June  2014, 9(2): 239-268. doi: 10.3934/nhm.2014.9.239

Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model

1. 

Department of Civil and Environmental Engineering, University of Illinois at Urbana Champaign, 205 N. Mathews Ave, Urbana, IL 61801, United States

2. 

Department of Mathematics, RWTH Aachen University, Templergraben 55, D-52056 Aachen, Germany

3. 

Temple University, Department of Mathematics, 1805 North Broad Street Philadelphia, PA 19122

Received  October 2013 Revised  June 2014 Published  July 2014

The Aw-Rascle-Zhang (ARZ) model can be interpreted as a generalization of the Lighthill-Whitham-Richards (LWR) model, possessing a family of fundamental diagram curves, each of which represents a class of drivers with a different empty road velocity. A weakness of this approach is that different drivers possess vastly different densities at which traffic flow stagnates. This drawback can be overcome by modifying the pressure relation in the ARZ model, leading to the generalized Aw-Rascle-Zhang (GARZ) model. We present an approach to determine the parameter functions of the GARZ model from fundamental diagram measurement data. The predictive accuracy of the resulting data-fitted GARZ model is compared to other traffic models by means of a three-detector test setup, employing two types of data: vehicle trajectory data, and sensor data. This work also considers the extension of the ARZ and the GARZ models to models with a relaxation term, and conducts an investigation of the optimal relaxation time.
Citation: Shimao Fan, Michael Herty, Benjamin Seibold. Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model. Networks & Heterogeneous Media, 2014, 9 (2) : 239-268. doi: 10.3934/nhm.2014.9.239
References:
[1]

T. Alperovich and A. Sopasakis, Modeling highway traffic with stochastic dynamics,, J. Stat. Phys, 133 (2008), 1083.  doi: 10.1007/s10955-008-9652-6.  Google Scholar

[2]

S. Amin, et al., Mobile century - Using GPS mobile phones as traffic sensors: A field experiment,, in 15th World Congress on Intelligent Transportation Systems, (2008).   Google Scholar

[3]

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

[4]

M. Bando, Hesebem K., A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation,, Phys. Rev. E, 51 (1995), 1035.   Google Scholar

[5]

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.  doi: 10.1109/TAC.2010.2041976.  Google Scholar

[6]

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.  doi: 10.1137/090778754.  Google Scholar

[7]

N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives,, SIAM Rev., 53 (2011), 409.  doi: 10.1137/090746677.  Google Scholar

[8]

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.  doi: 10.1007/s00205-007-0061-9.  Google Scholar

[9]

S. Blandin, G. Bretti, A. Cutolo and B. Piccoli, Numerical simulations of traffic data via fluid dynamic approach,, Appl. Math. Comput., 210 (2009), 441.  doi: 10.1016/j.amc.2009.01.057.  Google Scholar

[10]

S. Blandin, A. Coque and A. Bayen, On sequential data assimilation for scalar macroscopic traffic flow models,, Physica D, (2012), 1421.   Google Scholar

[11]

S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic,, SIAM J. Appl. Math., 71 (2011), 107.  doi: 10.1137/090754467.  Google Scholar

[12]

R. Borsche, M. Kimathi and A. Klar, A class of multiphase traffic theories for microscopic, kinetic and continuum traffic models,, Comp. Math. Appl., 64 (2012), 2939.  doi: 10.1016/j.camwa.2012.08.013.  Google Scholar

[13]

C. Chalons and P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling,, Commun. Math. Sci., 5 (2007), 533.  doi: 10.4310/CMS.2007.v5.n3.a2.  Google Scholar

[14]

G. Q. Chen, C. D. Levermore and T. P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy,, Comm. Pure Appl. Math., 47 (1994), 787.  doi: 10.1002/cpa.3160470602.  Google Scholar

[15]

R. M. Colombo, Hyperbolic phase transitions in traffic flow,, SIAM J. Appl. Math., 63 (2003), 708.  doi: 10.1137/S0036139901393184.  Google Scholar

[16]

R. M. Colombo and P. Goatin, Traffic flow models with phase transitions,, Flow Turbulence Combust., 76 (2006), 383.   Google Scholar

[17]

