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September  2014, 19(7): 1869-1888. doi: 10.3934/dcdsb.2014.19.1869

On the multiscale modeling of vehicular traffic: From kinetic to hydrodynamics

Nicola Bellomo 1, , Abdelghani Bellouquid 2, , Juanjo Nieto 3, and  Juan Soler 3,

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

Received  April 2013 Revised  May 2013 Published  August 2014

This paper deals with the multiscale modeling of vehicular traffic according to a kinetic theory approach, where the microscopic state of vehicles is described by position, velocity and activity, namely a variable suitable to model the quality of the driver-vehicle micro-system. Interactions at the microscopic scale are modeled by methods of game theory, thus leading to the derivation of mathematical models within the framework of the kinetic theory. Macroscopic equations are derived by asymptotic limits from the underlying description at the lower scale. This approach shows the hypothesis under which macroscopic models known in the literature can be derived and how new models can be developed.
Keywords: closure method., nonlinear interactions, Vehicular traffic, micro-macro, kinetic theory.
Mathematics Subject Classification: 35Q91, 35L65, 35Q7.
Citation: Nicola Bellomo, Abdelghani Bellouquid, Juanjo Nieto, Juan Soler. On the multiscale modeling of vehicular traffic: From kinetic to hydrodynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1869-1888. doi: 10.3934/dcdsb.2014.19.1869
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

S. Buchmuller and U. Weidman, Parameters of Pedestrians, Pedestrian Traffic and Walking Facilities, ETH Report Nr.132, October, 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), 411-421. doi: 10.1016/j.ijnonlinmec.2006.02.008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

D. Helbing, Traffic and related self-driven many-particle systems, Review Modern Phys., 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. 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), 567-591. doi: 10.1142/S0218202502001799.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

S. Buchmuller and U. Weidman, Parameters of Pedestrians, Pedestrian Traffic and Walking Facilities, ETH Report Nr.132, October, 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), 411-421. doi: 10.1016/j.ijnonlinmec.2006.02.008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

D. Helbing, Traffic and related self-driven many-particle systems, Review Modern Phys., 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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