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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.
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.
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.
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.
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.
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.
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.
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.
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).
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).
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).
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.
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, (2006). Google Scholar |
[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.
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.
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.
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.
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.
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.
doi: 10.1007/s00285-011-0452-2. |
[20] |
D. Helbing, Traffic and related self-driven many-particle systems,, Review Modern Phys., 73 (2001), 1067.
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.
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.
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.
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.
doi: 10.1137/S0036139999356181. |
[25] |
A. Klar and R. Wegener, Vehicular traffic: From microscopic to macroscopic description,, Transp. Theory Statist. Phys., 29 (2000), 479.
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.
|
[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.
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, 1 (1971), 51. Google Scholar |
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.
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.
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.
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.
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.
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.
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.
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.
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).
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).
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).
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.
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, (2006). Google Scholar |
[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.
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.
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.
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.
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.
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.
doi: 10.1007/s00285-011-0452-2. |
[20] |
D. Helbing, Traffic and related self-driven many-particle systems,, Review Modern Phys., 73 (2001), 1067.
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.
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.
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.
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.
doi: 10.1137/S0036139999356181. |
[25] |
A. Klar and R. Wegener, Vehicular traffic: From microscopic to macroscopic description,, Transp. Theory Statist. Phys., 29 (2000), 479.
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.
|
[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.
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, 1 (1971), 51. Google Scholar |
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