R. M. Colombo, F. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound,, SIAM J. Appl. Math., 70 (2010), 2652.  doi: 10.1137/090752468.  Google Scholar

[18]

R. Courant, K. Friedrichs and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik,, Mathematische Annalen, 100 (1928), 32.  doi: 10.1007/BF01448839.  Google Scholar

[19]

C. F. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory,, Transp. Res. B, 28 (1994), 269.  doi: 10.1016/0191-2615(94)90002-7.  Google Scholar

[20]

C. F. Daganzo, Requiem for second-order fluid approximations of traffic flow,, Transp. Res. B, 29 (1995), 277.  doi: 10.1016/0191-2615(95)00007-Z.  Google Scholar

[21]

C. F. Daganzo, Fundamentals of Transportation and Traffic Operations,, Emerald Group Pub Ltd, (1997).   Google Scholar

[22]

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

[23]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).   Google Scholar

[24]

S. Fan, Data-fitted Generic Second Order Macroscopic Traffic Flow Models,, Dissertation, (2013).   Google Scholar

[25]

S. Fan, B. Piccoli and B. Seibold, The Collapsed Generalized Aw-Rascle-Zhang Model of Traffic Flow,, in preparation, (2014).   Google Scholar

[26]

S. Fan and B. Seibold, A comparison of data-fitted first order traffic models and their second order generalizations via trajectory and sensor data,, in 93rd Annual Meeting of Transportation Research Board, (2013), 13.   Google Scholar

[27]

S. Fan and B. Seibold, Effect of the choice of stagnation density in data-fitted first- and second-order traffic models,, , (2013).   Google Scholar

[28]

Federal Highway Administration US Department of Transportation, Interstate 80 freeway dataset,, Website, ().   Google Scholar

[29]

Federal Highway Administration US Department of Transportation, Next Generation Simulation (NGSIM),, Website, ().   Google Scholar

[30]

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).  doi: 10.1103/PhysRevE.79.056113.  Google Scholar

[31]

M. Fukui and Y. Ishibashi, Traffic flow in 1D cellular automaton model including cars moving with high speed,, J. Phys. Soc. Japan, 65 (1996), 1868.  doi: 10.1143/JPSJ.65.1868.  Google Scholar

[32]

M. Garavello and B. Piccoli, Traffic Flow on Networks,, American Institute of Mathematical Sciences, (2006).   Google Scholar

[33]

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

[34]

S. K. Godunov, A difference scheme for the numerical computation of a discontinuous solution of the hydrodynamic equations,, Math. Sbornik, 47 (1959), 271.   Google Scholar

[35]

J. M. Greenberg, Extension and amplification of the Aw-Rascle model,, SIAM J. Appl. Math., 62 (2001), 729.  doi: 10.1137/S0036139900378657.  Google Scholar

[36]

J. M. Greenberg, Congestion redux,, SIAM J. Appl. Math., 64 (2004), 1175.  doi: 10.1137/S0036139903431737.  Google Scholar

[37]

B. D. Greenshields, A study of traffic capacity,, Proceedings of the Highway Research Record, 14 (1935), 448.   Google Scholar

[38]

A. Harten, P. D. Lax and B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws,, SIAM Rev., 25 (1983), 35.  doi: 10.1137/1025002.  Google Scholar

[39]

D. Helbing, Improved fluid-dynamic model for vehicular traffic,, Phys. Rev. E, 51 (1995), 3164.  doi: 10.1103/PhysRevE.51.3164.  Google Scholar

[40]

D. Helbing, Traffic and related self-driven many-particle systems,, Reviews of Modern Physics, 73 (2001), 1067.  doi: 10.1103/RevModPhys.73.1067.  Google Scholar

[41]

R. Herman and I. Prigogine, Kinetic Theory of Vehicular Traffic,, Elsevier, (1971).   Google Scholar

[42]

M. Herty and R. Illner, Analytical and numerical investigations of refined macroscopic traffic flow models,, Kinet. Relat. Models, 3 (2010), 311.  doi: 10.3934/krm.2010.3.311.  Google Scholar

[43]

M. Herty and L. Pareschi, Fokker-Planck asymptotics for traffic flow models,, Kinet. Relat. Models, 3 (2010), 165.  doi: 10.3934/krm.2010.3.165.  Google Scholar

