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December  2012, 5(4): 843-855. doi: 10.3934/krm.2012.5.843

Coupling of non-local driving behaviour with fundamental diagrams

1. 

RWTH Aachen University, Templergraben 55, D-52065 Aachen, Germany

2. 

University of Victoria, Department of Mathematics and Statistics, PO Box 3060 STN CSC, Victoria, B.C., Canada V8W 3R4., Canada

Received  July 2012 Revised  September 2012 Published  November 2012

We present an extended discussion of a macroscopic traffic flow model [18] which includes non-local and relaxation terms for vehicular traffic flow on unidirectional roads. The braking and acceleration forces are based on a behavioural model and on free flow dynamics. The latter are modelled by using different fundamental diagrams. Numerical investigations for different situations illustrate the properties of the mathematical model. In particular, the emergence of stop-and-go waves is observed for suitable parameter ranges.
Citation: Michael Herty, Reinhard Illner. Coupling of non-local driving behaviour with fundamental diagrams. Kinetic & Related Models, 2012, 5 (4) : 843-855. doi: 10.3934/krm.2012.5.843
References:
[1]

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

[2]

A. Aw, A. Klar, Th. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models,, SIAM J. Appl. Math., 63 (2002), 259.  doi: 10.1137/S0036139900380955.  Google Scholar

[3]

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

[4]

A. Chertock, A. Kurganov and Y. Rykov, A new sticky particle method for pressureless gas dynamics,, SIAM J. on Numerical Analysis, 45 (2007), 2408.  doi: 10.1137/050644124.  Google Scholar

[5]

G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network,, SIAM J. Math. Anal., 36 (2005), 1862.  doi: 10.1137/S0036141004402683.  Google Scholar

[6]

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

[7]

I. Gasser, T. Seidel, G. Sirito and B. Werner, Bifurcation Analysis of a Class of Car Following Traffic Models II: Variable Reaction Times and Agressive Drivers,, Bulletin of the Institute of Mathematics, 2 (2007), 587.   Google Scholar

[8]

I. Gasser, G. Sirito and B. Werner, Bifurcation analysis of a class of 'car following' traffic models,, Physica D, 197 (2004), 222.  doi: 10.1016/j.physd.2004.07.008.  Google Scholar

[9]

J. Greenberg, Extensions and amplifications of a traffic model of Aw and Rascle,, SIAM J. Appl. Math., 62 (2001), 729.  doi: 10.1137/S0036139900378657.  Google Scholar

[10]

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

[11]

______, Traffic congestion - an instability in a hyperbolic system,, Bulletin of the Institute of Mathematics, 2 (2007), 123.   Google Scholar

[12]

J. Greenberg, A. Klar and M. Rascle, Congestion on multilane highways,, SIAM J. Appl. Math., 63 (2003), 818.  doi: 10.1137/S0036139901396309.  Google Scholar

[13]

B. D. Greenshields, A study of traffic capacity,, Proc. Highway Res., 14 (1935), 448.   Google Scholar

[14]

R. Herman and I. Prigogine, A two-fluid approach to twon traffic,, Science, 204 (1979), 148.  doi: 10.1126/science.204.4389.148.  Google Scholar

[15]

D. Helbing, Traffic dynamics. New physical concepts of modelling. (Verkehrsdynamik. Neue physikalische Modellierungskonzepte),, Berlin: Springer. xii, (1997).   Google Scholar

[16]

D. Helbing, A. Hennecke, V. Shvetsov and M. Treiber, Micro- and macro-simulation of freeway traffic,, Math. Comput. Modelling, 35 (2002), 517.  doi: 10.1016/S0895-7177(02)80019-X.  Google Scholar

[17]

M. Herty and R. Illner, On stop-and-go waves in dense traffic,, Kinetic and Related Models, 1 (2008), 437.   Google Scholar

[18]

M. Herty and R. Illner, Analytical and Numerical Investigations of Refined Macroscopic Traffic Flow Models,, Kinetic and Related Models, 3 (2010), 311.   Google Scholar

[19]

M. Herty, R. Illner, A. Klar and V. Panferov, Qualitative properties of solutions to systems of Fokker-Planck equations for multilane traffic flow,, Transp. Theory Stat. Phys., 35 (2006), 31.  doi: 10.1080/00411450600878573.  Google Scholar

[20]

M. Herty and A. Klar, Modelling, simulation and optimization of traffic flow networks,, SIAM J. Sci. Comp., 25 (2003), 1066.  doi: 10.1137/S106482750241459X.  Google Scholar

[21]

M. Herty and M. Rascle, Coupling conditions for a class of second-order models for traffic flow,, SIAM J. Math. Anal., 38 (2006), 595.  doi: 10.1137/05062617X.  Google Scholar

[22]

R. Illner and G. McGregor, On a functional-differential equation arising from a traffic flow model,, SIAM J. Appl. Math., 72 (2012), 623.   Google Scholar

