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June  2018, 13(2): 217-240. doi: 10.3934/nhm.2018010

## A two-dimensional data-driven model for traffic flow on highways

 1 Institut für Geometrie und Praktische Mathematik (IGPM), RWTH Aachen University, Templergraben 55, 52062 Aachen, Germany 2 Institut für Straßenwesen (ISAC), RWTH Aachen University, Mies-van-der-Rohe-Str. 1, 52074 Aachen, Germany

* Corresponding author: Giuseppe Visconti

Received  June 2017 Revised  November 2017 Published  May 2018

Based on experimental traffic data obtained from German and US highways, we propose a novel two-dimensional first-order macroscopic traffic flow model. The goal is to reproduce a detailed description of traffic dynamics for the real road geometry. In our approach both the dynamics along the road and across the lanes is continuous. The closure relations, being necessary to complete the hydrodynamics equation, are obtained by regression on fundamental diagram data. Comparison with prediction of one-dimensional models shows the improvement in performance of the novel model.

Citation: 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
##### References:
 [1] Federal Highway Administration US Department of Transportation. Next Generation Simulation (NGSIM), Available from: http://ops.fhwa.dot.gov/trafficanalysistools/ngsim.htm.Google Scholar [2] Minnesota Department of Transportation. Mn/DOT Traffic Data, Available from: http://data.dot.state.mn.us/datatools.Google Scholar [3] Mobile Millennium Project, Available from: http://traffic.berkeley.edu.Google Scholar [4] 75 Years of the Fundamental Diagram for Traffic Flow Theory: Greenshields Symposium, Transportation Research Board, Circular E-C149. 2011.Google Scholar [5] S. Amin et al., Mobile century – Using GPS mobile phones as traffic sensors: A field experiment. 15th World Congress on Intelligent Transportation Systems, New York. Nov. 2008.Google Scholar [6] D. Aregba-Driollet and R. Natalini, Discrete Kinetic Schemes for Multidimensional Conservation Laws, SIAM J. Numer. Anal., 37 (2000), 1973-2004. doi: 10.1137/S0036142998343075. Google Scholar [7] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938 (electronic). doi: 10.1137/S0036139997332099. Google Scholar [8] A. M. Bayen and C. G. Claudel, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part Ⅰ: Theory, IEEE Trans. Automat. Contr., 55 (2010), 1142-1157. doi: 10.1109/TAC.2010.2041976. Google Scholar [9] A. M. Bayen and C. G. Claudel, Convex formulations of data assimilation problems for a class of hamilton-jacobi equations, SIAM J. Control Optim., 49 (2011), 383-402. doi: 10.1137/090778754. Google Scholar [10] N. Bellomo and C. Dogbé, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463. doi: 10.1137/090746677. Google Scholar [11] S. Benzoni-Gavage and R.M. Colombo, An $n$ -populations model for traffic flow, Eur. J. Appl. Math., 14 (2003), 587-612. doi: 10.1017/S0956792503005266. Google Scholar [12] F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, Arch. Ration. Mech. Anal., 187 (2008), 185-220. doi: 10.1007/s00205-007-0061-9. Google Scholar [13] S. Blandin, G. Bretti, A. Cutolo and B. Piccoli, Numerical simulations of traffic data via fluid dynamic approach, Appl. Math. Comput., 210 (2009), 441-454. doi: 10.1016/j.amc.2009.01.057. Google Scholar [14] R. Cao, A. Cuevas and W. G. Manteiga, A comparative study of several smoothing methods in density estimation, Comput. Stat. Data. An., 17 (1994), 153-176. doi: 10.1016/0167-9473(92)00066-Z. Google Scholar [15] B. N. Chetverushkin, N. G. Churbanova, I. R. Furmanov and M. A. Trapeznikova, 2D Micro-and Macroscopic Models for Simulation of Heterogeneous Traffic Flows, Proceedings of the ECCOMAS CFD 2010, V European Conference on Computational Fluid Dynamics, 2010.Google Scholar [16] B. N. Chetverushkin, N. G. Churbanova, A. B. Sukhinova and M. A. Trapeznikova, Congested Traffic Simulation Based on a 2D Hydrodynamical Model, Proceedings of 8th World Congress on Computational Mechanics and 5th European Congress on Computational Methods in Applied Science and Engineering, WCCM8 & ECCOMAS, 2008.Google Scholar [17] R. Courant, K. O. Friedrichs and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik, Mathematische Annalen, 100 (1928), 32-74. doi: 10.1007/BF01448839. Google Scholar [18] I. Cravero, G. Puppo, M. Semplice and G. Visconti, CWENO: Uniformly accurate reconstructions for balance laws, Math. Comp., 87 (2018), 1689-1719. doi: 10.1090/mcom/3273. Google Scholar [19] C. F. