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On the multiscale modeling of vehicular traffic: From kinetic to hydrodynamics
1. | King Abdulaziz University, Jeddah, Saudi Arabia, and Politecnico of Torino, Italy |
2. | Cadi Ayyad University, Ecole Nationale des Sciences Appliquées, Marrakech, Morocco |
3. | Departamento de Matemática Aplicada, Universidad de Granada |
References:
[1] |
L. Arlotti, N. Bellomo and E. De Angelis, Generalized kinetic (Boltzmann) models: Mathematical structures and applications, Math. Models Methods Appl. Sci., 12 (2002), 567-591.
doi: 10.1142/S0218202502001799. |
[2] |
L. Arlotti, E. De Angelis, L. Fermo, M. Lachowicz and N. Bellomo, On a class of integro-differential equations modeling complex systems with nonlinear interactions, Appl. Math. Letters, 25 (2012), 490-495.
doi: 10.1016/j.aml.2011.09.043. |
[3] |
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. |
[4] |
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. |
[5] |
M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Nat. Acad. Sci., 105 (2008), 1232-1237.
doi: 10.1073/pnas.0711437105. |
[6] |
N. Bellomo and A. Bellouquid, Global solution to the Cauchy problem for discrete velocity models of vehicular traffic, J. Diff. Equations, 252 (2012), 1350-1368.
doi: 10.1016/j.jde.2011.09.005. |
[7] |
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. |
[8] |
N. Bellomo, D. Knopoff and J. Soler, On the difficult interplay between life, "Complexity'', and mathematical sciences, Math. Models Methods Appl. Sci., 23 (2013), 1861-1913.
doi: 10.1142/S021820251350053X. |
[9] |
N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint, Math. Models Methods Appl. Sci., 22 (2012), Paper No.1230004 (29 pages).
doi: 10.1142/S0218202512300049. |
[10] |
N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci., 22 (2012), paper n.1140006 (29 pages).
doi: 10.1142/S0218202511400069. |
[11] |
A. Bellouquid, E. De Angelis and L. Fermo, Towards the modeling of vehicular traffic as a complex system: A kinetic theory approach, Math. Models Methods Appl. Sci., 22 (2012), paper No.1140003 (35 pages).
doi: 10.1142/S0218202511400033. |
[12] |
A. Bellouquid and M. Delitala, Asymptotic limits of a discrete Kinetic Theory model of vehicular traffic, Appl. Math. Lett., 24 (2011), 672-678.
doi: 10.1016/j.aml.2010.12.004. |
[13] |
S. Buchmuller and U. Weidman, Parameters of Pedestrians, Pedestrian Traffic and Walking Facilities, ETH Report Nr.132, October, 2006. |
[14] |
V. Coscia, M. Delitala and P. Frasca, On the mathematical theory of vehicular traffic flow models II. Discrete velocity kinetic models, Int. J. Non-linear Mechanics, 42 (2007), 411-421.
doi: 10.1016/j.ijnonlinmec.2006.02.008. |
[15] |
V. Coscia, L. Fermo and N. Bellomo, On the mathematical theory of living systems II: The interplay between mathematics and system biology, Comput. Math. Appl., 62 (2011), 3902-3911.
doi: 10.1016/j.camwa.2011.09.043. |
[16] |
C. F. Daganzo, Requiem for second order fluid approximations of traffic flow, Transp. Res. B, 29 (1995), 277-286.
doi: 10.1016/0191-2615(95)00007-Z. |
[17] |
E. De Angelis, Nonlinear hydrodynamic models of traffic flow modelling and mathematical problems, Mathl. Comp. Modelling, 29 (1999), 83-95.
doi: 10.1016/S0895-7177(99)00064-3. |
[18] |
M. Delitala and A. Tosin, Mathematical modelling of vehicular traffic: A discrete kinetic theory approach, Math. Models Methods Appl. Sci., 17 (2007), 901-932.
doi: 10.1142/S0218202507002157. |
[19] |
R. Eftimie, Hyperbolic and kinetic models for self-organized biological aggregations and movement: A brief review, J. Math. Biol., 65 (2012), 35-75.
doi: 10.1007/s00285-011-0452-2. |
[20] |
D. Helbing, Traffic and related self-driven many-particle systems, Review Modern Phys., 73 (2001), 1067-1141.
doi: 10.1103/RevModPhys.73.1067. |
[21] |
D. Helbing, Derivation of non-local macroscopic traffic equations and consistent traffic pressures from microscopic car-following models, Eur. Phys. J. B, 69 (2009), 539-548.
doi: 10.1140/epjb/e2009-00192-5. |
[22] |
M. Herty and R. Illner, Coupling of non-local driving behaviour with fundamental diagrams, Kinetic Rel. Models, 5 (2012), 843-855.
doi: 10.3934/krm.2012.5.843. |
[23] |
D. Helbing and A. Johansson, On the controversy around Daganzo's requiem and for the Aw-Rascle's resurrection of second-order traffic flow models, Eur. Phys. J., 69 (2009), 549-562.
doi: 10.1140/epjb/e2009-00182-7. |
[24] |
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. |
[25] |
A. Klar and R. Wegener, Vehicular traffic: From microscopic to macroscopic description, Transp. Theory Statist. Phys., 29 (2000), 479-493.
doi: 10.1080/00411450008205886. |
[26] |
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. |
[27] |
M. Moussaid, D. Helbing, S. Garnier, A. Johanson, M. Combe and G. Theraulaz, Experimental study of the behavioural underlying mechanism underlying self-organization in human crowd, Proc. Royal Society B: Biol. Sci., 276 (2009), 2755-2762.
