November  2019, 24(11): 6189-6207. doi: 10.3934/dcdsb.2019135

An interface-free multi-scale multi-order model for traffic flow

1. 

Istituto per le Applicazioni del Calcolo "M. Picone", Consiglio Nazionale delle Ricerche, Via dei Taurini 19, 00185 Rome, Italy

2. 

Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza, Università di Roma, Via Scarpa 16, 00161 Rome, Italy

*Corresponding author

Both authors are members of the INdAM Research group GNCS

Received  May 2018 Revised  January 2019 Published  July 2019

In this paper we present a new multi-scale method for reproducing traffic flow which couples a first-order macroscopic model with a second-order microscopic model, avoiding any interface or boundary conditions between them. The multi-scale model is characterized by the fact that microscopic and macroscopic descriptions are not spatially separated. On the contrary, the macro-scale is always active while the micro-scale is activated only if needed by the traffic conditions. The Euler-Godunov scheme associated to the model is conservative and it is able to reproduce typical traffic phenomena like stop & go waves.

Citation: Emiliano Cristiani, Elisa Iacomini. An interface-free multi-scale multi-order model for traffic flow. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6189-6207. doi: 10.3934/dcdsb.2019135
References:
[1]

A. AwA. KlarM. Rascle and T. Materne, Derivation of continuum flow traffic models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278. doi: 10.1137/S0036139900380955. Google Scholar

[2]

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. Google Scholar

[3]

E. Bourrel and J.-B. Lesort, Mixing microscopic and macroscopic representations of traffic flow: Hybrid model based on Lighthill-Whitham-Richards theory, Transportation Research Record, 1852 (2003), 193-200. doi: 10.3141/1852-24. Google Scholar

[4]

G. BrettiM. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: Numerical experiments, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 379-394. doi: 10.3934/dcdss.2014.7.379. Google Scholar

[5]

G. BrettiR. Natalini and B. Piccoli, A fluid-dynamic traffic model on road networks, Arch. Comput. Methods Eng., 14 (2007), 139-172. doi: 10.1007/s11831-007-9004-8. Google Scholar

[6]

M. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: Theoretical study, Netw. Heterog. Media, 9 (2014), 519-552. doi: 10.3934/nhm.2014.9.519. Google Scholar

[7]

R. M. Colombo and A. Groli, Minimising stop and go waves to optimise traffic flow, Appl. Math. Lett., 17 (2004), 697-701. doi: 10.1016/S0893-9659(04)90107-3. Google Scholar

[8]

R. M. Colombo and F. Marcellini, A mixed ODE-PDE model for vehicular traffic, Math. Meth. Appl. Sci., 38 (2015), 1292-1302. doi: 10.1002/mma.3146. Google Scholar

[9]

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

[10]

E. Cristiani, Blending Brownian motion and heat equation, J. Coupled Syst. Multiscale Dyn., 3 (2015), 351-356. doi: 10.1166/jcsmd.2015.1089. Google Scholar

[11]

E. CristianiB. Piccoli and A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics, Multiscale Model. Simul., 9 (2011), 155-182. doi: 10.1137/100797515. Google Scholar

[12]

————, How can macroscopic models reveal self-organization in traffic flow?, in 51st IEEE Conference on Decision and Control, 2012. Maui, Hawaii, December 10-13, 2012.Google Scholar

[13]

————, Multiscale Modeling of Pedestrian Dynamics, Modeling, Simulation & Applications, Springer, 2014. doi: 10.1007/978-3-319-06620-2. Google Scholar

[14]

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

[15]

E. Cristiani and A. Tosin, Reducing complexity of multiagent systems with symmetry breaking: an application to opinion dynamics with polls, Multiscale Model. Simul., 16 (2018), 528-549. doi: 10.1137/17M113397X. Google Scholar

[16]

M. Di FrancescoS. Fagioli and M. D. Rosini, Deterministic particle approximation of scalar conservation laws, Boll. Unione Mat. Ital., 10 (2017), 487-501. doi: 10.1007/s40574-017-0132-2. Google Scholar

[17]

————, Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic, Math. Biosci. Eng., 14 (2017), 127-141. doi: 10.3934/mbe.2017009. Google Scholar