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory, Transport. Res. B-Meth., 28 (1994), 269-287. doi: 10.1016/0191-2615(94)90002-7. Google Scholar [20] C. F. Daganzo, Requiem for second order fluid approximations of traffic flow, Transport. Res. B-Meth., 29 (1995), 277-286. doi: 10.1016/0191-2615(95)00007-Z. Google Scholar [21] C. F. Daganzo, In traffic flow, cellular automata = kinematic waves, Transport. Res. B-Meth., 40 (2006), 396-403. doi: 10.1016/j.trb.2005.05.004. Google Scholar [22] 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. Google Scholar [23] S. Fan and B. Seibold, A comparison of data-fitted first order traffic models and their second order generalizations via trajectory and sensor data, arXiv: 1208.0382, (2012).Google Scholar [24] M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. Google Scholar [25] P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Math. Comput. Modeling, 44 (2006), 287-303. doi: 10.1016/j.mcm.2006.01.016. Google Scholar [26] S. K. Godunov, Finite difference methods for numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sbornik, 47 (1959), 271-306. Google Scholar [27] B. D. Greenshields, A study of traffic capacity, Proc. Highway Res., 14 (1935), 448-477. Google Scholar [28] A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comp. Phys., 49 (1983), 357-393. doi: 10.1016/0021-9991(83)90136-5. Google Scholar [29] D. Helbing, Traffic and related self-driven many-particle systems, Reviews of Modern Physics, 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067. Google Scholar [30] M. Herty and R. Illner, Analytical and numerical investigations of refined macroscopic traffic flow models, Kinet. Relat. Models, 3 (2010), 311-333. doi: 10.3934/krm.2010.3.311. Google Scholar [31] M. Herty and L. Pareschi, Fokker-Planck asymptotics for traffic flow models, Kinet. Relat. Models, 3 (2010), 165-179. doi: 10.3934/krm.2010.3.165. Google Scholar [32] M. Herty, S. Moutari and G. Visconti, Macroscopic modeling of multi-lane motorways using a two-dimensional second-order model of traffic flow, Submitted, arXiv: 1710.07209, (2017).Google Scholar [33] M. Herty, A. Tosin, G. Visconti and M. Zanella, Hybrid stochastic kinetic description of two-dimensional traffic dynamics, Submitted, arXiv: 1711.02424, (2017).Google Scholar [34] M. Herty and G. Visconti, Analysis of risk levels for traffic on a multi-lane highway, Submitted, arXiv: 1710. 05752, (2017).Google Scholar [35] K. Heun, Neue Methoden zur approximativen Integration der Differentialgleichungen einer unabhängigen Veränderlichen, Z. Math. Phys, 45 (1900), 23-38. Google Scholar [36] S. P. Hoogendoorn, Traffic Flow Theory and Simulation: CT4821, Delft University of Technology, Faculty of Civil Engineering and Geosciences (TU Delft), 2007.Google Scholar [37] R. Illner, A. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow, Commun. Math. Sci., 1 (2003), 1–12, URL http://projecteuclid.org/euclid.cms/1118150395. doi: 10.4310/CMS.2003.v1.n1.a1. Google Scholar [38] M. C. Jones, J. S. Marron and S. J. Sheather, A brief survey of bandwidth selection for density estimation, J. Am. Statist. Assoc., 91 (1996), 401-407. doi: 10.1080/01621459.1996.10476701. Google Scholar [39] B. S. Kerner, The Physics of Traffic, Understanding Complex Systems. Springer, Berlin, 2004. doi: 10.1007/978-3-540-40986-1. Google Scholar [40] B. S. Kerner and P. Konhäuser, Cluster effect in initially homogeneous traffic flow, Phys. Rev. E, 48 (1993), R2335-R2338. doi: 10.1103/PhysRevE.48.R2335. Google Scholar [41] B. S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow, Phys. Rev. E, 50 (1994), 54-83. doi: 10.1103/PhysRevE.50.54. Google Scholar [42] A. Klar and R. Wegener, A hierarchy of models for multilane vehicular traffic. Ⅰ. Modeling, SIAM J. Appl. Math., 59 (1999), 983-1001 (electronic). doi: 10.1137/S0036139997326946. Google