June 2019, 14(2): 265-288. doi: 10.3934/nhm.2019011

Derivation of second order traffic flow models with time delays

1. 

Fraunhofer Institute ITWM, 67663 Kaiserslautern, Germany

2. 

University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany

 

Received  December 2017 Revised  December 2018 Published  April 2019

Fund Project: S. Göttlich acknowledges funding by the German Research Foundation under grant GO 1920/4-1

Starting from microscopic follow-the-leader models, we develop hyperbolic delay partial differential equations to govern the density and velocity of vehicular traffic. The proposed models can be seen as an extension of the classical Aw-Rascle-Zhang model, where the reaction time of drivers appears as an additional term in the velocity equation. We propose numerical methods based on first principles and present a numerical study, where we focus on the impact of time delays in comparison to undelayed models.

Citation: Michael Burger, Simone Göttlich, Thomas Jung. Derivation of second order traffic flow models with time delays. Networks & Heterogeneous Media, 2019, 14 (2) : 265-288. doi: 10.3934/nhm.2019011
References:
[1]

A. Ashyralyev and D. Agirseven, Well-posedness of delay parabolic difference equations, Adv. Difference Equ., 2014 (2014), 20pp. doi: 10.1186/1687-1847-2014-18.

[2]

A. Ashyralyev and D. Agirseven, Bounded Solutions of nonlinear hyperbolic equations with time delay, Electron. J. Differential Equations, 2018 (2018), Paper No. 21, 15 pp.

[3]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938. doi: 10.1137/S0036139997332099.

[4]

A. AwA. KlarT. 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.

[5] A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, 2013.
[6]

M. Brackstone and M. Mcdonald, Car-following: A historical review, Transportation Research Part F: Traffic Psychology and Behaviour, 2 (1999), 181-196. doi: 10.1016/S1369-8478(00)00005-X.

[7]

R. E. ChandlerR. Herman and E. W. Montroll, Traffic dynamics: Studies in car following, Operations Res., 6 (1958), 165-184. doi: 10.1287/opre.6.2.165.

[8]

R. M. CorlessG. H. GonnetD. E. G. HareD. J. Jeffrey and D. E. Knuth, On the Lambert W Function, Adv. Comput. Math, 5 (1996), 329-356. doi: 10.1007/BF02124750.

[9]

C. D'Apice and B. Piccoli, Vertex flow models for vehicular traffic on networks, Math. Models Methods Appl. Sci., 18 (2008), 1299-1315. doi: 10.1142/S0218202508003042.

[10]

C.F. Daganzo, Requiem for second-order fluid approximations of traffic flow, Transportation Research Part B: Methodological, 29 (1995), 277-286. doi: 10.1016/0191-2615(95)00007-Z.

[11]

N. DavoodiA. R. Soheili and S. M. Hashemi, A macro-model for traffic flow with consideration of driver's reaction time and distance, Nonlinear Dynam., 83 (2016), 1621-1628. doi: 10.1007/s11071-015-2435-0.

[12]

G. Emch, Coarse-graining in Liouville space and master equation, Helv. Phys. Acta, 37 (1964), 532-544.

[13]

S. Fan and B. Seibold, Effect of the choice of stagnation density in data-fitted first- and second-order traffic models, preprint, arXiv: 1308.0393.

[14]

S. FanM. Herty and B. Seibold, Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model, Netw. Heterog. Media, 9 (2014), 239-268. doi: 10.3934/nhm.2014.9.239.

[15]

M. Garavello and B. Piccoli, Traffic Flow on Networks, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.

[16]

S. Gottlieb and C.-W. Shu, Total variation diminishing Runge-Kutta schemes, Math. Comp., 67 (1998), 73-85. doi: 10.1090/S0025-5718-98-00913-2.

[17]

A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 135 (1997), 259-278. doi: 10.1006/jcph.1997.5725.

[18]

D. Helbing, Verkehrsdynamik: Neue Physikalische Modellierungskonzepte, Springer Berlin Heidelberg, 1997.

[19]

S. P. Hoogendoorn and P. H. Bovy, State-of-the-art of vehicular traffic flow modelling, Proceedings of the Institution of Mechanical Engineers, Part Ⅰ: Journal of Systems and Control Engineering, 215 (2001), 283–303. doi: 10.1177/095965180121500402.

[20]

W.-F. Jiang and Z. Wang, Developing an Aw-Rascle model of traffic flow, J. Engrg. Math., 97 (2016), 135-146. doi: 10.1007/s10665-015-9801-2.