[44]

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

[45]

R. J. Karunamuni and T. Alberts, A generalized reflection method of boundary correction in kernel density estimation,, Canad. J. Statist., 33 (2005), 497.  doi: 10.1002/cjs.5550330403.  Google Scholar

[46]

A. R. Kasimov, R. R. Rosales, B. Seibold and M. R. Flynn, Existence of jamitons in hyperbolic relaxation systems with application to traffic flow,, in preparation, (2014).   Google Scholar

[47]

B. S. Kerner and P. Konhäuser, Cluster effect in initially homogeneous traffic flow,, Phys. Rev. E, 48 (1993).  doi: 10.1103/PhysRevE.48.R2335.  Google Scholar

[48]

B. S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow,, Phys. Rev. E, 50 (1994), 54.  doi: 10.1103/PhysRevE.50.54.  Google Scholar

[49]

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

[50]

J.-P. Lebacque, Les modeles macroscopiques du traffic,, Annales des Ponts., 67 (1993), 24.   Google Scholar

[51]

J.-P. Lebacque, S. Mammar and H. Haj-Salem, Generic second order traffic flow modelling,, in Transportation and Traffic Theory (eds. R. E. Allsop, (2007), 755.   Google Scholar

[52]

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.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[53]

T. P. Liu, Hyperbolic conservation laws with relaxation,, Comm. Math. Phys., 108 (1987), 153.  doi: 10.1007/BF01210707.  Google Scholar

[54]

Minnesota Department of Transportation, Mn/DOT Traffic Data,, Website, ().   Google Scholar

[55]

Mobile Millennium project, Website,, , ().   Google Scholar

[56]

K. Nagel and M. Schreckenberg, A cellular automaton model for freeway traffic,, J. Phys. I France, 2 (1992), 2221.   Google Scholar

[57]

P. Nelson and A. Sopasakis, The Chapman-Enskog expansion: A novel approach to hierarchical extension of Lighthill-Whitham models,, in Proceedings of the 14th International Symposium on Transportation and Trafic Theory (ed. A. Ceder), (1999), 51.   Google Scholar

[58]

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

[59]

G. F. Newell, A simplified theory of kinematic waves in highway traffic II: Queueing at freeway bottlenecks,, Transp. Res. B, 27 (1993), 289.  doi: 10.1016/0191-2615(93)90039-D.  Google Scholar

[60]

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

[61]

H. J. Payne, Models of freeway traffic and control,, Proc. Simulation Council, 1 (1971), 51.   Google Scholar

[62]

H. J. Payne, FREEFLO: A macroscopic simulation model of freeway traffic,, Transp. Res. Rec., 722 (1979), 68.   Google Scholar

[63]

W. F. Phillips, A kinetic model for traffic flow with continuum implications,, Transportation Planning and Technology, 5 (1979), 131.  doi: 10.1080/03081067908717157.  Google Scholar

[64]

L. A. Pipes, An operational analysis of traffic dynamics,, Journal of Applied Physics, 24 (1953), 274.  doi: 10.1063/1.1721265.  Google Scholar

[65]

M. Rascle, An improved macroscopic model of traffic flow: Derivation and links with the Lightill-Whitham model,, Math. Comput. Modelling, 35 (2002), 581.  doi: 10.1016/S0895-7177(02)80022-X.  Google Scholar

[66]

P. I. Richards, Shock waves on the highway,, Operations Research, 4 (1956), 42.  doi: 10.1287/opre.4.1.42.  Google Scholar

[67]

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

[68]

S. Sakai, K. Nishinari and S. IIda, A new stochastic cellular automaton model on traffic flow and its jamming phase transition,, J. Phys. A: Math. Gen., 39 (2006), 15327.  doi: 10.1088/0305-4470/39/50/002.  Google Scholar

[69]

B. Seibold, M. R. Flynn, A. R. Kasimov and R. R. Rosales, Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models,, Netw. Heterog. Media, 8 (2013), 745.  doi: 10.3934/nhm.2013.8.745.  Google Scholar

[70]

