# American Institute of Mathematical Sciences

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

## 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

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.
Citation: Nicola Bellomo, Abdelghani Bellouquid, Juanjo Nieto, Juan Soler. On the multiscale modeling of vehicular traffic: From kinetic to hydrodynamics. Discrete and 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. [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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##### 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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