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  November 2019 Early access  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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[23]

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

[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.

[25]

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

[26]

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

[27]

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

[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.

[29]

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

[30]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

[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.

[40]

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

[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.

[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.

[43]

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

[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.

[45]

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

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[23]

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

[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.

[25]

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

[26]

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

[27]

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

[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.

[29]

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

[30]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

[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.

[40]

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

[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.

[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.

[43]

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

[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.

[45]

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

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

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} $
[1]

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 and Heterogeneous Media, 2018, 13 (3) : 409-421. doi: 10.3934/nhm.2018018

[2]

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