F. Siebel and W. Mauser, On the fundamental diagram of traffic flow,, SIAM J. Appl. Math., 66 (2006), 1150.  doi: 10.1137/050627113.  Google Scholar

[71]

B. Temple, Systems of conservation laws with coinciding shock and rarefaction curves,, Contemp. Math., 17 (1983), 143.   Google Scholar

[72]

R. Underwood, Speed, Volume, and Density Relationships: Quality and Theory of Traffic Flow,, Technical Report, (1961).   Google Scholar

[73]

G. B. Whitham, Linear and Nonlinear Waves,, John Wiley and Sons, (1974).   Google Scholar

[74]

D. Work, S. Blandin, O.-P. Tossavainen, B. Piccoli and A. Bayen, A traffic model for velocity data assimilation,, Appl. Math. Res. Express., 1 (2010), 1.   Google Scholar

[75]

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

show all references

References:
[1]

T. Alperovich and A. Sopasakis, Modeling highway traffic with stochastic dynamics,, J. Stat. Phys, 133 (2008), 1083.  doi: 10.1007/s10955-008-9652-6.  Google Scholar

[2]

S. Amin, et al., Mobile century - Using GPS mobile phones as traffic sensors: A field experiment,, in 15th World Congress on Intelligent Transportation Systems, (2008).   Google Scholar

[3]

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

[4]

M. Bando, Hesebem K., A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation,, Phys. Rev. E, 51 (1995), 1035.   Google Scholar

[5]

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.  doi: 10.1109/TAC.2010.2041976.  Google Scholar

[6]

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.  doi: 10.1137/090778754.  Google Scholar

[7]

N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives,, SIAM Rev., 53 (2011), 409.  doi: 10.1137/090746677.  Google Scholar

[8]

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.  doi: 10.1007/s00205-007-0061-9.  Google Scholar

[9]

S. Blandin, G. Bretti, A. Cutolo and B. Piccoli, Numerical simulations of traffic data via fluid dynamic approach,, Appl. Math. Comput., 210 (2009), 441.  doi: 10.1016/j.amc.2009.01.057.  Google Scholar

[10]

S. Blandin, A. Coque and A. Bayen, On sequential data assimilation for scalar macroscopic traffic flow models,, Physica D, (2012), 1421.   Google Scholar

[11]

S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic,, SIAM J. Appl. Math., 71 (2011), 107.  doi: 10.1137/090754467.  Google Scholar

[12]

R. Borsche, M. Kimathi and A. Klar, A class of multiphase traffic theories for microscopic, kinetic and continuum traffic models,, Comp. Math. Appl., 64 (2012), 2939.  doi: 10.1016/j.camwa.2012.08.013.  Google Scholar

[13]

C. Chalons and P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling,, Commun. Math. Sci., 5 (2007), 533.  doi: 10.4310/CMS.2007.v5.n3.a2.  Google Scholar

[14]

G. Q. Chen, C. D. Levermore and T. P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy,, Comm. Pure Appl. Math., 47 (1994), 787.  doi: 10.1002/cpa.3160470602.  Google Scholar

[15]

R. M. Colombo, Hyperbolic phase transitions in traffic flow,, SIAM J. Appl. Math., 63 (2003), 708.  doi: 10.1137/S0036139901393184.  Google Scholar

[16]

R. M. Colombo and P. Goatin, Traffic flow models with phase transitions,, Flow Turbulence Combust., 76 (2006), 383.   Google Scholar

[17]

R. M. Colombo, F. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound,, SIAM J. Appl. Math., 70 (2010), 2652.  doi: 10.1137/090752468.  Google Scholar

[18]

R. Courant, K. Friedrichs and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik,, Mathematische Annalen, 100 (1928), 32.  doi: 10.1007/BF01448839.  Google Scholar

[19]

C. F. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory,, Transp. Res. B, 28 (1994), 269.  doi: 10.1016/0191-2615(94)90002-7.  Google Scholar

[20]

C. F. Daganzo, Requiem for second-order fluid approximations of traffic flow,, Transp. Res. B, 29 (1995), 277.  doi: 10.1016/0191-2615(95)00007-Z.  Google Scholar

[21]