[21]

E. Kometani and T. Sasaki, On the stability of traffic flow (Report-I), J. Op. Res. Japan, 2 (1958), 11-26.

[22]

H. K. Lee, H.-W. Lee and D. Kim, Macroscopic traffic models from microscopic car-following models, Physical Review E, 64 (2001), 056126. doi: 10.1103/PhysRevE.64.056126.

[23]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, 31st edition, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.

[24]

R. J. LeVeque, Numerical Methods for Conservation Laws, 2nd edition, Birkhäuser Verlag, Basel, 1992. doi: 10.1007/978-3-0348-8629-1.

[25]

F. Maghami Asl and A. Ulsoy, Analysis of a system of linear delay differential equations, Journal of Dynamic Systems, Measurement, and Control, 125 (2003), 215-223. doi: 10.1115/1.1568121.

[26]

D. Ngoduy, Generalized macroscopic traffic model with time delay, Nonlinear Dynam., 77 (2014), 289-296. doi: 10.1007/s11071-014-1293-5.

[27]

H. J. Payne, Models of freeway traffic and control, Mathematical Models of Public Systems, Simulation Council Proceedings, (1971), 51–61.

[28]

RTMC Data Set, Available from: http://data.dot.state.mn.us/datatools/.

[29]

C.-W. Shu, TVB uniformly high-order schemes for conservation laws, Math. Comp., 49 (1987), 105-121. doi: 10.1090/S0025-5718-1987-0890256-5.

[30]

A. TordeuxG. CostesequeM. Herty and A. Seyfried, From traffic and pedestrian follow-the-leader models with reaction time to first order convection-diffusion flow models, SIAM J. Appl. Math., 78 (2018), 63-79. doi: 10.1137/16M110695X.

[31]

C. Travis and G. Webb, Existence and Stability for Partial Functional Differential Equations, Transactions of the American Mathematical Society, 200 (1974), 395-418. doi: 10.1090/S0002-9947-1974-0382808-3.

[32]

M. Treiber and A. Kesting, Traffic Flow Dynamics, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-32460-4.

[33]

G. B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9781118032954.

show all references

References:
[1]

A. Ashyralyev and D. Agirseven, Well-posedness of delay parabolic difference equations, Adv. Difference Equ., 2014 (2014), 20pp. doi: 10.1186/1687-1847-2014-18.

[2]

A. Ashyralyev and D. Agirseven, Bounded Solutions of nonlinear hyperbolic equations with time delay, Electron. J. Differential Equations, 2018 (2018), Paper No. 21, 15 pp.

[3]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938. doi: 10.1137/S0036139997332099.

[4]

A. AwA. KlarT. 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.

[5] A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, 2013.
[6]

M. Brackstone and M. Mcdonald, Car-following: A historical review, Transportation Research Part F: Traffic Psychology and Behaviour, 2 (1999), 181-196. doi: 10.1016/S1369-8478(00)00005-X.

[7]

R. E. ChandlerR. Herman and E. W. Montroll, Traffic dynamics: Studies in car following, Operations Res., 6 (1958), 165-184. doi: 10.1287/opre.6.2.165.

[8]

R. M. CorlessG. H. GonnetD. E. G. HareD. J. Jeffrey and D. E. Knuth, On the Lambert W Function, Adv. Comput. Math, 5 (1996), 329-356. doi: 10.1007/BF02124750.

[9]

C. D'Apice and B. Piccoli, Vertex flow models for vehicular traffic on networks, Math. Models Methods Appl. Sci., 18 (2008), 1299-1315. doi: 10.1142/S0218202508003042.

[10]

C.F. Daganzo, Requiem for second-order fluid approximations of traffic flow, Transportation Research Part B: Methodological, 29 (1995), 277-286. doi: 10.1016/0191-2615(95)00007-Z.

[11]

N. DavoodiA. R. Soheili and S. M. Hashemi, A macro-model for traffic flow with consideration of driver's reaction time and distance, Nonlinear Dynam., 83 (2016), 1621-1628. doi: 10.1007/s11071-015-2435-0.

[12]

G. Emch, Coarse-graining in Liouville space and master equation, Helv. Phys. Acta, 37 (1964), 532-544.

[13]

S. Fan and B. Seibold, Effect of the choice of stagnation density in data-fitted first- and second-order traffic models, preprint, arXiv: 1308.0393.

[14]

S. FanM. Herty and B. Seibold, Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model, Netw. Heterog. Media, 9 (2014), 239-268. doi: 10.3934/nhm.2014.9.239.