Scholar [43] A. Klar and R. Wegener, A hierarchy of models for multilane vehicular traffic. Ⅱ. Numerical investigations, SIAM J. Appl. Math., 59 (1999), 1002-1011 (electronic). doi: 10.1137/S0036139997326958. Google Scholar [44] A. Klar and R. Wegener, Kinetic derivation of macroscopic anticipation models for vehicular traffic, SIAM J. Appl. Math., 60 (2000), 1749-1766 (electronic). doi: 10.1137/S0036139999356181. Google Scholar [45] J. P. Lebacque, Les modeles macroscopiques du traffic, Annales des Ponts., 67 (1993), 24-45. Google Scholar [46] W. Leutzbach, Introduction to the Theory of Traffic Flow, Springer, New York, 1988. doi: 10.1007/978-3-642-61353-1. Google Scholar [47] R. J. LeVeque, Finite-Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253. Google Scholar [48] M. J. Lighthill and G. B. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. Google Scholar [49] S. Maerivoet and B. De Moor, Traffic Flow Theory, Katholieke Universiteit Leuven, 2005.Google Scholar [50] G. F. Newell, A simplified theory of kinematic waves in highway traffic Ⅱ: Queueing at freeway bottlenecks, Transport. Res. B-Meth., 27 (1993), 289-303. doi: 10.1016/0191-2615(93)90039-D. Google Scholar [51] E. Parzen, On estimation of a probability density function and model, Ann. Math. Statist., 33 (1962), 1065-1076. doi: 10.1214/aoms/1177704472. Google Scholar [52] H. J. Payne, Models of freeway traffic and control, Math. Models Publ. Sys., Simul. Council Proc., 28 (1971), 51-61. Google Scholar [53] H. J. Payne, FREFLO: A macroscopic simulation model for freeway traffic, Transportation Research Record, 722 (1979), 68-77. Google Scholar [54] W. F. Phillips, A kinetic model for traffic flow with continuum implications, Transportation Planning and Technology, 5 (1979), 131-138. doi: 10.1080/03081067908717157. Google Scholar [55] B. Piccoli and A. Tosin, Vehicular traffic: A review of continuum mathematical models, in Mathematics of Complexity and Dynamical Systems. Vols. 1–3, Springer, New York, 2012, 1748–1770. doi: 10.1007/978-1-4614-1806-1_112. Google Scholar [56] I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic, American Elsevier Publishing Co., New York, 1971.Google Scholar [57] G. Puppo, M. Semplice, A. 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 [58] G. Puppo, M. Semplice, A. Tosin and G. Visconti, Analysis of a multi-population kinetic model for traffic flow, Commun. Math. Sci., 15 (2017), 379-412. doi: 10.4310/CMS.2017.v15.n2.a5. Google Scholar [59] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. Google Scholar [60] 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 [61] M. D. Rosini, Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Understanding Complex Systems, Springer, Heidelberg, 2013. doi: 10.1007/978-3-319-00155-5. Google Scholar [62] C. W. Shu, High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Rev., 51 (2009), 82-126. doi: 10.1137/070679065. Google Scholar [63] G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), 506-517. doi: 10.1137/0705041. Google Scholar [64] A.B. Sukhinova, M.A. Trapeznikova, B.N. Chetverushkin and N.G. 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show all references

##### References:
 [1] Federal Highway Administration US Department of Transportation. Next Generation Simulation (NGSIM), Available from: http://ops.fhwa.dot.gov/trafficanalysistools/ngsim.htm.Google Scholar [2] Minnesota Department of Transportation. Mn/DOT Traffic Data, Available from: http://data.dot.state.mn.us/datatools.Google Scholar [3] Mobile Millennium Project, Available from: http://traffic.berkeley.edu.Google Scholar [4] 75 Years of the Fundamental Diagram for Traffic Flow Theory: Greenshields Symposium, Transportation Research Board, Circular E-C149. 2011.Google Scholar [5] S. Amin et al., Mobile century – Using GPS mobile phones as traffic sensors: A field experiment. 