[18]

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. Rational Mech. Anal., 217 (2015), 831-871. doi: 10.1007/s00205-015-0843-4. Google Scholar

[19]

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. Google Scholar

[20]

S. Fan and B. Seibold, Data-fitted first-order traffic models and their second-order generalizations. Comparison by trajectory and sensor data, Transportation Research Record, 2391 (2013), 32-43. Google Scholar

[21]

S. Fan, Y. Sun, B. Piccoli, B. Seibold and D. B. Work, A collapsed generalized Aw-Rascle-Zhang model and its model accuracy., arXiv: 1702.03624.Google Scholar

[22]

M. R. Flynn, A. R. Kasimov, J.-C. Nave, R. R. Rosales and B. Seibold, Self-sustained nonlinear waves in traffic flow, Phys. Rev. E, 79 (2009), 056113, 13pp. doi: 10.1103/PhysRevE.79.056113. Google Scholar

[23]

N. Forcadel and W. Salazar, Homogenization of second order discrete model and application to traffic flow, Differential Integral Equations, 28 (2015), 1039-1068. Google Scholar

[24]

N. ForcadelW. Salazar and M. Zaydan, Homogenization of second order discrete model with local perturbation and application to traffic, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 1437-1487. doi: 10.3934/dcds.2017060. Google Scholar

[25]

M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 2006. Google Scholar

[26]

————, Coupling of microscopic and phase transition models at boundary, Netw. Heterog. Media, 8 (2013), 649-661. doi: 10.3934/nhm.2013.8.649. Google Scholar

[27]

————, Boundary coupling of microscopic and first order macroscopic traffic model, Nonlinear Differ. Equ. Appl., 24 (2017), p43.Google Scholar

[28]

J. M. Greenberg, Extensions and amplifications of a traffic model of Aw and Rascle, SIAM J. Appl. Math., 62 (2001), 729-745. doi: 10.1137/S0036139900378657. Google Scholar

[29]

R. Haberman, Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow, Classics in Applied Mathematics, 21, SIAM, Philadelphia, 1998. Google Scholar

[30]

D. Helbing, Traffic and related self-driven many-particle systems, Rev. Mod. Phys., 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067. Google Scholar

[31]

M. Herty and R. Illner, Coupling of non-local driving behaviour with fundamental diagrams, Kinetic Rel. Mod., 5 (2012), 843-855. doi: 10.3934/krm.2012.5.843. Google Scholar

[32]

M. Hilliges and W. Weidlich, A phenomenological model for dynamic traffic flow in networks, Transportation Res. Part B, 29 (1995), 407-431. doi: 10.1016/0191-2615(95)00018-9. Google Scholar

[33]

M. JoueiaiL. LeclercqH. van Lint and S. P. Hoogendoorn, Multiscale traffic flow model based on the mesoscopic Lighthill-Whitham and Richards models, Transportation Research Record, 2491 (2015), 98-106. doi: 10.3141/2491-11. Google Scholar

[34]

B. S. Kerner, Synchronized flow as a new traffic phase and related problems for traffic flow modelling, Math. Comput. Modelling, 35 (2002), 481-508. doi: 10.1016/S0895-7177(02)80017-6. Google Scholar

[35]

A. KlarM. GüntherR. Wegener and T. Materne, Multivalued fundamental diagrams and stop and go waves for continuum traffic flow equations, SIAM J. Appl. Math., 64 (2004), 468-483. doi: 10.1137/S0036139902404700. Google Scholar

[36]

C. Lattanzio and B. Piccoli, Coupling of microscopic and macroscopic traffic models at boundaries, Math. Models Methods Appl. Sci., 20 (2010), 2349-2370. doi: 10.1142/S0218202510004945. Google Scholar

[37]

L. Leclercq, Hybrid approaches to the solutions of the "Lighthill-Whitham-Richards" model, Transportation Research Part B, 41 (2007), 701-709. doi: 10.1016/j.trb.2006.11.004. Google Scholar

[38]

R. J. LeVeque, Numerical Methods for Conservation Laws, Springer Basel AG, 1992. doi: 10.1007/978-3-0348-8629-1. Google Scholar