[15]

M. Garavello and B. Piccoli, Traffic Flow on Networks, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.

[16]

S. Gottlieb and C.-W. Shu, Total variation diminishing Runge-Kutta schemes, Math. Comp., 67 (1998), 73-85. doi: 10.1090/S0025-5718-98-00913-2.

[17]

A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 135 (1997), 259-278. doi: 10.1006/jcph.1997.5725.

[18]

D. Helbing, Verkehrsdynamik: Neue Physikalische Modellierungskonzepte, Springer Berlin Heidelberg, 1997.

[19]

S. P. Hoogendoorn and P. H. Bovy, State-of-the-art of vehicular traffic flow modelling, Proceedings of the Institution of Mechanical Engineers, Part Ⅰ: Journal of Systems and Control Engineering, 215 (2001), 283–303. doi: 10.1177/095965180121500402.

[20]

W.-F. Jiang and Z. Wang, Developing an Aw-Rascle model of traffic flow, J. Engrg. Math., 97 (2016), 135-146. doi: 10.1007/s10665-015-9801-2.

[21]

E. Kometani and T. Sasaki, On the stability of traffic flow (Report-I), J. Op. Res. Japan, 2 (1958), 11-26.

[22]

H. K. Lee, H.-W. Lee and D. Kim, Macroscopic traffic models from microscopic car-following models, Physical Review E, 64 (2001), 056126. doi: 10.1103/PhysRevE.64.056126.

[23]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, 31st edition, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.

[24]

R. J. LeVeque, Numerical Methods for Conservation Laws, 2nd edition, Birkhäuser Verlag, Basel, 1992. doi: 10.1007/978-3-0348-8629-1.

[25]

F. Maghami Asl and A. Ulsoy, Analysis of a system of linear delay differential equations, Journal of Dynamic Systems, Measurement, and Control, 125 (2003), 215-223. doi: 10.1115/1.1568121.

[26]

D. Ngoduy, Generalized macroscopic traffic model with time delay, Nonlinear Dynam., 77 (2014), 289-296. doi: 10.1007/s11071-014-1293-5.

[27]

H. J. Payne, Models of freeway traffic and control, Mathematical Models of Public Systems, Simulation Council Proceedings, (1971), 51–61.

[28]

RTMC Data Set, Available from: http://data.dot.state.mn.us/datatools/.

[29]

C.-W. Shu, TVB uniformly high-order schemes for conservation laws, Math. Comp., 49 (1987), 105-121. doi: 10.1090/S0025-5718-1987-0890256-5.

[30]

A. TordeuxG. CostesequeM. Herty and A. Seyfried, From traffic and pedestrian follow-the-leader models with reaction time to first order convection-diffusion flow models, SIAM J. Appl. Math., 78 (2018), 63-79. doi: 10.1137/16M110695X.

[31]

C. Travis and G. Webb, Existence and Stability for Partial Functional Differential Equations, Transactions of the American Mathematical Society, 200 (1974), 395-418. doi: 10.1090/S0002-9947-1974-0382808-3.

[32]

M. Treiber and A. Kesting, Traffic Flow Dynamics, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-32460-4.

[33]

G. B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9781118032954.

Figure 5.  Comparison for contact discontinuity: numerical RSD and ARZ solutions vs. analytical ARZ
Figure 6.  Comparison for shock solution: numerical RSD and ARZ solutions vs. analytical ARZ
Figure 7.  Comparison for rarefaction wave: numerical RSD and ARZ solutions vs. analytical ARZ
Figure 1.  Real parts $ \lambda_2 $ for $ \gamma = 0. $
Figure 2.  Time evolution of density and flux of the delayed microscopic and macroscopic models for $ T = 0.5 $
Figure 3.  Zoom: Time evolution of density and flux of the delayed microscopic and macroscopic models for $ T = 0.5 $
Figure 4.  Convergence of the microscopic to the macroscopic models at time $ t = 10 $ and fixed delay $ T = 0.5 $
Figure 8.  Comparison of macroscopic models for $ T = 0.5 $ with Zoom
Figure 9.  Comparison of macroscopic models for $ T = 5 $ with Zoom
Figure 10.  Comparison TE model and convection-diffusion flow model for $ T = 0.5 $
Figure 11.  Traffic light scenario
Figure 12.  Density for the delayed model
Figure 13.  Density for the undelayed model
Figure 14.  Speed over time at the end of the road in the Traffic Light Situation
Figure 15.  Fundamental relation between density and flux and fitted function
Figure 16.  Comparison of the data-fitted delayed and undelayed ARZ model to real data
Table 1.  Comparison of errors for different $ \Delta x $
$ \Delta x $ 5 1 0.5 0.1
RSD $ ||\cdot||_{2} $-Error 0.074 0.0759 0.0706 0.0335
RSD $ ||\cdot||_{\infty} $-Error 0.038 0.0337 0.027 0.0288
CG $ ||\cdot||_{2} $-Error 0.0727 0.0746 0.0692 0.0308
CG $ ||\cdot||_{\infty} $-Error 0.036 0.0305 0.0236 0.0288
TE $ ||\cdot||_{2} $-Error 0.0651 0.0683 0.0657 0.0486
TE $ ||\cdot||_{\infty} $-Error 0.0223 0.0181 0.0236 0.0288
$ \Delta x $ 5 1 0.5 0.1
RSD $ ||\cdot||_{2} $-Error 0.074 0.0759 0.0706 0.0335
RSD $ ||\cdot||_{\infty} $-Error 0.038 0.0337 0.027 0.0288
CG $ ||\cdot||_{2} $-Error 0.0727 0.0746 0.0692 0.0308
CG $ ||\cdot||_{\infty} $-Error 0.036 0.0305 0.0236 0.0288
TE $ ||\cdot||_{2} $-Error 0.0651 0.0683 0.0657 0.0486
TE $ ||\cdot||_{\infty} $-Error 0.0223 0.0181 0.0236 0.0288
[1]