15th World Congress on Intelligent Transportation Systems, New York. Nov. 2008.Google Scholar [6] D. Aregba-Driollet and R. Natalini, Discrete Kinetic Schemes for Multidimensional Conservation Laws, SIAM J. Numer. Anal., 37 (2000), 1973-2004. doi: 10.1137/S0036142998343075. Google Scholar [7] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938 (electronic). doi: 10.1137/S0036139997332099. Google Scholar [8] A. M. Bayen and C. G. Claudel, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part Ⅰ: Theory, IEEE Trans. Automat. Contr., 55 (2010), 1142-1157. doi: 10.1109/TAC.2010.2041976. Google Scholar [9] A. M. Bayen and C. G. Claudel, Convex formulations of data assimilation problems for a class of hamilton-jacobi equations, SIAM J. Control Optim., 49 (2011), 383-402. doi: 10.1137/090778754. Google Scholar [10] N. Bellomo and C. Dogbé, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463. doi: 10.1137/090746677. Google Scholar [11] S. Benzoni-Gavage and R.M. Colombo, An $n$ -populations model for traffic flow, Eur. J. Appl. Math., 14 (2003), 587-612. doi: 10.1017/S0956792503005266. Google Scholar [12] F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, Arch. Ration. Mech. Anal., 187 (2008), 185-220. doi: 10.1007/s00205-007-0061-9. Google Scholar [13] S. Blandin, G. Bretti, A. Cutolo and B. Piccoli, Numerical simulations of traffic data via fluid dynamic approach, Appl. Math. Comput., 210 (2009), 441-454. doi: 10.1016/j.amc.2009.01.057. Google Scholar [14] R. Cao, A. Cuevas and W. G. Manteiga, A comparative study of several smoothing methods in density estimation, Comput. Stat. Data. An., 17 (1994), 153-176. doi: 10.1016/0167-9473(92)00066-Z. Google Scholar [15] B. N. Chetverushkin, N. G. Churbanova, I. R. Furmanov and M. A. Trapeznikova, 2D Micro-and Macroscopic Models for Simulation of Heterogeneous Traffic Flows, Proceedings of the ECCOMAS CFD 2010, V European Conference on Computational Fluid Dynamics, 2010.Google Scholar [16] B. N. Chetverushkin, N. G. Churbanova, A. B. Sukhinova and M. A. Trapeznikova, Congested Traffic Simulation Based on a 2D Hydrodynamical Model, Proceedings of 8th World Congress on Computational Mechanics and 5th European Congress on Computational Methods in Applied Science and Engineering, WCCM8 & ECCOMAS, 2008.Google Scholar [17] R. Courant, K. O. Friedrichs and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik, Mathematische Annalen, 100 (1928), 32-74. doi: 10.1007/BF01448839. Google Scholar [18] I. Cravero, G. Puppo, M. Semplice and G. Visconti, CWENO: Uniformly accurate reconstructions for balance laws, Math. Comp., 87 (2018), 1689-1719. doi: 10.1090/mcom/3273. Google Scholar [19] C. F. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory, Transport. Res. B-Meth., 28 (1994), 269-287. doi: 10.1016/0191-2615(94)90002-7. Google Scholar [20] C. F. Daganzo, Requiem for second order fluid approximations of traffic flow, Transport. Res. B-Meth., 29 (1995), 277-286. doi: 10.1016/0191-2615(95)00007-Z. Google Scholar [21] C. F. Daganzo, In traffic flow, cellular automata = kinematic waves, Transport. Res. B-Meth., 40 (2006), 396-403. doi: 10.1016/j.trb.2005.05.004. Google Scholar [22] 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. Google Scholar [23] S. Fan and B. Seibold, A comparison of data-fitted first order traffic models and their second order generalizations via trajectory and sensor data, arXiv: 1208.0382, (2012).Google Scholar [24] M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. Google Scholar [25] P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Math. Comput. Modeling, 44 (2006), 287-303. doi: 10.1016/j.mcm.2006.01.016. Google Scholar [26] S. K. Godunov, Finite difference methods for numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sbornik, 47 (1959), 271-306. Google Scholar [27] B. D. Greenshields, A study of traffic capacity, Proc. Highway Res., 14 (1935), 448-477. Google Scholar [28] A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comp. Phys., 49 (1983), 357-393. doi: 10.1016/0021-9991(83)90136-5. Google Scholar [29] D. Helbing, Traffic and related self-driven many-particle systems, Reviews of Modern Physics, 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067. Google Scholar [30] M. Herty and R. Illner, Analytical and numerical investigations of refined macroscopic traffic flow models, Kinet. Relat. Models, 3 (2010), 311-333. doi: 10.3934/krm.2010.3.311. Google Scholar [31] M. Herty and L. Pareschi, Fokker-Planck asymptotics for traffic flow models, Kinet. Relat. Models, 3 (2010), 165-179. doi: 10.3934/krm.2010.3.165. Google Scholar [32] M. Herty, S. Moutari and G. Visconti, Macroscopic modeling of multi-lane motorways using a two-dimensional second-order model of traffic flow, Submitted, arXiv: 1710.07209, (2017).Google Scholar [33] M. Herty, A. Tosin, G. Visconti and M. Zanella, Hybrid stochastic kinetic description of two-dimensional traffic dynamics, Submitted, arXiv: 1711.02424, (2017).Google Scholar [34] M. Herty and G. Visconti, Analysis of risk levels for traffic on a multi-lane highway, Submitted, arXiv: 1710. 