[23]

R. Illner, C. Kirchner and R. Pinnau, A derivation of the Aw-Rascle traffic models from Fokker-Planck type kinetic models,, Quarterly Appl. Math., 67 (2009), 39.   Google Scholar

[24]

R. Illner, A. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow,, Commun. Math. Sci., 1 (2003), 1.   Google Scholar

[25]

A. Klar and R. Wegener, A hierarchy of models for multilane vehicular traffic. I. Modeling,, SIAM J. Appl. Math., 3 (1999), 983.   Google Scholar

[26]

B. Kerner, "The Physics of Traffic,", Springer, (2004).   Google Scholar

[27]

M. E. M. Kimathi, "Mathematical Models for 3-Phase Traffic Flow Theory,", Ph. D. Thesis, (2012).   Google Scholar

[28]

J. P. Lebacque and M. Khoshyaran, First-order macroscopic traffic flow models for networks in the context of dynamic assignment,, Transportation Planning and Applied Optimization, 64 (2004), 119.   Google Scholar

[29]

R. LeVeque, The dynamics of pressureless dust clouds and delta waves,, Journal Hyperbolic Differential Equations, 1 (2004), 315.   Google Scholar

[30]

T. Li, Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow,, J. Diff. Eqn., 190 (2003), 131.  doi: 10.1016/S0022-0396(03)00014-7.  Google Scholar

[31]

M. Lighthill and J. Whitham, On kinematic waves,, Proc. Roy. Soc. London Ser. A, 229 (1955), 281.  doi: 10.1098/rspa.1955.0088.  Google Scholar

[32]

E. Ben-Naim, P. L. Krapinsky and S. Redner, Kinetics of clustering in traffic flows,, Physical Rev. E, 50 (1994), 822.   Google Scholar

[33]

P. I. Richards, Shock waves on the highway,, Oper. Res., 4 (1956), 42.   Google Scholar

[34]

L. Santen, A. Schadschneider and M. Schreckenberg, Towards a realistic microscopic description of highway traffic,, J. Phys A, 33 (2000), 477.   Google Scholar

[35]

S. Marinosson, R. Chrobok, A. Pottmeier, J. Wahle and M. Schreckenberg, Simulation framework for the autobahn traffic in North Rhine-Westphalia,, Cellular automata, (2002), 315.   Google Scholar

[36]

T. Alperovich and A. Sopasakis, Stochastic description of traffic flow,, J. Stat. Phys., 133 (2008), 1083.  doi: 10.1007/s10955-008-9652-6.  Google Scholar

[37]

A. Sopasakis and M. A. Katsoulakis, Stochastic modeling and simulation of traffic flow: asymmetric single exclusion process with Arrhenius look-ahead dynamics,, SIAM J. Appl. Math., 66 (2006), 921.  doi: 10.1137/040617790.  Google Scholar

[38]

M. Treiber and D. Helbing, Macroscopic simulation of widely scattered synchronized traffic states,, J. Phys. A, (1999).   Google Scholar

[39]

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

show all references

References:
[1]

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

[2]

A. Aw, A. Klar, Th. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models,, SIAM J. Appl. Math., 63 (2002), 259.  doi: 10.1137/S0036139900380955.  Google Scholar

[3]

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

[4]

A. Chertock, A. Kurganov and Y. Rykov, A new sticky particle method for pressureless gas dynamics,, SIAM J. on Numerical Analysis, 45 (2007), 2408.  doi: 10.1137/050644124.  Google Scholar

[5]

G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network,, SIAM J. Math. Anal., 36 (2005), 1862.  doi: 10.1137/S0036141004402683.  Google Scholar

[6]

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

[7]

I. Gasser, T. Seidel, G. Sirito and B. Werner, Bifurcation Analysis of a Class of Car Following Traffic Models II: Variable Reaction Times and Agressive Drivers,, Bulletin of the Institute of Mathematics, 2 (2007), 587.   Google Scholar

[8]

I. Gasser, G. Sirito and B. Werner, Bifurcation analysis of a class of 'car following' traffic models,, Physica D, 197 (2004), 222.  doi: 10.1016/j.physd.2004.07.008.  Google Scholar

[9]

J. Greenberg, Extensions and amplifications of a traffic model of Aw and Rascle,, SIAM J. Appl. Math., 62 (2001), 729.  doi: 10.1137/S0036139900378657.  Google Scholar

[10]

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

[11]

______, Traffic congestion - an instability in a hyperbolic system,, Bulletin of the Institute of Mathematics, 2 (2007), 123.   Google Scholar

[12]

J. Greenberg, A. Klar and M. Rascle, Congestion on multilane highways,, SIAM J. Appl. Math., 63 (2003), 818.  doi: 10.1137/S0036139901396309.  Google Scholar

[13]

B. D. Greenshields, A study of traffic capacity,, Proc. Highway Res., 14 (1935), 448.   Google Scholar

[14]

