March  2021, 16(1): 69-90. doi: 10.3934/nhm.2020034

A two-dimensional multi-class traffic flow model

1. 

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

2. 

Department of Mathematics, University of Mannheim, B6 28-29, 68159 Mannheim, Germany

* Corresponding author: Caterina Balzotti

Received  June 2020 Revised  October 2020 Published  December 2020

Fund Project: Part of this work was carried out while the first author was visiting the University of Mannheim within the program IPID4all funded by the German Academic Exchange Service (DAAD)

The aim of this work is to introduce a two-dimensional macroscopic traffic model for multiple populations of vehicles. Starting from the paper [21], where a two-dimensional model for a single class of vehicles is proposed, we extend the dynamics to a multi-class model leading to a coupled system of conservation laws in two space dimensions. Besides the study of the Riemann problems we also present a Lax-Friedrichs type discretization scheme recovering the theoretical results by means of numerical tests. We calibrate the multi-class model with real data and compare the fitted model to the real trajectories. Finally, we test the ability of the model to simulate the overtaking of vehicles.

Citation: Caterina Balzotti, Simone Göttlich. A two-dimensional multi-class traffic flow model. Networks & Heterogeneous Media, 2021, 16 (1) : 69-90. doi: 10.3934/nhm.2020034
References:
[1]

G. AlbiN. BellomoL. FermoS.-Y. HaJ. KimL. PareschiD. Poyato and J. Soler, Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374.  Google Scholar

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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.  Google Scholar

[3]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-944.  doi: 10.1137/S0036139997332099.  Google Scholar

[4]

S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, European J. Appl. Math., 14 (2003), 587-612.  doi: 10.1017/S0956792503005266.  Google Scholar

[5]

S. Bertoluzza, S. Falletta, G. Russo and C.-W. Shu, Numerical Solutions of Partial Differential Equations, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2009.  Google Scholar

[6]

B. N. Chetverushkin, N. G. Churbanova, Y. N. Karamzin and M. A. Trapeznikova, A two-dimensional macroscopic model of traffic flows based on KCFD-schemes, in ECCOMAS CFD 2006: Proceedings of the European Conference on Computational Fluid Dynamics, Egmond aan Zee, The Netherlands, September 5-8, 2006, Citeseer, 2006. Google Scholar

[7]

R. M. Colombo, A $2\times 2$ hyperbolic traffic flow model, Math. Comput. Modelling, 35 (2002), 683-688.  doi: 10.1016/S0895-7177(02)80029-2.  Google Scholar

[8]

R. M. Colombo and F. Marcellini, A traffic model aware of real time data, Math. Models Methods Appl. Sci., 26 (2016), 445-467.  doi: 10.1142/S0218202516500081.  Google Scholar

[9]

E. CristianiC. de Fabritiis and B. Piccoli, A fluid dynamic approach for traffic forecast from mobile sensors data, Commun. Appl. Ind. Math., 1 (2010), 54-71.  doi: 10.1685/2010CAIM487.  Google Scholar

[10]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4$^{th}$ edition, Grundlehren der Mathematischen Wissenschaften, 325, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-49451-6.  Google Scholar

[11]

C. F. Daganzo, A continuum theory of traffic dynamics for freeways with special lanes, Transport. Res. B-Meth., 31 (1997), 83-102.  doi: 10.1016/S0191-2615(96)00017-3.  Google Scholar

[12]

C. F. Daganzo, In traffic flow, cellular automata = kinematic waves, Transp. Res. B-Meth., 40 (2006), 396-403.  doi: 10.1016/j.trb.2005.05.004.  Google Scholar

[13]

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

[14]

S. Fan and B. Seibold, Data-fitted first-order traffic models and their second-order generalizations: Comparison by trajectory and sensor data, Transport. Res. Rec., 2391 (2013), 32-43.  doi: 10.3141/2391-04.  Google Scholar

[15]

M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks, American Institute of Mathematical Sciences, 2016.  Google Scholar

[16]

D. C. GazisR. Herman and R. W. Rothery, Nonlinear follow-the-leader models of traffic flow, Oper. Res., 9 (1961), 545-567.  doi: 10.1287/opre.9.4.545.  Google Scholar

[17]

P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Math. Comput. Modelling, 44 (2006), 287-303.  doi: 10.1016/j.mcm.2006.01.016.  Google Scholar

