_1$, $\rho^+_1$, $\rho^-_2$, $\rho^+_2$, and $\rho^*$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Case 1A: (1) Plot of the flux functions $f_-, f_+$ and the locations of $\rho_-, \rho_+$; (2) Plots of various profiles of $Q(x)$, with different values of $Q(0)$; (3) Plots of various viscous traveling waves $\rho^\varepsilon(x)$, with different values of $\rho^\varepsilon(0)$; (4) Plots of various solutions of the FtL model $\{z_i(t), \rho_i(t)\}$, with 3 different initial Riemann data. Here the thick dots denote the locations of cars at $t = 2$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Case 1B. (1) Plots of the flux functions and the locations of $\rho_-, \rho_+$; (2) Plot of the unique stationary profile $Q(x)$ with $Q(0) = \rho_+$; (3) Plot of the unique viscous profile $\rho^\varepsilon(x)$ with $\rho^\varepsilon(0) = \rho_+$; (4) Plot of the solution of the FtL model $\{z_i(t), \rho_i(t)\}$ with a Riemann initial data. Here the thick dots are the locations of cars at $t = 2$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Case 1C. (1) Plots of the flux functions and the locations of $\rho_-, \rho_+$; (2) Plot of the unique viscous profile $\rho^\varepsilon(x)$ with $\rho^\varepsilon(0) = \rho_-$; (3) Plot of the solution of the FtL model $\{z_i(t), \rho_i(t)\}$ with a Riemann initial data. Here the thick dots are the locations of cars at $t = 2$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Case 1D. (1): Plots of the flux functions and the locations of $\rho_-, \rho_+$; (2): Plot of the solution of the FtL model $\{z_i(t), \rho_i(t)\}$ with a Riemann initial data. Here the thick dots are the locations of cars at $t = 2$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Flux functions $f_-, f_+$, and the locations of $\rho^-_1, \rho^+_1, \rho^-_2, \rho^+_2$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Case 2A. (1) Flux functions and the locations of $\rho_-, \rho_+$; (2) Plots of various profiles of $Q(x)$, with different values of $Q(0)$; (3) Plots of various viscous traveling waves $\rho^\varepsilon(x)$, with different values of $\rho^\varepsilon(0)$; (4) Plots of various solutions of the FtL model $\{z_i(t), \rho_i(t)\}$, with 3 different initial Riemann data. Here the thick dots denote the locations of cars at $t = 2$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  Case 2B. (1) Flux functions and the locations of $\rho_-, \rho_+$; (2) Plots of the unique profile of $Q(x)$, with $Q(0) = \rho_+$; (3) Plots of various viscous traveling waves $\rho^\varepsilon(x)$, with $\rho^\varepsilon(0) = \rho_+$; (4) Plots of the solution of the FtL model $\{z_i(t), \rho_i(t)\}$ with a Riemann initial data. Here the thick dots denote the locations of cars at $t = 2$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10.  Case 2C. (1) Plot of the flux functions $f_-, f_+$ and the locations of $\rho_-, \rho_+$; (2) Plot of the unique viscous profile $\rho^\varepsilon$ with $\rho^\varepsilon(0) = \rho_-$; (3) Plot of the solution of the FtL model $\{z_i(t), \rho_i(t)\}$ with a Riemann initial data. Here the thick dots are the locations of cars at $t = 2$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 11.  Case 2D. (1): Plots of the flux functions and the locations of $\rho_-, \rho_+$; (2): Plot of the solution of the FtL model $\{z_i(t), \rho_i(t)\}$ with a Riemann initial data. Here the thick dots are the locations of cars at $t = 2$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 12.  (1). Plots of the flux functions and the location of the left (L), right (R) and middle (M) states in the solution of the Riemann problem; (2). Numerical simulation results $\{z_i(t), \rho_i(t)\}$ with FtL model with Riemann initial data, at $t = 1$; (3) Numerical simulation results $\rho^\varepsilon(t)$ for the viscous conservation law at $t = 1$, with the same Riemann initial data
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

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
[2]

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
[3]

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
[4]

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
[5]

Dan Stanescu, Benito Chen-Charpentier. Random coefficient differential equation models for Monod kinetics. Conference Publications, 2009, 2009 (Special) : 719-728. doi: 10.3934/proc.2009.2009.719
[6]

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
[7]

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
[8]

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
[9]

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
[10]

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
[11]

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
[12]

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
[13]

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
[14]

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
[15]

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
[16]

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
[17]

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
[18]

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
[19]

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
[20]

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

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (62)
  • HTML views (238)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences