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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 |
References:
[1] |
A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938.
doi: 10.1137/S0036139997332099. |
[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), 259-278.
doi: 10.1137/S0036139900380955. |
[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), 185-220.
doi: 10.1007/s00205-007-0061-9. |
[4] |
A. Chertock, A. Kurganov and Y. Rykov, A new sticky particle method for pressureless gas dynamics, SIAM J. on Numerical Analysis, 45 (2007), 2408-2441.
doi: 10.1137/050644124. |
[5] |
G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886.
doi: 10.1137/S0036141004402683. |
[6] |
C. F. Daganzo, The cell transmission model : A dynamic representation of highway traffic consistent with the hydrodynamic theory, Transp. Res. B, 28 (1994), 269-287.
doi: 10.1016/0191-2615(94)90002-7. |
[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, Academica Sinica (New Series), 2 (2007), 587-607. |
[8] |
I. Gasser, G. Sirito and B. Werner, Bifurcation analysis of a class of 'car following' traffic models, Physica D, 197 (2004), 222-241.
doi: 10.1016/j.physd.2004.07.008. |
[9] |
J. Greenberg, Extensions and amplifications of a traffic model of Aw and Rascle, SIAM J. Appl. Math., 62 (2001), 729-745.
doi: 10.1137/S0036139900378657. |
[10] |
______, Congestion redux, SIAM J. Appl. Math., 64 (2004), 1175-1185.
doi: 10.1137/S0036139903431737. |
[11] |
______, Traffic congestion - an instability in a hyperbolic system, Bulletin of the Institute of Mathematics, Academica Sinica (New Series), 2 (2007), 123-138. |
[12] |
J. Greenberg, A. Klar and M. Rascle, Congestion on multilane highways, SIAM J. Appl. Math., 63 (2003), 818-833.
doi: 10.1137/S0036139901396309. |
[13] |
B. D. Greenshields, A study of traffic capacity, Proc. Highway Res., 14 (1935), 448-477. |
[14] |
R. Herman and I. Prigogine, A two-fluid approach to twon traffic, Science, 204 (1979), 148-151.
doi: 10.1126/science.204.4389.148. |
[15] |
D. Helbing, Traffic dynamics. New physical concepts of modelling. (Verkehrsdynamik. Neue physikalische Modellierungskonzepte), Berlin: Springer. xii, 308 p. DM 128.00; öS934.40; sFr 113.00, 1997. |
[16] |
D. Helbing, A. Hennecke, V. Shvetsov and M. Treiber, Micro- and macro-simulation of freeway traffic, Math. Comput. Modelling, 35 (2002), 517-547.
doi: 10.1016/S0895-7177(02)80019-X. |
[17] |
M. Herty and R. Illner, On stop-and-go waves in dense traffic, Kinetic and Related Models, 1 (2008), 437-452. |
[18] |
M. Herty and R. Illner, Analytical and Numerical Investigations of Refined Macroscopic Traffic Flow Models, Kinetic and Related Models, 3 (2010), 311-333. |
[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), 31-54.
doi: 10.1080/00411450600878573. |
[20] |
M. Herty and A. Klar, Modelling, simulation and optimization of traffic flow networks, SIAM J. Sci. Comp., 25 (2003), 1066-1087.
doi: 10.1137/S106482750241459X. |
[21] |
M. Herty and M. Rascle, Coupling conditions for a class of second-order models for traffic flow, SIAM J. Math. Anal., 38 (2006), 595-616.