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S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Matematicheskii Sbornik, 89 (1959), 271-306.   Google Scholar

[19]

D. Helbing, From microscopic to macroscopic traffic models, in A Perspective Look at Nonlinear Media, Lecture Notes in Physics, 503, Springer, Berlin, 1998, 122–139. doi: 10.1007/BFb0104959.  Google Scholar

[20]

M. HertyC. Kirchner and S. Moutari, Multi-class traffic models on road networks, Commun. Math. Sci., 4 (2006), 591-608.  doi: 10.4310/CMS.2006.v4.n3.a6.  Google Scholar

[21]

M. HertyA. Fazekas and G. Visconti, A two-dimensional data-driven model for traffic flow on highways, Netw. Heterog. Media, 13 (2018), 217-240.  doi: 10.3934/nhm.2018010.  Google Scholar

[22]

M. HertyS. Moutari and G. Visconti, Macroscopic modeling of multilane motorways using a two-dimensional second-order model of traffic flow, SIAM J. Appl. Math., 78 (2018), 2252-2278.  doi: 10.1137/17M1151821.  Google Scholar

[23]

H. Holden and N. H. Risebro, Models for dense multilane vehicular traffic, SIAM J. Math. Anal., 51 (2019), 3694-3713.  doi: 10.1137/19M124318X.  Google Scholar

[24]

R. IllnerA. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow, Commun. Math. Sci., 1 (2003), 1-12.  doi: 10.4310/CMS.2003.v1.n1.a1.  Google Scholar

[25]

E. Kallo, A. Fazekas, S. Lamberty and M. Oeser, Microscopic traffic data obtained from videos recorded on a German motorway, https://data.mendeley.com/datasets/tzckcsrpn6/1, 2019. Google Scholar

[26]

A. Klar and R. Wegener, A hierarchy of models for multilane vehicular traffic. Ⅰ. Modeling, SIAM J. Appl. Math., 59 (1999), 983-1001.  doi: 10.1137/S0036139997326946.  Google Scholar

[27]

A. Klar and R. Wegener, A hierarchy of models for multilane vehicular traffic. Ⅱ. Numerical investigations, SIAM J. Appl. Math., 59 (1999), 1002-1011.  doi: 10.1137/S0036139997326958.  Google Scholar

[28]

A. Klar and R. Wegener, Kinetic derivation of macroscopic anticipation models for vehicular traffic, SIAM J. Appl. Math., 60 (2000), 1749-1766.  doi: 10.1137/S0036139999356181.  Google Scholar

[29]

J. A. Laval and L. Leclercq, A mechanism to describe the formation and propagation of stop-and-go waves in congested freeway traffic, Philos. T. R. Soc. A, 368 (2010), 4519-4541.  doi: 10.1098/rsta.2010.0138.  Google Scholar

[30]

J.-P. Lebacque, Two-phase bounded-acceleration traffic flow model: Analytical solutions and applications, Transp. Res. Rec., 1852 (2003), 220-230.  doi: 10.3141/1852-27.  Google Scholar

[31] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511791253.  Google Scholar
[32]

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

[33]

P. Nelson, A kinetic model of vehicular traffic and its associated bimodal equilibrium solutions, in Special Issue Devoted to the Proceedings of the 13th International Conference on Transport Theory (Riccione, 1993), 24, 1995,383–409. doi: 10.1080/00411459508205136.  Google Scholar

[34]

G. F. Newell, Nonlinear effects in the dynamics of car following, Oper. Res., 9 (1961), 209-229.  doi: 10.1287/opre.9.2.209.  Google Scholar

[35]

E. Parzen, On estimation of a probability density function and mode, Ann. Math. Statist., 33 (1962), 1065-1076.  doi: 10.1214/aoms/1177704472.  Google Scholar

[36]

W. F. Phillips, A kinetic model for traffic flow with continuum implications, Transport. Plan. Techn., 5 (1979), 131-138.  doi: 10.1080/03081067908717157.  Google Scholar

[37]

B. PiccoliK. HanT. L. FrieszT. Yao and J. Tang, Second-order models and traffic data from mobile sensors, Transp. Res. C-Emer., 52 (2015), 32-56.  doi: 10.1016/j.trc.2014.12.013.  Google Scholar