doi: 10.1098/rspb.2009.0405. |
[28] |
H. J. Payne, Models of freeway traffic and control, in Mathematical Models of Public Systems. Simulation Councils Proceed. Series, (Ed. G. A. Bekey), 1 1971, 51-60. |
show all references
References:
[1] |
L. Arlotti, N. Bellomo and E. De Angelis, Generalized kinetic (Boltzmann) models: Mathematical structures and applications, Math. Models Methods Appl. Sci., 12 (2002), 567-591.
doi: 10.1142/S0218202502001799. |
[2] |
L. Arlotti, E. De Angelis, L. Fermo, M. Lachowicz and N. Bellomo, On a class of integro-differential equations modeling complex systems with nonlinear interactions, Appl. Math. Letters, 25 (2012), 490-495.
doi: 10.1016/j.aml.2011.09.043. |
[3] |
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. |
[4] |
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. |
[5] |
M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Nat. Acad. Sci., 105 (2008), 1232-1237.
doi: 10.1073/pnas.0711437105. |
[6] |
N. Bellomo and A. Bellouquid, Global solution to the Cauchy problem for discrete velocity models of vehicular traffic, J. Diff. Equations, 252 (2012), 1350-1368.
doi: 10.1016/j.jde.2011.09.005. |
[7] |
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. |
[8] |
N. Bellomo, D. Knopoff and J. Soler, On the difficult interplay between life, "Complexity'', and mathematical sciences, Math. Models Methods Appl. Sci., 23 (2013), 1861-1913.
doi: 10.1142/S021820251350053X. |
[9] |
N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint, Math. Models Methods Appl. Sci., 22 (2012), Paper No.1230004 (29 pages).
doi: 10.1142/S0218202512300049. |
[10] |
N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci., 22 (2012), paper n.1140006 (29 pages).
doi: 10.1142/S0218202511400069. |
[11] |
A. Bellouquid, E. De Angelis and L. Fermo, Towards the modeling of vehicular traffic as a complex system: A kinetic theory approach, Math. Models Methods Appl. Sci., 22 (2012), paper No.1140003 (35 pages).
doi: 10.1142/S0218202511400033. |
[12] |
A. Bellouquid and M. Delitala, Asymptotic limits of a discrete Kinetic Theory model of vehicular traffic, Appl. Math. Lett., 24 (2011), 672-678.
doi: 10.1016/j.aml.2010.12.004. |
[13] |
S. Buchmuller and U. Weidman, Parameters of Pedestrians, Pedestrian Traffic and Walking Facilities, ETH Report Nr.132, October, 2006. |
[14] |
V. Coscia, M. Delitala and P. Frasca, On the mathematical theory of vehicular traffic flow models II. Discrete velocity kinetic models, Int. J. Non-linear Mechanics, 42 (2007), 411-421.
doi: 10.1016/j.ijnonlinmec.2006.02.008. |
[15] |
V. Coscia, L. Fermo and N. Bellomo, On the mathematical theory of living systems II: The interplay between mathematics and system biology, Comput. Math. Appl., 62 (2011), 3902-3911.
doi: 10.1016/j.camwa.2011.09.043. |
[16] |
C. F. Daganzo, Requiem for second order fluid approximations of traffic flow, Transp. Res. B, 29 (1995), 277-286.
doi: 10.1016/0191-2615(95)00007-Z. |
[17] |
E. De Angelis, Nonlinear hydrodynamic models of traffic flow modelling and mathematical problems, Mathl. Comp. Modelling, 29 (1999), 83-95.
doi: 10.1016/S0895-7177(99)00064-3. |
[18] |
M. Delitala and A. Tosin, Mathematical modelling of vehicular traffic: A discrete kinetic theory approach, Math. Models Methods Appl. Sci., 17 (2007), 901-932.
doi: 10.1142/S0218202507002157. |
[19] |
R. Eftimie, Hyperbolic and kinetic models for self-organized biological aggregations and movement: A brief review, J. Math. Biol., 65 (2012), 35-75.
doi: 10.1007/s00285-011-0452-2. |
[20] |
D. Helbing, Traffic and related self-driven many-particle systems, Review Modern Phys., 73 (2001), 1067-1141.
doi: 10.1103/RevModPhys.73.1067. |
[21] |
D. Helbing, Derivation of non-local macroscopic traffic equations and consistent traffic pressures from microscopic car-following models, Eur. Phys. J. B, 69 (2009), 539-548.
doi: 10.1140/epjb/e2009-00192-5. |
[22] |
M. Herty and R. Illner, Coupling of non-local driving behaviour with fundamental diagrams, Kinetic Rel. Models, 5 (2012), 843-855.
doi: 10.3934/krm.2012.5.843. |
[23] |
D. Helbing and A. Johansson, On the controversy around Daganzo's requiem and for the Aw-Rascle's resurrection of second-order traffic flow models, Eur. Phys. J., 69 (2009), 549-562.
doi: 10.1140/epjb/e2009-00182-7. |
[24] |
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. |
[25] |
A. Klar and R. Wegener, Vehicular traffic: From microscopic to macroscopic description, Transp. Theory Statist. Phys., 29 (2000), 479-493.
doi: 10.1080/00411450008205886. |
[26] |
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. |
[27] |
M. Moussaid, D. Helbing, S. Garnier, A. Johanson, M. Combe and G. Theraulaz, Experimental study of the behavioural underlying mechanism underlying self-organization in human crowd, Proc. Royal Society B: Biol. Sci., 276 (2009), 2755-2762.
doi: 10.1098/rspb.2009.0405. |
[28] |
H. J. Payne, Models of freeway traffic and control, in Mathematical Models of Public Systems. Simulation Councils Proceed. Series, (Ed. G. A. Bekey), 1 1971, 51-60. |
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