# American Institute of Mathematical Sciences

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  March 2021 Early access  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 and Heterogeneous Media, 2021, 16 (1) : 69-90. doi: 10.3934/nhm.2020034
##### References:
 [1] G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. 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. [2] A. Aw, A. Klar, T. 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. [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. [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. [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. [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. [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. [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. [9] E. Cristiani, C. 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. [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. [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. [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. [13] S. Fan, M. 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. [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. [15] M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks, American Institute of Mathematical Sciences, 2016. [16] D. C. Gazis, R. 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. [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. [18] S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Matematicheskii Sbornik, 89 (1959), 271-306. [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. [20] M. Herty, C. 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. [21] M. Herty, A. 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. [22] M. Herty, S. 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. [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. [24] R. Illner, A. 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. [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. [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. [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. [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. [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. [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. [31] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511791253. [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. [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. [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. [35] E. Parzen, On estimation of a probability density function and mode, Ann. Math. Statist., 33 (1962), 1065-1076.  doi: 10.1214/aoms/1177704472. [36] W. F. Phillips, A kinetic model for traffic flow with continuum implications, Transport. Plan. Techn., 5 (1979), 131-138.  doi: 10.1080/03081067908717157. [37] B. Piccoli, K. Han, T. L. Friesz, T. 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. [38] L. A. Pipes, An operational analysis of traffic dynamics, J. Appl. Phys., 24 (1953), 274-281.  doi: 10.1063/1.1721265. [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. [40] G. Puppo, M. Semplice, A. 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. [41] P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42. [42] M. Rosenblatt, Remarks on some nonparametric estimates of a density function, Ann. Math. Statist., 27 (1956), 832-837.  doi: 10.1214/aoms/1177728190. [43] M. Schönhof and D. Helbing, Empirical features of congested traffic states and their implications for traffic modeling, Transport. Sci., 41 (2007), 135-166. [44] US Department of Transportation and Federal Highway Administration, Next Generation Simulation (NGSIM), http://ops.fhwa.dot.gov/trafficanalysistools/ngsim.htm. [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. [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. [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.

show all references

##### References:
 [1] G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. 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. [2] A. Aw, A. Klar, T. 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. [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. [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. [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. [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. [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. [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. [9] E. Cristiani, C. 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. [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. [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. [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. [13] S. Fan, M. 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. [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. [15] M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks, American Institute of Mathematical Sciences, 2016. [16] D. C. Gazis, R. 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. [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. [18] S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Matematicheskii Sbornik, 89 (1959), 271-306. [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. [20] M. Herty, C. 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. [21] M. Herty, A. 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. [22] M. Herty, S. 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. [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. [24] R. Illner, A. 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. [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. [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. [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. [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. [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. [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. [31] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511791253. [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. [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. [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. [35] E. Parzen, On estimation of a probability density function and mode, Ann. Math. Statist., 33 (1962), 1065-1076.  doi: 10.1214/aoms/1177704472. [36] W. F. Phillips, A kinetic model for traffic flow with continuum implications, Transport. Plan. Techn., 5 (1979), 131-138.  doi: 10.1080/03081067908717157. [37] B. Piccoli, K. Han, T. L. Friesz, T. 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. [38] L. A. Pipes, An operational analysis of traffic dynamics, J. Appl. Phys., 24 (1953), 274-281.  doi: 10.1063/1.1721265. [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. [40] G. Puppo, M. Semplice, A. 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. [41] P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42. [42] M. Rosenblatt, Remarks on some nonparametric estimates of a density function, Ann. Math. Statist., 27 (1956), 832-837.  doi: 10.1214/aoms/1177728190. [43] M. Schönhof and D. Helbing, Empirical features of congested traffic states and their implications for traffic modeling, Transport. Sci., 41 (2007), 135-166. [44] US Department of Transportation and Federal Highway Administration, Next Generation Simulation (NGSIM), http://ops.fhwa.dot.gov/trafficanalysistools/ngsim.htm. [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. [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. [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.
Representation of no shocks (A) and no rarefaction waves (B)
Representation of exactly one shock, where $\mathit{R}$ denotes rarefaction waves and $\mathit{S}$ shocks
Representation of exactly one rarefaction wave, where $\mathit{R}$ denotes rarefaction waves, $\mathit{S}$ shocks and $\mathit{C.D.}$ contact discontinuities
Representation of two shocks and two rarefaction waves, where $\mathit{R}$ denotes rarefaction waves and $\mathit{S}$ shocks
Numerical solutions of Riemann problems depending on the initial datum
German motorway A3 structure, cf. [25]
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
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
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)
Error between real and numerical density of cars and trucks during 10 seconds of simulation, computed every 0.5 seconds
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)
Error between real density and numerical density of cars and trucks during 10 seconds of simulation, computed every 0.5 seconds
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 $T$
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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