[38]

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

[39]

I. Prigogine and F. C. Andrews, A Boltzmann-like approach for traffic flow, Oper. Res., 8 (1960), 789-797.  doi: 10.1287/opre.8.6.789.  Google Scholar

[40]

G. PuppoM. SempliceA. Tosin and G. Visconti, Kinetic models for traffic flow resulting in a reduced space of microscopic velocities, Kinet. Relat. Models, 10 (2017), 823-854.  doi: 10.3934/krm.2017033.  Google Scholar

[41]

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

[42]

M. Rosenblatt, Remarks on some nonparametric estimates of a density function, Ann. Math. Statist., 27 (1956), 832-837.  doi: 10.1214/aoms/1177728190.  Google Scholar

[43]

M. Schönhof and D. Helbing, Empirical features of congested traffic states and their implications for traffic modeling, Transport. Sci., 41 (2007), 135-166.   Google Scholar

[44]

US Department of Transportation and Federal Highway Administration, Next Generation Simulation (NGSIM), http://ops.fhwa.dot.gov/trafficanalysistools/ngsim.htm. Google Scholar

[45]

G. Wong and S. Wong, A multi-class traffic flow model–an extension of LWR model with heterogeneous drivers, Transport. Res. A-Pol, 36 (2002), 827-841.  doi: 10.1016/S0965-8564(01)00042-8.  Google Scholar

[46]

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

[47]

T. Zhang and Y. X. Zheng, Two-dimensional Riemann problem for a single conservation law, Trans. Amer. Math. Soc., 312 (1989), 589-619.  doi: 10.1090/S0002-9947-1989-0930070-3.  Google Scholar

show all references

References:
[1]

G. AlbiN. BellomoL. FermoS.-Y. HaJ. KimL. PareschiD. Poyato and J. Soler, Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374.  Google Scholar

[2]

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.  Google Scholar

[3]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-944.  doi: 10.1137/S0036139997332099.  Google Scholar

[4]

S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, European J. Appl. Math., 14 (2003), 587-612.  doi: 10.1017/S0956792503005266.  Google Scholar

[5]

S. Bertoluzza, S. Falletta, G. Russo and C.-W. Shu, Numerical Solutions of Partial Differential Equations, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2009.  Google Scholar

[6]

B. N. Chetverushkin, N. G. Churbanova, Y. N. Karamzin and M. A. Trapeznikova, A two-dimensional macroscopic model of traffic flows based on KCFD-schemes, in ECCOMAS CFD 2006: Proceedings of the European Conference on Computational Fluid Dynamics, Egmond aan Zee, The Netherlands, September 5-8, 2006, Citeseer, 2006. Google Scholar

[7]

R. M. Colombo, A $2\times 2$ hyperbolic traffic flow model, Math. Comput. Modelling, 35 (2002), 683-688.  doi: 10.1016/S0895-7177(02)80029-2.  Google Scholar

[8]

R. M. Colombo and F. Marcellini, A traffic model aware of real time data, Math. Models Methods Appl. Sci., 26 (2016), 445-467.  doi: 10.1142/S0218202516500081.  Google Scholar

[9]

E. CristianiC. de Fabritiis and B. Piccoli, A fluid dynamic approach for traffic forecast from mobile sensors data, Commun. Appl. Ind. Math., 1 (2010), 54-71.  doi: 10.1685/2010CAIM487.  Google Scholar

[10]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4$^{th}$ edition, Grundlehren der Mathematischen Wissenschaften, 325, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-49451-6.  Google Scholar

[11]

C. F. Daganzo, A continuum theory of traffic dynamics for freeways with special lanes, Transport. Res. B-Meth., 31 (1997), 83-102.  doi: 10.1016/S0191-2615(96)00017-3.  Google Scholar

[12]

C. F. Daganzo, In traffic flow, cellular automata = kinematic waves, Transp. Res. B-Meth., 40 (2006), 396-403.  doi: 10.1016/j.trb.2005.05.004.  Google Scholar

[13]

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

[14]

S. Fan and B. Seibold, Data-fitted first-order traffic models and their second-order generalizations: Comparison by trajectory and sensor data, Transport. Res. Rec., 2391 (2013), 32-43.  doi: 10.3141/2391-04.  Google Scholar

[15]

