On the multiscale modeling of vehicular traffic: from kinetic to hydrodynamics. (English) Zbl 1302.35372
Summary: 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.
MSC:
| 35Q91 | PDEs in connection with game theory, economics, social and behavioral sciences |
| 35L65 | Hyperbolic conservation laws |
| 35Q70 | PDEs in connection with mechanics of particles and systems of particles |
References:
| [1] | L. Arlotti, Generalized kinetic (Boltzmann) models: Mathematical structures and applications,, Math. Models Methods Appl. Sci., 12, 567 (2002) · Zbl 1174.82325 · doi:10.1142/S0218202502001799 |
| [2] | L. Arlotti, On a class of integro-differential equations modeling complex systems with nonlinear interactions,, Appl. Math. Letters, 25, 490 (2012) · Zbl 1243.82046 · doi:10.1016/j.aml.2011.09.043 |
| [3] | A. Aw, Derivation of continuum traffic flow models from microscopic follow-the-leader models,, SIAM J. Appl. Math., 63, 259 (2002) · Zbl 1023.35063 · doi:10.1137/S0036139900380955 |
| [4] | A. Aw, Resurrection of “second-order” models of traffic flow,, SIAM J. Appl. Math., 60, 916 (2000) · Zbl 0957.35086 · doi:10.1137/S0036139997332099 |
| [5] | M. Ballerini, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study,, Proc. Nat. Acad. Sci., 105, 1232 (2008) · doi:10.1073/pnas.0711437105 |
| [6] | N. Bellomo, Global solution to the Cauchy problem for discrete velocity models of vehicular traffic,, J. Diff. Equations, 252, 1350 (2012) · Zbl 1230.35056 · doi:10.1016/j.jde.2011.09.005 |
| [7] | N. Bellomo, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives,, SIAM Rev., 53, 409 (2011) · Zbl 1231.90123 · doi:10.1137/090746677 |
| [8] | N. Bellomo, On the difficult interplay between life, “Complexity”, and mathematical sciences,, Math. Models Methods Appl. Sci., 23, 1861 (2013) · Zbl 1315.35137 · doi:10.1142/S021820251350053X |
| [9] | N. Bellomo, Modeling crowd dynamics from a complex system viewpoint,, Math. Models Methods Appl. Sci., 22 (2012) · Zbl 1252.35192 · doi:10.1142/S0218202512300049 |
| [10] | N. Bellomo, On the mathematical theory of the dynamics of swarms viewed as complex systems,, Math. Models Methods Appl. Sci., 22 (2012) · Zbl 1242.92065 · doi:10.1142/S0218202511400069 |
| [11] | A. Bellouquid, Towards the modeling of vehicular traffic as a complex system: A kinetic theory approach,, Math. Models Methods Appl. Sci., 22 (2012) · Zbl 1243.35157 · doi:10.1142/S0218202511400033 |
| [12] | A. Bellouquid, Asymptotic limits of a discrete Kinetic Theory model of vehicular traffic,, Appl. Math. Lett., 24, 672 (2011) · Zbl 1210.90047 · doi:10.1016/j.aml.2010.12.004 |
| [13] | S. Buchmuller, <em>Parameters of Pedestrians, Pedestrian Traffic and Walking Facilities</em>,, ETH Report Nr.132 (2006) |
| [14] | V. Coscia, On the mathematical theory of vehicular traffic flow models II. Discrete velocity kinetic models,, Int. J. Non-linear Mechanics, 42, 411 (2007) · Zbl 1200.76163 · doi:10.1016/j.ijnonlinmec.2006.02.008 |
| [15] | V. Coscia, On the mathematical theory of living systems II: The interplay between mathematics and system biology,, Comput. Math. Appl., 62, 3902 (2011) · Zbl 1236.92005 · 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, 277 (1995) · 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, 83 (1999) · Zbl 0987.90012 · doi:10.1016/S0895-7177(99)00064-3 |
| [18] | M. Delitala, Mathematical modelling of vehicular traffic: A discrete kinetic theory approach,, Math. Models Methods Appl. Sci., 17, 901 (2007) · Zbl 1117.35320 · 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, 35 (2012) · Zbl 1252.92012 · doi:10.1007/s00285-011-0452-2 |
| [20] | D. Helbing, Traffic and related self-driven many-particle systems,, Review Modern Phys., 73, 1067 (2001) · 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, 539 (2009) · doi:10.1140/epjb/e2009-00192-5 |
| [22] | M. Herty, Coupling of non-local driving behaviour with fundamental diagrams,, Kinetic Rel. Models, 5, 843 (2012) · Zbl 1262.90037 · doi:10.3934/krm.2012.5.843 |
| [23] | D. Helbing, On the controversy around Daganzo’s requiem and for the Aw-Rascle’s resurrection of second-order traffic flow models,, Eur. Phys. J., 69, 549 (2009) · doi:10.1140/epjb/e2009-00182-7 |
| [24] | A. Klar, Kinetic derivation of macroscopic anticipation models for vehicular traffic,, SIAM J. Appl. Math., 60, 1749 (2000) · Zbl 0953.90007 · doi:10.1137/S0036139999356181 |
| [25] | A. Klar, Vehicular traffic: From microscopic to macroscopic description,, Transp. Theory Statist. Phys., 29, 479 (2000) · Zbl 1174.82322 · doi:10.1080/00411450008205886 |
| [26] | R. Illner, A derivation of the AW-Rascle traffic models from Fokker-Planck type kinetic models,, Quarterly Appl. Math., 67, 39 (2009) · Zbl 1163.35492 |
| [27] | M. Moussaid, Experimental study of the behavioural underlying mechanism underlying self-organization in human crowd,, Proc. Royal Society B: Biol. Sci., 276, 2755 (2009) · 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, 51 (1971) |
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