doi: 10.1137/05062617X. |
[22] |
R. Illner and G. McGregor, On a functional-differential equation arising from a traffic flow model, SIAM J. Appl. Math., 72 (2012), 623-645. |
[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), 39-45. |
[24] |
R. Illner, A. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow, Commun. Math. Sci., 1 (2003), 1-12. |
[25] |
A. Klar and R. Wegener, A hierarchy of models for multilane vehicular traffic. I. Modeling, SIAM J. Appl. Math., 3 (1999), 983-1001. |
[26] |
B. Kerner, "The Physics of Traffic," Springer, Berlin, 2004. |
[27] |
M. E. M. Kimathi, "Mathematical Models for 3-Phase Traffic Flow Theory," Ph. D. Thesis, Kaiserslautern 2012 |
[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), 119-140. |
[29] |
R. LeVeque, The dynamics of pressureless dust clouds and delta waves, Journal Hyperbolic Differential Equations, 1 (2004), 315-327. |
[30] |
T. Li, Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow, J. Diff. Eqn., 190 (2003), 131-149.
doi: 10.1016/S0022-0396(03)00014-7. |
[31] |
M. Lighthill and J. Whitham, On kinematic waves, Proc. Roy. Soc. London Ser. A, 229 (1955), 281-345.
doi: 10.1098/rspa.1955.0088. |
[32] |
E. Ben-Naim, P. L. Krapinsky and S. Redner, Kinetics of clustering in traffic flows, Physical Rev. E, 50 (1994), 822-829. |
[33] |
P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. |
[34] |
L. Santen, A. Schadschneider and M. Schreckenberg, Towards a realistic microscopic description of highway traffic, J. Phys A, 33 (2000), 477-485. |
[35] |
S. Marinosson, R. Chrobok, A. Pottmeier, J. Wahle and M. Schreckenberg, Simulation framework for the autobahn traffic in North Rhine-Westphalia, Cellular automata, 315-324, Lecture Notes in Comput. Sci., 2493, Springer, Berlin (2002). |
[36] |
T. Alperovich and A. Sopasakis, Stochastic description of traffic flow, J. Stat. Phys., 133 (2008), 1083-1105.
doi: 10.1007/s10955-008-9652-6. |
[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), 921-944.
doi: 10.1137/040617790. |
[38] |
M. Treiber and D. Helbing, Macroscopic simulation of widely scattered synchronized traffic states, J. Phys. A, Math. Gen., (1999). |
[39] |
H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Tans. Res. B, 36 (2002), 275-290.
doi: 10.1016/S0191-2615(00)00050-3. |
show all references
References:
[1] |
A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938.
doi: 10.1137/S0036139997332099. |
[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), 259-278.
doi: 10.1137/S0036139900380955. |
[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), 185-220.
doi: 10.1007/s00205-007-0061-9. |
[4] |
A. Chertock, A. Kurganov and Y. Rykov, A new sticky particle method for pressureless gas dynamics, SIAM J. on Numerical Analysis, 45 (2007), 2408-2441.
doi: 10.1137/050644124. |
[5] |
G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886.
doi: 10.1137/S0036141004402683. |
[6] |
C. F. Daganzo, The cell transmission model : A dynamic representation of highway traffic consistent with the hydrodynamic theory, Transp. Res. B, 28 (1994), 269-287.
doi: 10.1016/0191-2615(94)90002-7. |
[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, Academica Sinica (New Series), 2 (2007), 587-607. |
[8] |
I. Gasser, G. Sirito and B. Werner, Bifurcation analysis of a class of 'car following' traffic models, Physica D, 197 (2004), 222-241.
doi: 10.1016/j.physd.2004.07.008. |
[9] |
J. Greenberg, Extensions and amplifications of a traffic model of Aw and Rascle, SIAM J. Appl. Math., 62 (2001), 729-745.
doi: 10.1137/S0036139900378657. |
[10] |
______, Congestion redux, SIAM J. Appl. Math., 64 (2004), 1175-1185.
doi: 10.1137/S0036139903431737. |
[11] |
______, Traffic congestion - an instability in a hyperbolic system, Bulletin of the Institute of Mathematics, Academica Sinica (New Series), 2 (2007), 123-138. |
[12] |
J. Greenberg, A. Klar and M. Rascle, Congestion on multilane highways, SIAM J. Appl. Math., 63 (2003), 818-833.
doi: 10.1137/S0036139901396309. |
[13] |
B. D. Greenshields, A study of traffic capacity, Proc. Highway Res., 14 (1935), 448-477. |
[14] |
R. Herman and I. Prigogine, A two-fluid approach to twon traffic, Science, 204 (1979), 148-151.
doi: 10.1126/science.204.4389.148. |
[15] |
D. Helbing, Traffic dynamics. New physical concepts of modelling. (Verkehrsdynamik. Neue physikalische Modellierungskonzepte), Berlin: Springer. xii, 308 p. DM 128.00; öS934.40; sFr 113.00, 1997. |
[16] |
D. Helbing, A. Hennecke, V. Shvetsov and M. Treiber, Micro- and macro-simulation of freeway traffic, Math. Comput. Modelling, 35 (2002), 517-547.