[39]

M. J. Lighthill and G. B. Whitham, On kinematic waves Ⅱ. A theory of traffic flow on long crowded roads, Proc. R. Soc. Lond. Ser. A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. Google Scholar

[40]

D. Ni, Multiscale modeling of traffic flow, Mathematica Aeterna, 1 (2011), 27-54. Google Scholar

[41]

D. NiH. K. Hsieh and T. Jiang, Modeling phase diagrams as stochastic processes with application in vehicular traffic flow, Appl. Math. Model., 53 (2018), 106-117. doi: 10.1016/j.apm.2017.08.029. Google Scholar

[42]

B. Piccoli and A. Tosin, Vehicular traffic: A review of continuum mathematical models, Mathematics of Complexity and Dynamical Systems, Vols. 1C3, 1748-1770, Springer, New York, 2012. doi: 10.1007/978-1-4614-1806-1_112. Google Scholar

[43]

L. A. Pipes, An operational analysis of traffic dynamics, J. Appl. Phys., 24 (1953), 274-281. doi: 10.1063/1.1721265. Google Scholar

[44]

G. PuppoM. SempliceA. Tosin and G. Visconti, Fundamental diagrams in traffic flow: The case of heterogeneous kinetic models, Commun. Math. Sci., 14 (2016), 643-669. doi: 10.4310/CMS.2016.v14.n3.a3. Google Scholar

[45]

P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. Google Scholar

[46]

B. G. RosV. L. KnoopB. van Arem and S. P. Hoogendoorn, Empirical analysis of the causes of stop-and-go waves at sags, IET Intell. Transp. Syst., 8 (2014), 499-506. Google Scholar

[47]

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

[48]

R. E. SternS. CuiM. L. Delle MonacheR. BhadaniM. BuntingM. ChurchillN. HamiltonR. HaulcyH. PohlmannF. WuB. PiccoliB. SeiboldJ. Sprinkle and D. B. Work, Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments, Transportation Res. Part C, 89 (2018), 205-221. doi: 10.1016/j.trc.2018.02.005. Google Scholar

[49]

G. ViscontiM. HertyG. Puppo and A. Tosin, Multivalued fundamental diagrams of traffic flow in the kinetic Fokker-Planck limit, Multiscale Model. Simul., 15 (2017), 1267-1293. doi: 10.1137/16M1087035. Google Scholar

[50]

H. WangD. NiQ.-Y. Chen and J. Li, Stochastic modeling of the equilibrium speed-density relationship, J. Adv. Transp., 47 (2013), 126-150. doi: 10.1002/atr.172. Google Scholar

[51]

H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Res. Part B, 36 (2002), 275-290. doi: 10.1016/S0191-2615(00)00050-3. Google Scholar

[52]

Y. Zhao and H. M. Zhang, A unified follow-the-leader model for vehicle, bicycle and pedestrian traffic, Transportation Res. Part B, 105 (2017), 315-327. doi: 10.1016/j.trb.2017.09.004. Google Scholar

show all references

References:
[1]

A. AwA. KlarM. Rascle and T. Materne, Derivation of continuum flow traffic models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278. doi: 10.1137/S0036139900380955. Google Scholar

[2]

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. Google Scholar

[3]

E. Bourrel and J.-B. Lesort, Mixing microscopic and macroscopic representations of traffic flow: Hybrid model based on Lighthill-Whitham-Richards theory, Transportation Research Record, 1852 (2003), 193-200. doi: 10.3141/1852-24. Google Scholar

[4]

G. BrettiM. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: Numerical experiments, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 379-394. doi: 10.3934/dcdss.2014.7.379. Google Scholar

[5]

G. BrettiR. Natalini and B. Piccoli, A fluid-dynamic traffic model on road networks, Arch. Comput. Methods Eng., 14 (2007), 139-172. doi: 10.1007/s11831-007-9004-8. Google Scholar

[6]

M. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: Theoretical study, Netw. Heterog. Media, 9 (2014), 519-552. doi: 10.3934/nhm.2014.9.519. Google Scholar

[7]