R. Herman and I. Prigogine, A two-fluid approach to twon traffic,, Science, 204 (1979), 148.  doi: 10.1126/science.204.4389.148.  Google Scholar

[15]

D. Helbing, Traffic dynamics. New physical concepts of modelling. (Verkehrsdynamik. Neue physikalische Modellierungskonzepte),, Berlin: Springer. xii, (1997).   Google Scholar

[16]

D. Helbing, A. Hennecke, V. Shvetsov and M. Treiber, Micro- and macro-simulation of freeway traffic,, Math. Comput. Modelling, 35 (2002), 517.  doi: 10.1016/S0895-7177(02)80019-X.  Google Scholar

[17]

M. Herty and R. Illner, On stop-and-go waves in dense traffic,, Kinetic and Related Models, 1 (2008), 437.   Google Scholar

[18]

M. Herty and R. Illner, Analytical and Numerical Investigations of Refined Macroscopic Traffic Flow Models,, Kinetic and Related Models, 3 (2010), 311.   Google Scholar

[19]

M. Herty, R. Illner, A. Klar and V. Panferov, Qualitative properties of solutions to systems of Fokker-Planck equations for multilane traffic flow,, Transp. Theory Stat. Phys., 35 (2006), 31.  doi: 10.1080/00411450600878573.  Google Scholar

[20]

M. Herty and A. Klar, Modelling, simulation and optimization of traffic flow networks,, SIAM J. Sci. Comp., 25 (2003), 1066.  doi: 10.1137/S106482750241459X.  Google Scholar

[21]

M. Herty and M. Rascle, Coupling conditions for a class of second-order models for traffic flow,, SIAM J. Math. Anal., 38 (2006), 595.  doi: 10.1137/05062617X.  Google Scholar

[22]

R. Illner and G. McGregor, On a functional-differential equation arising from a traffic flow model,, SIAM J. Appl. Math., 72 (2012), 623.   Google Scholar

[23]

R. Illner, C. Kirchner and R. Pinnau, A derivation of the Aw-Rascle traffic models from Fokker-Planck type kinetic models,, Quarterly Appl. Math., 67 (2009), 39.   Google Scholar

[24]

R. Illner, A. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow,, Commun. Math. Sci., 1 (2003), 1.   Google Scholar

[25]

A. Klar and R. Wegener, A hierarchy of models for multilane vehicular traffic. I. Modeling,, SIAM J. Appl. Math., 3 (1999), 983.   Google Scholar

[26]

B. Kerner, "The Physics of Traffic,", Springer, (2004).   Google Scholar

[27]

M. E. M. Kimathi, "Mathematical Models for 3-Phase Traffic Flow Theory,", Ph. D. Thesis, (2012).   Google Scholar

[28]

J. P. Lebacque and M. Khoshyaran, First-order macroscopic traffic flow models for networks in the context of dynamic assignment,, Transportation Planning and Applied Optimization, 64 (2004), 119.   Google Scholar

[29]

R. LeVeque, The dynamics of pressureless dust clouds and delta waves,, Journal Hyperbolic Differential Equations, 1 (2004), 315.   Google Scholar

[30]

T. Li, Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow,, J. Diff. Eqn., 190 (2003), 131.  doi: 10.1016/S0022-0396(03)00014-7.  Google Scholar

[31]

M. Lighthill and J. Whitham, On kinematic waves,, Proc. Roy. Soc. London Ser. A, 229 (1955), 281.  doi: 10.1098/rspa.1955.0088.  Google Scholar

[32]

E. Ben-Naim, P. L. Krapinsky and S. Redner, Kinetics of clustering in traffic flows,, Physical Rev. E, 50 (1994), 822.   Google Scholar

[33]

P. I. Richards, Shock waves on the highway,, Oper. Res., 4 (1956), 42.   Google Scholar

[34]

L. Santen, A. Schadschneider and M. Schreckenberg, Towards a realistic microscopic description of highway traffic,, J. Phys A, 33 (2000), 477.   Google Scholar

[35]

S. Marinosson, R. Chrobok, A. Pottmeier, J. Wahle and M. Schreckenberg, Simulation framework for the autobahn traffic in North Rhine-Westphalia,, Cellular automata, (2002), 315.   Google Scholar

[36]

T. Alperovich and A. Sopasakis, Stochastic description of traffic flow,, J. Stat. Phys., 133 (2008), 1083.  doi: 10.1007/s10955-008-9652-6.  Google Scholar

[37]

A. Sopasakis and M. A. Katsoulakis, Stochastic modeling and simulation of traffic flow: asymmetric single exclusion process with Arrhenius look-ahead dynamics,, SIAM J. Appl. Math., 66 (2006), 921.  doi: 10.1137/040617790.  Google Scholar

[38]

M. Treiber and D. Helbing, Macroscopic simulation of widely scattered synchronized traffic states,, J. Phys. A, (1999).   Google Scholar

[39]

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

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