doi: 10.1016/S0895-7177(02)80019-X. |
[17] |
M. Herty and R. Illner, On stop-and-go waves in dense traffic, Kinetic and Related Models, 1 (2008), 437-452. |
[18] |
M. Herty and R. Illner, Analytical and Numerical Investigations of Refined Macroscopic Traffic Flow Models, Kinetic and Related Models, 3 (2010), 311-333. |
[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), 31-54.
doi: 10.1080/00411450600878573. |
[20] |
M. Herty and A. Klar, Modelling, simulation and optimization of traffic flow networks, SIAM J. Sci. Comp., 25 (2003), 1066-1087.
doi: 10.1137/S106482750241459X. |
[21] |
M. Herty and M. Rascle, Coupling conditions for a class of second-order models for traffic flow, SIAM J. Math. Anal., 38 (2006), 595-616.
doi: 10.1137/05062617X. |
[22] |
R. Illner and G. McGregor, On a functional-differential equation arising from a traffic flow model, SIAM J. Appl. Math., 72 (2012), 623-645. |
[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), 39-45. |
[24] |
R. Illner, A. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow, Commun. Math. Sci., 1 (2003), 1-12. |
[25] |
A. Klar and R. Wegener, A hierarchy of models for multilane vehicular traffic. I. Modeling, SIAM J. Appl. Math., 3 (1999), 983-1001. |
[26] |
B. Kerner, "The Physics of Traffic," Springer, Berlin, 2004. |
[27] |
M. E. M. Kimathi, "Mathematical Models for 3-Phase Traffic Flow Theory," Ph. D. Thesis, Kaiserslautern 2012 |
[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), 119-140. |
[29] |
R. LeVeque, The dynamics of pressureless dust clouds and delta waves, Journal Hyperbolic Differential Equations, 1 (2004), 315-327. |
[30] |
T. Li, Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow, J. Diff. Eqn., 190 (2003), 131-149.
doi: 10.1016/S0022-0396(03)00014-7. |
[31] |
M. Lighthill and J. Whitham, On kinematic waves, Proc. Roy. Soc. London Ser. A, 229 (1955), 281-345.
doi: 10.1098/rspa.1955.0088. |
[32] |
E. Ben-Naim, P. L. Krapinsky and S. Redner, Kinetics of clustering in traffic flows, Physical Rev. E, 50 (1994), 822-829. |
[33] |
P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. |
[34] |
L. Santen, A. Schadschneider and M. Schreckenberg, Towards a realistic microscopic description of highway traffic, J. Phys A, 33 (2000), 477-485. |
[35] |
S. Marinosson, R. Chrobok, A. Pottmeier, J. Wahle and M. Schreckenberg, Simulation framework for the autobahn traffic in North Rhine-Westphalia, Cellular automata, 315-324, Lecture Notes in Comput. Sci., 2493, Springer, Berlin (2002). |
[36] |
T. Alperovich and A. Sopasakis, Stochastic description of traffic flow, J. Stat. Phys., 133 (2008), 1083-1105.
doi: 10.1007/s10955-008-9652-6. |
[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), 921-944.
doi: 10.1137/040617790. |
[38] |
M. Treiber and D. Helbing, Macroscopic simulation of widely scattered synchronized traffic states, J. Phys. A, Math. Gen., (1999). |
[39] |
H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Tans. Res. B, 36 (2002), 275-290.
doi: 10.1016/S0191-2615(00)00050-3. |
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