R. M. Colombo and A. Groli, Minimising stop and go waves to optimise traffic flow, Appl. Math. Lett., 17 (2004), 697-701. doi: 10.1016/S0893-9659(04)90107-3. Google Scholar

[8]

R. M. Colombo and F. Marcellini, A mixed ODE-PDE model for vehicular traffic, Math. Meth. Appl. Sci., 38 (2015), 1292-1302. doi: 10.1002/mma.3146. Google Scholar

[9]

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

[10]

E. Cristiani, Blending Brownian motion and heat equation, J. Coupled Syst. Multiscale Dyn., 3 (2015), 351-356. doi: 10.1166/jcsmd.2015.1089. Google Scholar

[11]

E. CristianiB. Piccoli and A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics, Multiscale Model. Simul., 9 (2011), 155-182. doi: 10.1137/100797515. Google Scholar

[12]

————, How can macroscopic models reveal self-organization in traffic flow?, in 51st IEEE Conference on Decision and Control, 2012. Maui, Hawaii, December 10-13, 2012.Google Scholar

[13]

————, Multiscale Modeling of Pedestrian Dynamics, Modeling, Simulation & Applications, Springer, 2014. doi: 10.1007/978-3-319-06620-2. Google Scholar

[14]

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

[15]

E. Cristiani and A. Tosin, Reducing complexity of multiagent systems with symmetry breaking: an application to opinion dynamics with polls, Multiscale Model. Simul., 16 (2018), 528-549. doi: 10.1137/17M113397X. Google Scholar

[16]

M. Di FrancescoS. Fagioli and M. D. Rosini, Deterministic particle approximation of scalar conservation laws, Boll. Unione Mat. Ital., 10 (2017), 487-501. doi: 10.1007/s40574-017-0132-2. Google Scholar

[17]

————, Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic, Math. Biosci. Eng., 14 (2017), 127-141. doi: 10.3934/mbe.2017009. Google Scholar

[18]

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. Rational Mech. Anal., 217 (2015), 831-871. doi: 10.1007/s00205-015-0843-4. Google Scholar

[19]

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. Google Scholar

[20]

S. Fan and B. Seibold, Data-fitted first-order traffic models and their second-order generalizations. Comparison by trajectory and sensor data, Transportation Research Record, 2391 (2013), 32-43. Google Scholar

[21]

S. Fan, Y. Sun, B. Piccoli, B. Seibold and D. B. Work, A collapsed generalized Aw-Rascle-Zhang model and its model accuracy., arXiv: 1702.03624.Google Scholar

[22]

M. R. Flynn, A. R. Kasimov, J.-C. Nave, R. R. Rosales and B. Seibold, Self-sustained nonlinear waves in traffic flow, Phys. Rev. E, 79 (2009), 056113, 13pp. doi: 10.1103/PhysRevE.79.056113. Google Scholar

[23]

N. Forcadel and W. Salazar, Homogenization of second order discrete model and application to traffic flow, Differential Integral Equations, 28 (2015), 1039-1068. Google Scholar

[24]

N. ForcadelW. Salazar and M. Zaydan, Homogenization of second order discrete model with local perturbation and application to traffic, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 1437-1487. doi: 10.3934/dcds.2017060. Google Scholar

[25]

M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 2006. Google Scholar

[26]

————, Coupling of microscopic and phase transition models at boundary, Netw. Heterog. Media, 8 (2013), 649-661. doi: 10.3934/nhm.2013.8.649. Google Scholar

[27]

————, Boundary coupling of microscopic and first order macroscopic traffic model, Nonlinear Differ. Equ. Appl., 24 (2017), p43.Google Scholar

[28]

J. M. Greenberg, Extensions and amplifications of a traffic model of Aw and Rascle, SIAM J. Appl. Math., 62 (2001), 729-745. doi: 10.1137/S0036139900378657. Google Scholar

[29]

R. Haberman, Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow, Classics in Applied Mathematics, 21, SIAM, Philadelphia, 1998. Google Scholar

[30]

D. Helbing, Traffic and related self-driven many-particle systems, Rev. Mod. Phys., 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067. Google Scholar

[31]