C. F. Daganzo, Fundamentals of Transportation and Traffic Operations,, Emerald Group Pub Ltd, (1997).   Google Scholar

[22]

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

[23]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).   Google Scholar

[24]

S. Fan, Data-fitted Generic Second Order Macroscopic Traffic Flow Models,, Dissertation, (2013).   Google Scholar

[25]

S. Fan, B. Piccoli and B. Seibold, The Collapsed Generalized Aw-Rascle-Zhang Model of Traffic Flow,, in preparation, (2014).   Google Scholar

[26]

S. Fan and B. Seibold, A comparison of data-fitted first order traffic models and their second order generalizations via trajectory and sensor data,, in 93rd Annual Meeting of Transportation Research Board, (2013), 13.   Google Scholar

[27]

S. Fan and B. Seibold, Effect of the choice of stagnation density in data-fitted first- and second-order traffic models,, , (2013).   Google Scholar

[28]

Federal Highway Administration US Department of Transportation, Interstate 80 freeway dataset,, Website, ().   Google Scholar

[29]

Federal Highway Administration US Department of Transportation, Next Generation Simulation (NGSIM),, Website, ().   Google Scholar

[30]

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).  doi: 10.1103/PhysRevE.79.056113.  Google Scholar

[31]

M. Fukui and Y. Ishibashi, Traffic flow in 1D cellular automaton model including cars moving with high speed,, J. Phys. Soc. Japan, 65 (1996), 1868.  doi: 10.1143/JPSJ.65.1868.  Google Scholar

[32]

M. Garavello and B. Piccoli, Traffic Flow on Networks,, American Institute of Mathematical Sciences, (2006).   Google Scholar

[33]

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

[34]

S. K. Godunov, A difference scheme for the numerical computation of a discontinuous solution of the hydrodynamic equations,, Math. Sbornik, 47 (1959), 271.   Google Scholar

[35]

J. M. Greenberg, Extension and amplification of the Aw-Rascle model,, SIAM J. Appl. Math., 62 (2001), 729.  doi: 10.1137/S0036139900378657.  Google Scholar

[36]

J. M. Greenberg, Congestion redux,, SIAM J. Appl. Math., 64 (2004), 1175.  doi: 10.1137/S0036139903431737.  Google Scholar

[37]

B. D. Greenshields, A study of traffic capacity,, Proceedings of the Highway Research Record, 14 (1935), 448.   Google Scholar

[38]

A. Harten, P. D. Lax and B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws,, SIAM Rev., 25 (1983), 35.  doi: 10.1137/1025002.  Google Scholar

[39]

D. Helbing, Improved fluid-dynamic model for vehicular traffic,, Phys. Rev. E, 51 (1995), 3164.  doi: 10.1103/PhysRevE.51.3164.  Google Scholar

[40]

D. Helbing, Traffic and related self-driven many-particle systems,, Reviews of Modern Physics, 73 (2001), 1067.  doi: 10.1103/RevModPhys.73.1067.  Google Scholar

[41]

R. Herman and I. Prigogine, Kinetic Theory of Vehicular Traffic,, Elsevier, (1971).   Google Scholar

[42]

M. Herty and R. Illner, Analytical and numerical investigations of refined macroscopic traffic flow models,, Kinet. Relat. Models, 3 (2010), 311.  doi: 10.3934/krm.2010.3.311.  Google Scholar

[43]

M. Herty and L. Pareschi, Fokker-Planck asymptotics for traffic flow models,, Kinet. Relat. Models, 3 (2010), 165.  doi: 10.3934/krm.2010.3.165.  Google Scholar

[44]

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

[45]

R. J. Karunamuni and T. Alberts, A generalized reflection method of boundary correction in kernel density estimation,, Canad. J. Statist., 33 (2005), 497.  doi: 10.1002/cjs.5550330403.  Google Scholar

[46]

A. R. Kasimov, R. R. Rosales, B. Seibold and M. R. Flynn, Existence of jamitons in hyperbolic relaxation systems with application to traffic flow,, in preparation, (2014).   Google Scholar

[47]

B. S. Kerner and P. Konhäuser, Cluster effect in initially homogeneous traffic flow,, Phys. Rev. E, 48 (1993).  doi: 10.1103/PhysRevE.48.R2335.  Google Scholar