05752, (2017).Google Scholar [35] K. Heun, Neue Methoden zur approximativen Integration der Differentialgleichungen einer unabhängigen Veränderlichen, Z. Math. Phys, 45 (1900), 23-38. Google Scholar [36] S. P. Hoogendoorn, Traffic Flow Theory and Simulation: CT4821, Delft University of Technology, Faculty of Civil Engineering and Geosciences (TU Delft), 2007.Google Scholar [37] R. Illner, A. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow, Commun. Math. Sci., 1 (2003), 1–12, URL http://projecteuclid.org/euclid.cms/1118150395. doi: 10.4310/CMS.2003.v1.n1.a1. Google Scholar [38] M. C. Jones, J. S. Marron and S. J. Sheather, A brief survey of bandwidth selection for density estimation, J. Am. Statist. Assoc., 91 (1996), 401-407. doi: 10.1080/01621459.1996.10476701. Google Scholar [39] B. S. Kerner, The Physics of Traffic, Understanding Complex Systems. Springer, Berlin, 2004. doi: 10.1007/978-3-540-40986-1. Google Scholar [40] B. S. Kerner and P. Konhäuser, Cluster effect in initially homogeneous traffic flow, Phys. Rev. E, 48 (1993), R2335-R2338. doi: 10.1103/PhysRevE.48.R2335. Google Scholar [41] B. S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow, Phys. Rev. E, 50 (1994), 54-83. doi: 10.1103/PhysRevE.50.54. Google Scholar [42] A. Klar and R. Wegener, A hierarchy of models for multilane vehicular traffic. Ⅰ. Modeling, SIAM J. Appl. Math., 59 (1999), 983-1001 (electronic). doi: 10.1137/S0036139997326946. Google Scholar [43] A. Klar and R. Wegener, A hierarchy of models for multilane vehicular traffic. Ⅱ. Numerical investigations, SIAM J. Appl. Math., 59 (1999), 1002-1011 (electronic). doi: 10.1137/S0036139997326958. Google Scholar [44] A. Klar and R. Wegener, Kinetic derivation of macroscopic anticipation models for vehicular traffic, SIAM J. Appl. Math., 60 (2000), 1749-1766 (electronic). doi: 10.1137/S0036139999356181. Google Scholar [45] J. P. Lebacque, Les modeles macroscopiques du traffic, Annales des Ponts., 67 (1993), 24-45. Google Scholar [46] W. Leutzbach, Introduction to the Theory of Traffic Flow, Springer, New York, 1988. doi: 10.1007/978-3-642-61353-1. Google Scholar [47] R. J. LeVeque, Finite-Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253. Google Scholar [48] M. J. Lighthill and G. B. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. Google Scholar [49] S. Maerivoet and B. De Moor, Traffic Flow Theory, Katholieke Universiteit Leuven, 2005.Google Scholar [50] G. F. Newell, A simplified theory of kinematic waves in highway traffic Ⅱ: Queueing at freeway bottlenecks, Transport. Res. B-Meth., 27 (1993), 289-303. doi: 10.1016/0191-2615(93)90039-D. Google Scholar [51] E. Parzen, On estimation of a probability density function and model, Ann. Math. Statist., 33 (1962), 1065-1076. doi: 10.1214/aoms/1177704472. Google Scholar [52] H. J. Payne, Models of freeway traffic and control, Math. Models Publ. Sys., Simul. Council Proc., 28 (1971), 51-61. Google Scholar [53] H. J. Payne, FREFLO: A macroscopic simulation model for freeway traffic, Transportation Research Record, 722 (1979), 68-77. Google Scholar [54] W. F. Phillips, A kinetic model for traffic flow with continuum implications, Transportation Planning and Technology, 5 (1979), 131-138. doi: 10.1080/03081067908717157. Google Scholar [55] B. Piccoli and A. Tosin, Vehicular traffic: A review of continuum mathematical models, in Mathematics of Complexity and Dynamical Systems. Vols. 1–3, Springer, New York, 2012, 1748–1770. doi: 10.1007/978-1-4614-1806-1_112. Google Scholar [56] I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic, American Elsevier Publishing Co., New York, 1971.Google Scholar [57] G. Puppo, M. Semplice, A. 