M. Herty and R. Illner, Coupling of non-local driving behaviour with fundamental diagrams, Kinetic Rel. Mod., 5 (2012), 843-855. doi: 10.3934/krm.2012.5.843. Google Scholar

[32]

M. Hilliges and W. Weidlich, A phenomenological model for dynamic traffic flow in networks, Transportation Res. Part B, 29 (1995), 407-431. doi: 10.1016/0191-2615(95)00018-9. Google Scholar

[33]

M. JoueiaiL. LeclercqH. van Lint and S. P. Hoogendoorn, Multiscale traffic flow model based on the mesoscopic Lighthill-Whitham and Richards models, Transportation Research Record, 2491 (2015), 98-106. doi: 10.3141/2491-11. Google Scholar

[34]

B. S. Kerner, Synchronized flow as a new traffic phase and related problems for traffic flow modelling, Math. Comput. Modelling, 35 (2002), 481-508. doi: 10.1016/S0895-7177(02)80017-6. Google Scholar

[35]

A. KlarM. GüntherR. Wegener and T. Materne, Multivalued fundamental diagrams and stop and go waves for continuum traffic flow equations, SIAM J. Appl. Math., 64 (2004), 468-483. doi: 10.1137/S0036139902404700. Google Scholar

[36]

C. Lattanzio and B. Piccoli, Coupling of microscopic and macroscopic traffic models at boundaries, Math. Models Methods Appl. Sci., 20 (2010), 2349-2370. doi: 10.1142/S0218202510004945. Google Scholar

[37]

L. Leclercq, Hybrid approaches to the solutions of the "Lighthill-Whitham-Richards" model, Transportation Research Part B, 41 (2007), 701-709. doi: 10.1016/j.trb.2006.11.004. Google Scholar

[38]

R. J. LeVeque, Numerical Methods for Conservation Laws, Springer Basel AG, 1992. doi: 10.1007/978-3-0348-8629-1. Google Scholar

[39]

M. J. Lighthill and G. B. Whitham, On kinematic waves Ⅱ. A theory of traffic flow on long crowded roads, Proc. R. Soc. Lond. Ser. A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. Google Scholar

[40]

D. Ni, Multiscale modeling of traffic flow, Mathematica Aeterna, 1 (2011), 27-54. Google Scholar

[41]

D. NiH. K. Hsieh and T. Jiang, Modeling phase diagrams as stochastic processes with application in vehicular traffic flow, Appl. Math. Model., 53 (2018), 106-117. doi: 10.1016/j.apm.2017.08.029. Google Scholar

[42]

B. Piccoli and A. Tosin, Vehicular traffic: A review of continuum mathematical models, Mathematics of Complexity and Dynamical Systems, Vols. 1C3, 1748-1770, Springer, New York, 2012. doi: 10.1007/978-1-4614-1806-1_112. Google Scholar

[43]

L. A. Pipes, An operational analysis of traffic dynamics, J. Appl. Phys., 24 (1953), 274-281. doi: 10.1063/1.1721265. Google Scholar

[44]

G. PuppoM. SempliceA. Tosin and G. Visconti, Fundamental diagrams in traffic flow: The case of heterogeneous kinetic models, Commun. Math. Sci., 14 (2016), 643-669. doi: 10.4310/CMS.2016.v14.n3.a3. Google Scholar

[45]

P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. Google Scholar

[46]

B. G. RosV. L. KnoopB. van Arem and S. P. Hoogendoorn, Empirical analysis of the causes of stop-and-go waves at sags, IET Intell. Transp. Syst., 8 (2014), 499-506. Google Scholar

[47]

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

[48]

R. E. SternS. CuiM. L. Delle MonacheR. BhadaniM. BuntingM. ChurchillN. HamiltonR. HaulcyH. PohlmannF. WuB. PiccoliB. SeiboldJ. Sprinkle and D. B. Work, Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments, Transportation Res. Part C, 89 (2018), 205-221. doi: 10.1016/j.trc.2018.02.005. Google Scholar

[49]

