Applications of Mathematics, first online, pp. 1-32


Kinetic BGK model for a crowd: Crowd characterized by a state of equilibrium

Abdelghani El Mousaoui, Pierre Argoul, Mohammed El Rhabi, Abdelilah Hakim

Received June 17, 2019.   Published online October 21, 2020.

Abstract:  This article focuses on dynamic description of the collective pedestrian motion based on the kinetic model of Bhatnagar-Gross-Krook. The proposed mathematical model is based on a tendency of pedestrians to reach a state of equilibrium within a certain time of relaxation. An approximation of the Maxwellian function representing this equilibrium state is determined. A result of the existence and uniqueness of the discrete velocity model is demonstrated. Thus, the convergence of the solution to that of the continuous BGK equation is proven. Numerical simulations are presented to validate the proposed mathematical model.
Keywords:  discrete kinetic theory; crowd dynamics; BGK model; semi-Lagrangian schemes
Classification MSC:  35A01, 35A02, 97M70, 97N40
DOI:  10.21136/AM.2020.0153-19

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References:
[1] J. P. Agnelli, F. Colasuonno, D. Knopoff: A kinetic theory approach to the dynamics of crowd evacuation from bounded domains. Math. Models Methods Appl. Sci. 25 (2015), 109-129. DOI 10.1142/S0218202515500049 | MR 3277286 | Zbl 1309.35176
[2] N. Bellomo: Modeling Complex Living Systems: A Kinetic Theory and Stochastic Game Approach. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston (2008). DOI 10.1007/978-0-8176-4600-4 | MR 2359781 | Zbl 1140.91007
[3] N. Bellomo, A. Bellouquid: On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms. Netw. Heterog. Media 6 (2011), 383-399. DOI 10.3934/nhm.2011.6.383 | MR 2826751 | Zbl 1260.90052
[4] N. Bellomo, L. Gibelli: Toward a mathematical theory of behavioral-social dynamics for pedestrian crowds. Math. Models Methods Appl. Sci. 25 (2015), 2417-2437. DOI 10.1142/S0218202515400138 | MR 3397538 | Zbl 1325.91042
[5] N. Bellomo, L. Gibelli: Behavioral crowds: Modeling and Monte Carlo simulations toward validation. Comput. Fluids 141 (2016), 13-21. DOI 10.1016/j.compfluid.2016.04.022 | MR 3569212 | Zbl 1390.65030
[6] F. Bouchut: On zero pressure gas dynamics. Advances in Kinetic Theory and Computing. Series on Advances in Mathematics for Applied Sciences 22. World Scientific, Singapore, 1994, 171-190. DOI 10.1142/9789814354165_0006 | MR 1323183 | Zbl 0863.76068
[7] F. Bouchut, F. James: Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness. Commun. Partial Differ. Equations 24 (1999), 2173-2189. DOI 10.1080/03605309908821498 | MR 1720754 | Zbl 0937.35098
[8] C. Buet: A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics. Transp. Theory Stat. Phys. 25 (1996), 33-60. DOI 10.1080/00411459608204829 | MR 1380030 | Zbl 0857.76079
[9] D. Burini, L. Gibelli, N. Outada: A kinetic theory approach to the modeling of complex living systems. Modeling and Simulation in Science, Engineering and Technology. Active Particles. Vol. 1. Birkhäuser, Cham, 2017, 229-258. DOI 10.1007/978-3-319-49996-3_6 | MR 3644592 | Zbl 1368.00045
[10] G. Dimarco, R. Loubere: Towards an ultra efficient kinetic scheme. I: Basics on the BGK equation. J. Comput. Phys. 255 (2013), 680-698. DOI 10.1016/j.jcp.2012.10.058 | MR 3109810 | Zbl 1349.76674
[11] G. Dimarco, S. Motsch: Self-alignment driven by jump processes: Macroscopic limit and numerical investigation. Math. Models Methods Appl. Sci. 26 (2016), 1385-1410. DOI 10.1142/S0218202516500330 | MR 3494681 | Zbl 1341.35170
[12] N. Dunford, J. T. Schwartz: Linear Operators. Part I: General Theory. Wiley Classics Library. John Wiley & Sons, New York (1988). MR 1009162 | Zbl 0635.47001
[13] R. M. Colombo, M. D. Rosini: Existence of nonclassical solutions in a pedestrian flow model. Nonlinear Anal., Real World Appl. 10 (2009), 2716-2728. DOI 10.1016/j.nonrwa.2008.08.002 | MR 2523235 | Zbl 1169.35360
[14] E. Cristiani, B. Piccoli, A. Tosin: Multiscale Modeling of Pedestrian Dynamics. MS&A. Modeling, Simulation and Applications 12. Springer, Cham (2014). DOI 10.1007/978-3-319-06620-2 | MR 3308728 | Zbl 1314.00081
[15] M. El-Amrani, M. Seaïd: A finite element modified method of characteristics for convective heat transport. Numer. Methods Partial Differ. Equations 24 (2008), 776-798. DOI 10.1002/num.20288 | MR 2402574 | Zbl 1143.65075
[16] A. Elmoussaoui, P. Argoul, M. El Rhabi, A. Hakim: Discrete kinetic theory for 2D modeling of a moving crowd: Application to the evacuation of a non-connected bounded domain. Comput. Math. Appl. 75 (2018), 1159-1180. DOI 10.1016/j.camwa.2017.10.023 | MR 3766510 | Zbl 1409.82013