M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks, American Institute of Mathematical Sciences, 2016.  Google Scholar

[16]

D. C. GazisR. Herman and R. W. Rothery, Nonlinear follow-the-leader models of traffic flow, Oper. Res., 9 (1961), 545-567.  doi: 10.1287/opre.9.4.545.  Google Scholar

[17]

P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Math. Comput. Modelling, 44 (2006), 287-303.  doi: 10.1016/j.mcm.2006.01.016.  Google Scholar

[18]

S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Matematicheskii Sbornik, 89 (1959), 271-306.   Google Scholar

[19]

D. Helbing, From microscopic to macroscopic traffic models, in A Perspective Look at Nonlinear Media, Lecture Notes in Physics, 503, Springer, Berlin, 1998, 122–139. doi: 10.1007/BFb0104959.  Google Scholar

[20]

M. HertyC. Kirchner and S. Moutari, Multi-class traffic models on road networks, Commun. Math. Sci., 4 (2006), 591-608.  doi: 10.4310/CMS.2006.v4.n3.a6.  Google Scholar

[21]

M. HertyA. Fazekas and G. Visconti, A two-dimensional data-driven model for traffic flow on highways, Netw. Heterog. Media, 13 (2018), 217-240.  doi: 10.3934/nhm.2018010.  Google Scholar

[22]

M. HertyS. Moutari and G. Visconti, Macroscopic modeling of multilane motorways using a two-dimensional second-order model of traffic flow, SIAM J. Appl. Math., 78 (2018), 2252-2278.  doi: 10.1137/17M1151821.  Google Scholar

[23]

H. Holden and N. H. Risebro, Models for dense multilane vehicular traffic, SIAM J. Math. Anal., 51 (2019), 3694-3713.  doi: 10.1137/19M124318X.  Google Scholar

[24]

R. IllnerA. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow, Commun. Math. Sci., 1 (2003), 1-12.  doi: 10.4310/CMS.2003.v1.n1.a1.  Google Scholar

[25]

E. Kallo, A. Fazekas, S. Lamberty and M. Oeser, Microscopic traffic data obtained from videos recorded on a German motorway, https://data.mendeley.com/datasets/tzckcsrpn6/1, 2019. Google Scholar

[26]

A. Klar and R. Wegener, A hierarchy of models for multilane vehicular traffic. Ⅰ. Modeling, SIAM J. Appl. Math., 59 (1999), 983-1001.  doi: 10.1137/S0036139997326946.  Google Scholar

[27]

A. Klar and R. Wegener, A hierarchy of models for multilane vehicular traffic. Ⅱ. Numerical investigations, SIAM J. Appl. Math., 59 (1999), 1002-1011.  doi: 10.1137/S0036139997326958.  Google Scholar

[28]

A. Klar and R. Wegener, Kinetic derivation of macroscopic anticipation models for vehicular traffic, SIAM J. Appl. Math., 60 (2000), 1749-1766.  doi: 10.1137/S0036139999356181.  Google Scholar

[29]

J. A. Laval and L. Leclercq, A mechanism to describe the formation and propagation of stop-and-go waves in congested freeway traffic, Philos. T. R. Soc. A, 368 (2010), 4519-4541.  doi: 10.1098/rsta.2010.0138.  Google Scholar

[30]

J.-P. Lebacque, Two-phase bounded-acceleration traffic flow model: Analytical solutions and applications, Transp. Res. Rec., 1852 (2003), 220-230.  doi: 10.3141/1852-27.  Google Scholar

[31] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511791253.  Google Scholar
[32]

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

[33]

P. Nelson, A kinetic model of vehicular traffic and its associated bimodal equilibrium solutions, in Special Issue Devoted to the Proceedings of the 13th International Conference on Transport Theory (Riccione, 1993), 24, 1995,383–409. doi: 10.1080/00411459508205136.  Google Scholar

[34]

G. F. Newell, Nonlinear effects in the dynamics of car following, Oper. Res., 9 (1961), 209-229.  doi: 10.1287/opre.9.2.209.  Google Scholar

[35]

E. Parzen, On estimation of a probability density function and mode, Ann. Math. Statist., 33 (1962), 1065-1076.  doi: 10.1214/aoms/1177704472.  Google Scholar

[36]