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 [58] G. Puppo, M. Semplice, A. Tosin and G. Visconti, Analysis of a multi-population kinetic model for traffic flow, Commun. Math. Sci., 15 (2017), 379-412. doi: 10.4310/CMS.2017.v15.n2.a5. Google Scholar [59] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. Google Scholar [60] 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 [61] M. D. Rosini, Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Understanding Complex Systems, Springer, Heidelberg, 2013. doi: 10.1007/978-3-319-00155-5. Google Scholar [62] C. W. Shu, High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Rev., 51 (2009), 82-126. doi: 10.1137/070679065. Google Scholar [63] G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), 506-517. doi: 10.1137/0705041. Google Scholar [64] A.B. Sukhinova, M.A. Trapeznikova, B.N. Chetverushkin and N.G. 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Experimental diagrams from the A3 German highway using $20$ minutes of recorded video. Top row: flux-density (left) and speed-density (right) diagrams in $x$-direction. Bottom row: flux-density (left) and speed-density (right) diagrams in $y$-direction
Two-dimensional trajectories extrapolated from the German data-set. In red we show the trajectories of vehicles crossing a lane while traveling
Fundamental diagrams in $x$ direction. Left: macroscopic data computed each $0.2$ seconds and aggregated over $60$ seconds. Right: macroscopic data computed each $1$ second and aggregated over $30$ seconds
Data-fitting of the experimental diagrams. Top row: functions approximating the flux-density (left) and speed-density (right) diagrams in $x$-direction. Bottom row: functions approximating flux-density (left) and speed-density (right) diagrams in $y$-direction
Log-log graph of the convergence test on the Gaussian initial datum (left panel) and on the double sine wave (right panel)
Top-left: initial density profile at time $T_{\text{init}} = 407.4$ seconds. Top-right: simulated density profile after $0.5$ seconds. Bottom-left: real density profile at final time. Bottom-right: difference between the simulated and the real density profiles
Top-left: initial density profile at time $T_{\text{init}} = 870.9$ seconds. Top-right: simulated density profile after $0.5$ seconds. Bottom-left: real density profile at final time. Bottom-right: difference between the simulated and the real density profiles
Left: $15$ seconds of simulation between $400-415$ seconds, showing the $1$-norm error each $0.5$ seconds (the top panel). Right: $15$ seconds of simulation between $863-878$ seconds, showing the $1$-norm error each $0.5$ seconds (the top panel). The mid and the bottom panels show the variation of density in the time intervals and on the whole recorded time period, respectively
Error plots comparing the predictive accuracy of the 1D model (2) (red data) and of the 2D model (3) (blue data). Each panel refers to different initial density profiles computed by using the kernel density estimation approach. On the $x$-axis we show the percentage of the traveling time with respect to the total time to cover the road section at the maximum speed
Error plots comparing the predictive prediction of traveling time of the 1D model (2) (red data) and of the 2D model (3) (blue data). On the $x$-axis we show the percentage of the traveling time with respect to the total time to cover the road section at the maximum speed
Relative error between $\boldsymbol{E}_{\boldsymbol{\alpha}^{\boldsymbol{y}}, \boldsymbol{p}^{\boldsymbol{y}}}$ and $E_{\alpha^y_{\text{opt}}, p^y_{\text{opt}}}(T_{\text{fin}})$ for each pair of possible parameters in the vectors $\boldsymbol{\alpha}^{\boldsymbol{y}}$ and $\boldsymbol{p}^{\boldsymbol{y}}$
Experimental diagrams in $y$-direction resulting from the US101 highway and using $15$ minutes of recorded data (07:50 - 08:05 a.m.). The macroscopic quantities are obtained by aggregating each $100$ meter sections and every $1$ second. Left panel: flux-density diagram. Right: speed-density diagram
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