[48]

B. S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow,, Phys. Rev. E, 50 (1994), 54.  doi: 10.1103/PhysRevE.50.54.  Google Scholar

[49]

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

[50]

J.-P. Lebacque, Les modeles macroscopiques du traffic,, Annales des Ponts., 67 (1993), 24.   Google Scholar

[51]

J.-P. Lebacque, S. Mammar and H. Haj-Salem, Generic second order traffic flow modelling,, in Transportation and Traffic Theory (eds. R. E. Allsop, (2007), 755.   Google Scholar

[52]

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.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[53]

T. P. Liu, Hyperbolic conservation laws with relaxation,, Comm. Math. Phys., 108 (1987), 153.  doi: 10.1007/BF01210707.  Google Scholar

[54]

Minnesota Department of Transportation, Mn/DOT Traffic Data,, Website, ().   Google Scholar

[55]

Mobile Millennium project, Website,, , ().   Google Scholar

[56]

K. Nagel and M. Schreckenberg, A cellular automaton model for freeway traffic,, J. Phys. I France, 2 (1992), 2221.   Google Scholar

[57]

P. Nelson and A. Sopasakis, The Chapman-Enskog expansion: A novel approach to hierarchical extension of Lighthill-Whitham models,, in Proceedings of the 14th International Symposium on Transportation and Trafic Theory (ed. A. Ceder), (1999), 51.   Google Scholar

[58]

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

[59]

G. F. Newell, A simplified theory of kinematic waves in highway traffic II: Queueing at freeway bottlenecks,, Transp. Res. B, 27 (1993), 289.  doi: 10.1016/0191-2615(93)90039-D.  Google Scholar

[60]

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

[61]

H. J. Payne, Models of freeway traffic and control,, Proc. Simulation Council, 1 (1971), 51.   Google Scholar

[62]

H. J. Payne, FREEFLO: A macroscopic simulation model of freeway traffic,, Transp. Res. Rec., 722 (1979), 68.   Google Scholar

[63]

W. F. Phillips, A kinetic model for traffic flow with continuum implications,, Transportation Planning and Technology, 5 (1979), 131.  doi: 10.1080/03081067908717157.  Google Scholar

[64]

L. A. Pipes, An operational analysis of traffic dynamics,, Journal of Applied Physics, 24 (1953), 274.  doi: 10.1063/1.1721265.  Google Scholar

[65]

M. Rascle, An improved macroscopic model of traffic flow: Derivation and links with the Lightill-Whitham model,, Math. Comput. Modelling, 35 (2002), 581.  doi: 10.1016/S0895-7177(02)80022-X.  Google Scholar

[66]

P. I. Richards, Shock waves on the highway,, Operations Research, 4 (1956), 42.  doi: 10.1287/opre.4.1.42.  Google Scholar

[67]

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

[68]

S. Sakai, K. Nishinari and S. IIda, A new stochastic cellular automaton model on traffic flow and its jamming phase transition,, J. Phys. A: Math. Gen., 39 (2006), 15327.  doi: 10.1088/0305-4470/39/50/002.  Google Scholar

[69]

B. Seibold, M. R. Flynn, A. R. Kasimov and R. R. Rosales, Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models,, Netw. Heterog. Media, 8 (2013), 745.  doi: 10.3934/nhm.2013.8.745.  Google Scholar

[70]

F. Siebel and W. Mauser, On the fundamental diagram of traffic flow,, SIAM J. Appl. Math., 66 (2006), 1150.  doi: 10.1137/050627113.  Google Scholar

[71]

B. Temple, Systems of conservation laws with coinciding shock and rarefaction curves,, Contemp. Math., 17 (1983), 143.   Google Scholar

[72]

R. Underwood, Speed, Volume, and Density Relationships: Quality and Theory of Traffic Flow,, Technical Report, (1961).   Google Scholar

[73]

G. B. Whitham, Linear and Nonlinear Waves,, John Wiley and Sons, (1974).   Google Scholar

[74]

D. Work, S. Blandin, O.-P. Tossavainen, B. Piccoli and A. Bayen, A traffic model for velocity data assimilation,, Appl. Math. Res. Express., 1 (2010), 1.   Google Scholar