G. ViscontiM. HertyG. Puppo and A. Tosin, Multivalued fundamental diagrams of traffic flow in the kinetic Fokker-Planck limit, Multiscale Model. Simul., 15 (2017), 1267-1293. doi: 10.1137/16M1087035. Google Scholar

[50]

H. WangD. NiQ.-Y. Chen and J. Li, Stochastic modeling of the equilibrium speed-density relationship, J. Adv. Transp., 47 (2013), 126-150. doi: 10.1002/atr.172. Google Scholar

[51]

H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Res. Part B, 36 (2002), 275-290. doi: 10.1016/S0191-2615(00)00050-3. Google Scholar

[52]

Y. Zhao and H. M. Zhang, A unified follow-the-leader model for vehicle, bicycle and pedestrian traffic, Transportation Res. Part B, 105 (2017), 315-327. doi: 10.1016/j.trb.2017.09.004. Google Scholar

Figure 1.  Space-time trajectories of vehicles obeying to the system (1)-(6)-(7) with $ N = 34 $, $ \alpha = 0.6 $, $ \Delta_{ \rm{min}} = 7.89 $, $ V_{ \rm{max}} = 1 $, $ \tau = 4.86 $, $ L = 314 $
Figure 2.  Zoom of the trajectories shown in Fig. 1 around initial time. It is well visible the emergence of the stop & go wave from the interaction between the first and the last vehicle
Figure 3.  Step 1: Vehicles appear around large jumps of the macroscopic velocity (corresponding to large jumps of the macroscopic density)
Figure 4.  Step 3: Green vehicles are leaders
Figure 5.  Step 4: Red vehicles are going to be deactivated
Figure 6.  Step 9: Update of density $ \rho_j $ using microscopic flux on the left boundary and macroscopic flux on the right boundary of the cell $ j $ (case $ \Gamma_{j-1},\ \Gamma_j>0 $ & $ \Gamma_{j+1} = 0 $, $ \theta = 0 $)
Figure 7.  Test 1: a. $ n = 1 $, b. $ n = 100 $
Figure 8.  Test 1: CPU time for the fully microscopic model and the multi-scale model
Figure 9.  Test 2: a. $ n = 1 $, b. $ n = 100 $, c. $ n = 400 $, $ \tau = 0.01 $, d. $ n = 400 $, $ \tau = 3 $
Figure 10.  Test 2: Fundamental diagram of the multi-scale model compared with that of the LWR model. a. $ \tau = 0.01 $, b. $ \tau = 3 $
Figure 11.  Test 3: a. $ n = 1 $, b. $ n = 22 $, c. $ n = 441 $, d. $ n = 926 $
Figure 12.  Test 4: a. $ n = 1 $, b. $ n = 277 $, c. $ n = 1666 $, d. $ n = 2191 $
Table 1.  Model and algorithm parameters used for the numerical tests
$ T $ $ L $ $ N_x $ $ N_t $ $ \tau $ $ \Gamma_{ \rm{max}} $ $ \delta v $ $ \delta t $ $ \delta V $ $ \alpha $ $ \Delta_{ \rm{min}} $
T1 3 20 100 300 0.01 20 0.08 15$ \Delta t $ 0.3 - -
T2 3 20 100 600 0.01Ƀ3 30 0.1 15$ \Delta t $ 0.5 - -
T3 12 20 100 1200 0.1 30 0.1 30$ \Delta t $ 0.2 - -
T4 500 314 35 4000 4.86 16 0.3 250$ \Delta t $ 0.07 0.47 2.6$ \ell_{N} $
$ T $ $ L $ $ N_x $ $ N_t $ $ \tau $ $ \Gamma_{ \rm{max}} $ $ \delta v $ $ \delta t $ $ \delta V $ $ \alpha $ $ \Delta_{ \rm{min}} $
T1 3 20 100 300 0.01 20 0.08 15$ \Delta t $ 0.3 - -
T2 3 20 100 600 0.01Ƀ3 30 0.1 15$ \Delta t $ 0.5 - -
T3 12 20 100 1200 0.1 30 0.1 30$ \Delta t $ 0.2 - -
T4 500 314 35 4000 4.86 16 0.3 250$ \Delta t $ 0.07 0.47 2.6$ \ell_{N} $
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