Michael Herty, Reinhard Illner. Analytical and numerical investigations of refined macroscopic traffic flow models. Kinetic & Related Models, 2010, 3 (2) : 311-333. doi: 10.3934/krm.2010.3.311

[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]

Simone Göttlich, Stephan Knapp, Peter Schillen. A pedestrian flow model with stochastic velocities: Microscopic and macroscopic approaches. Kinetic & Related Models, 2018, 11 (6) : 1333-1358. doi: 10.3934/krm.2018052

[4]

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

[5]

Heiko Enderling, Alexander R.A. Anderson, Mark A.J. Chaplain, Glenn W.A. Rowe. Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology. Mathematical Biosciences & Engineering, 2006, 3 (4) : 571-582. doi: 10.3934/mbe.2006.3.571

[6]

Tadahisa Funaki, Hirofumi Izuhara, Masayasu Mimura, Chiyori Urabe. A link between microscopic and macroscopic models of self-organized aggregation. Networks & Heterogeneous Media, 2012, 7 (4) : 705-740. doi: 10.3934/nhm.2012.7.705

[7]

Seyedeh Marzieh Ghavidel, Wolfgang M. Ruess. Flow invariance for nonautonomous nonlinear partial differential delay equations. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2351-2369. doi: 10.3934/cpaa.2012.11.2351

[8]

Marte Godvik, Harald Hanche-Olsen. Car-following and the macroscopic Aw-Rascle traffic flow model. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 279-303. doi: 10.3934/dcdsb.2010.13.279

[9]

Mohamed Benyahia, Massimiliano D. Rosini. A macroscopic traffic model with phase transitions and local point constraints on the flow. Networks & Heterogeneous Media, 2017, 12 (2) : 297-317. doi: 10.3934/nhm.2017013

[10]

Roberto Camassa, Pao-Hsiung Chiu, Long Lee, W.-H. Sheu. A particle method and numerical study of a quasilinear partial differential equation. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1503-1515. doi: 10.3934/cpaa.2011.10.1503

[11]

Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9

[12]

Helge Holden, Nils Henrik Risebro. Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow. Networks & Heterogeneous Media, 2018, 13 (3) : 409-421. doi: 10.3934/nhm.2018018

[13]

Gabriella Bretti, Roberto Natalini, Benedetto Piccoli. Numerical approximations of a traffic flow model on networks. Networks & Heterogeneous Media, 2006, 1 (1) : 57-84. doi: 10.3934/nhm.2006.1.57

[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]

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

[19]

Gabriella Bretti, Maya Briani, Emiliano Cristiani. An easy-to-use algorithm for simulating traffic flow on networks: Numerical experiments. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 379-394. doi: 10.3934/dcdss.2014.7.379

[20]

Michael Herty, Adrian Fazekas, Giuseppe Visconti. A two-dimensional data-driven model for traffic flow on highways. Networks & Heterogeneous Media, 2018, 13 (2) : 217-240. doi: 10.3934/nhm.2018010

2017 Impact Factor: 1.187

Metrics

  • PDF downloads (60)
  • HTML views (149)
  • Cited by (0)

Other articles
by authors

[Back to Top]