W. F. Phillips, A kinetic model for traffic flow with continuum implications, Transport. Plan. Techn., 5 (1979), 131-138.  doi: 10.1080/03081067908717157.  Google Scholar

[37]

B. PiccoliK. HanT. L. FrieszT. Yao and J. Tang, Second-order models and traffic data from mobile sensors, Transp. Res. C-Emer., 52 (2015), 32-56.  doi: 10.1016/j.trc.2014.12.013.  Google Scholar

[38]

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

[39]

I. Prigogine and F. C. Andrews, A Boltzmann-like approach for traffic flow, Oper. Res., 8 (1960), 789-797.  doi: 10.1287/opre.8.6.789.  Google Scholar

[40]

G. PuppoM. SempliceA. Tosin and G. Visconti, Kinetic models for traffic flow resulting in a reduced space of microscopic velocities, Kinet. Relat. Models, 10 (2017), 823-854.  doi: 10.3934/krm.2017033.  Google Scholar

[41]

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

[42]

M. Rosenblatt, Remarks on some nonparametric estimates of a density function, Ann. Math. Statist., 27 (1956), 832-837.  doi: 10.1214/aoms/1177728190.  Google Scholar

[43]

M. Schönhof and D. Helbing, Empirical features of congested traffic states and their implications for traffic modeling, Transport. Sci., 41 (2007), 135-166.   Google Scholar

[44]

US Department of Transportation and Federal Highway Administration, Next Generation Simulation (NGSIM), http://ops.fhwa.dot.gov/trafficanalysistools/ngsim.htm. Google Scholar

[45]

G. Wong and S. Wong, A multi-class traffic flow model–an extension of LWR model with heterogeneous drivers, Transport. Res. A-Pol, 36 (2002), 827-841.  doi: 10.1016/S0965-8564(01)00042-8.  Google Scholar

[46]

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

[47]

T. Zhang and Y. X. Zheng, Two-dimensional Riemann problem for a single conservation law, Trans. Amer. Math. Soc., 312 (1989), 589-619.  doi: 10.1090/S0002-9947-1989-0930070-3.  Google Scholar

Figure 1.  Representation of no shocks (A) and no rarefaction waves (B)
Figure 2.  Representation of exactly one shock, where $ \mathit{R} $ denotes rarefaction waves and $ \mathit{S} $ shocks
Figure 3.  Representation of exactly one rarefaction wave, where $ \mathit{R} $ denotes rarefaction waves, $ \mathit{S} $ shocks and $ \mathit{C.D.} $ contact discontinuities
Figure 4.  Representation of two shocks and two rarefaction waves, where $ \mathit{R} $ denotes rarefaction waves and $ \mathit{S} $ shocks
Figure 5.  Numerical solutions of Riemann problems depending on the initial datum
25]">Figure 6.  German motorway A3 structure, cf. [25]
Figure 7.  Speed-density and flux-density diagrams for the two classes related to the $ x $-direction in the first row, and to the $ y $-direction on the second row
Figure 8.  Speed-density and flux-density diagrams for the two classes defined from real data (green and blue circles) and family of speed and velocity functions related to the $ x $-direction in the first row, and to the $ y $-direction in the second row
Figure 9.  Contours of the density of cars (top) and trucks (bottom): initial condition at time $ t = 0 $ (left), simulated results at time $ t = 5\,\mathrm{s} $ (middle) and reconstructed real data at time $ t = 5\,\mathrm{s} $ (right)
Figure 10.  Error between real and numerical density of cars and trucks during 10 seconds of simulation, computed every 0.5 seconds
Figure 11.  Speed-density and flow-density diagrams for the two classes defined from real data (green and blue circles) and family of speed and flux functions defined by (19)
Figure 12.  Error between real density and numerical density of cars and trucks during 10 seconds of simulation, computed every 0.5 seconds
Figure 13.  Contours of the density of cars and truck at time $ t = 0 $ (left), $ t = T/4 $ (middle) and $ t = T $ (right). The truck does not move, Car 1 leaves the road during the simulation and Car 2 overtakes the truck and is exiting the road at time $ T $
Figure 14.  Plot of the density of cars and truck on Lane 1 (first row) and Lane 2 (second row) at time $ t = 0 $ (left), $ t = T/4 $ (middle) and $ t = T $ (right)
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