[75]

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

[1]

Marco Di Francesco, Simone Fagioli, Massimiliano D. Rosini. Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic. Mathematical Biosciences & Engineering, 2017, 14 (1) : 127-141. doi: 10.3934/mbe.2017009

[2]

Stefano Villa, Paola Goatin, Christophe Chalons. Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3921-3952. doi: 10.3934/dcdsb.2017202

[3]

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

[4]

Boris P. Andreianov, Carlotta Donadello, Ulrich Razafison, Julien Y. Rolland, Massimiliano D. Rosini. Solutions of the Aw-Rascle-Zhang system with point constraints. Networks & Heterogeneous Media, 2016, 11 (1) : 29-47. doi: 10.3934/nhm.2016.11.29

[5]

Marte Godvik, Harald Hanche-Olsen. Car-following and the macroscopic Aw-Rascle traffic flow model. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 279-303. doi: 10.3934/dcdsb.2010.13.279

[6]

Benjamin Seibold, Morris R. Flynn, Aslan R. Kasimov, Rodolfo R. Rosales. Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models. Networks & Heterogeneous Media, 2013, 8 (3) : 745-772. doi: 10.3934/nhm.2013.8.745

[7]

Oliver Kolb, Simone Göttlich, Paola Goatin. Capacity drop and traffic control for a second order traffic model. Networks & Heterogeneous Media, 2017, 12 (4) : 663-681. doi: 10.3934/nhm.2017027

[8]

Bertrand Haut, Georges Bastin. A second order model of road junctions in fluid models of traffic networks. Networks & Heterogeneous Media, 2007, 2 (2) : 227-253. doi: 10.3934/nhm.2007.2.227

[9]

Nicolas Forcadel, Wilfredo Salazar, Mamdouh Zaydan. Homogenization of second order discrete model with local perturbation and application to traffic flow. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1437-1487. doi: 10.3934/dcds.2017060

[10]

Michael Burger, Simone Göttlich, Thomas Jung. Derivation of second order traffic flow models with time delays. Networks & Heterogeneous Media, 2019, 14 (2) : 265-288. doi: 10.3934/nhm.2019011

[11]

Luisa Fermo, Andrea Tosin. Fundamental diagrams for kinetic equations of traffic flow. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 449-462. doi: 10.3934/dcdss.2014.7.449

[12]

Rodrigo Samprogna, Cláudia B. Gentile Moussa, Tomás Caraballo, Karina Schiabel. Trajectory and global attractors for generalized processes. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3995-4020. doi: 10.3934/dcdsb.2019047

[13]

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

[14]

Genni Fragnelli, Gisèle Ruiz Goldstein, Jerome Goldstein, Rosa Maria Mininni, Silvia Romanelli. Generalized Wentzell boundary conditions for second order operators with interior degeneracy. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 697-715. doi: 10.3934/dcdss.2016023

[15]

W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209

[16]

Z. G. Feng, Kok Lay Teo, N. U. Ahmed, Yulin Zhao, W. Y. Yan. Optimal fusion of sensor data for Kalman filtering. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 483-503. doi: 10.3934/dcds.2006.14.483

[17]

Martin Redmann, Peter Benner. Approximation and model order reduction for second order systems with Levy-noise. Conference Publications, 2015, 2015 (special) : 945-953. doi: 10.3934/proc.2015.0945

[18]

H. Thomas Banks, Shuhua Hu, Zackary R. Kenz, Carola Kruse, Simon Shaw, John Whiteman, Mark P. Brewin, Stephen E. Greenwald, Malcolm J. Birch. Model validation for a noninvasive arterial stenosis detection problem. Mathematical Biosciences & Engineering, 2014, 11 (3) : 427-448. doi: 10.3934/mbe.2014.11.427

[19]

Bachir Bar, Tewfik Sari. The operating diagram for a model of competition in a chemostat with an external lethal inhibitor. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019203

[20]

Yaqing Liu, Liancun Zheng. Second-order slip flow of a generalized Oldroyd-B fluid through porous medium. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2031-2046. doi: 10.3934/dcdss.2016083

2018 Impact Factor: 0.871

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]