[17] F. Golse, P.-L. Lions, B. Perthame, R. Sentis: Regularity of the moments of the solution of a transport equation. J. Funct. Anal. 76 (1988), 110-125. DOI 10.1016/0022-1236(88)90051-1 | MR 0923047 | Zbl 0652.47031
[18] D. Helbing: A mathematical model for the behavior of pedestrians. Behavioral Science 36 (1991), 298-310. DOI 10.1002/bs.3830360405
[19] D. Helbing: A fluid-dynamic model for the movement of pedestrians. Complex Syst. 6 (1992), 391-415. MR 1211939 | Zbl 0776.92016
[20] D. Helbing, P. Molnár: Social force model for pedestrian dynamics. Phys. Rev. E 51 (1995), Article ID 4282. DOI 10.1103/PhysRevE.51.4282
[21] D. Helbing, P. Molnár: Self-organization phenomena in pedestrian crowds. Self-Organization of Complex Structures: From Individual to Collective Dynamics (F. Schweitzer, ed.). Gordon and Breach, Reading, 1997, 569-577. Zbl 0926.91068
[22] L. F. Henderson: The statistics of crowd fluids. Nature 229 (1971), 381-383. DOI 10.1038/229381a0
[23] L. F. Henderson: On the fluid mechanics of human crowd motion. Transp. Research 8 (1974), 509-515. DOI 10.1016/0041-1647(74)90027-6
[24] L. F. Henderson, D. J. Lyons: Sexual differences in human crowd motion. Nature 240 (1972), 353-355. DOI 10.1038/240353a0
[25] S. Hoogendoorn, P. H. Bovy: Gas-kinetic modeling and simulation of pedestrian flows. Transp. Research Record 1710 (2000), 28-36. DOI 10.3141/1710-04
[26] D. Issautier: Convergence of a weighted particle method for solving the Boltzmann (B.G.K.) equation. SIAM J. Numer. Anal. 33 (1996), 2099-2119. DOI 10.1137/S0036142994266856 | MR 1427455 | Zbl 0861.65128
[27] M. Lentine, J. T. Grétarsson, R. Fedkiw: An unconditionally stable fully conservative semi-Lagrangian method. J. Comput. Phys. 230 (2011), 2857-2879. DOI 10.1016/j.jcp.2010.12.036 | MR 2774321 | Zbl 1316.76076
[28] L. Mieussens: Discrete-velocity models and numerical schemes for the Boltzmann-BGK equation in plane and axisymmetric geometries. J. Comput. Phys. 162 (2000), 429-466. DOI 10.1006/jcph.2000.6548 | MR 1774264 | Zbl 0984.76070
[29] L. Mieussens: Convergence of a discrete-velocity model for the Boltzmann-BGK equation. Comput. Math. Appl. 41 (2001), 83-96. DOI 10.1016/S0898-1221(01)85008-2 | MR 1808507 | Zbl 0980.82027
[30] S. Mischler: Convergence of discrete-velocity schemes for the Boltzmann equation. Arch. Ration. Mech. Anal. 140 (1997), 53-77. DOI 10.1007/s002050050060 | MR 1482928 | Zbl 0898.76089
[31] K. Nishinari, A. Kirchner, A. Namazi, A. Schadschneider: Extended floor field CA model for evacuation dynamics. IEICE Trans. Inf. Syst. E87-D (2004), 726-732.
[32] V. A. Panferov, A. G. Heintz: A new consistent discrete-velocity model for the Boltzmann equation. Math. Methods Appl. Sci. 25 (2002), 571-593. DOI 10.1002/mma.303 | MR 1895119 | Zbl 0997.82036
[33] B. Perthame: Global existence to the BGK model of Boltzmann equation. J. Differ. Equations 82 (1989), 191-205. DOI 10.1016/0022-0396(89)90173-3 | MR 1023307 | Zbl 0694.35134
[34] B. Perthame, M. Pulvirenti: Weighted $L^\infty$ bounds and uniqueness for the Boltzmann BGK model. Arch. Ration. Mech. Anal. 125 (1993), 289-295. DOI 10.1007/BF00383223 | MR 1245074 | Zbl 0786.76072
[35] C. W. Reynolds: Steering behaviors for autonomous characters. Proceedings of Game Developers Conference. San Francisco, Miller Freeman Game Group, 1999, 763-782.
[36] E. Ringeisen: Contributions a l'étude Mathématique des Equations Cinétiques: Thèse de Doctorat en Mathématique. Université Paris-Saclay, Paris (1991). (In French.)
[37] F. Rogier, J. Schneider: A direct method for solving the Boltzmann equation. Transp. Theory Stat. Phys. 23 (1994), 313-338. DOI 10.1080/00411459408203868 | MR 1257657 | Zbl 0811.76050
[38] G. K. Still: Introduction to Crowd Science. CRC Press, Boca Raton (2014). DOI 10.1201/b17097
[39] G. Stracquadanio: High Order Semi-Lagrangian Methods for BGK-Type Models in the Kinetic Theory of Rarefied Gases: PhD Thesis. Università degli Studi di Parma, Dipartimento di Matematica e Informatica, Parma (2015).
[40] J. Y. Yang, J. C. Huang: Rarefied flow computations using nonlinear model Boltzmann equations. J. Comput. Phys. 120 (1995), 323-339. DOI 10.1006/jcph.1995.1168 | Zbl 0845.76064

Affiliations:   Abdelghani El Mousaoui (corresponding author), School of Industrial Management, Mohammed VI Polytechnic University, Lot 660, Hay Moulay Rachid, 43150 Ben Guerir, Morocco, e-mail: abdelghani.elmousaoui@emines.um6p.ma; Pierre Argoul, Laboratoire Ville Mobilité Transport, Université Gustave Eiffel, IFSTTAR, ENPC, F-77447 Marne-la-Vallée, France, e-mail: pierre.argoul@ifsttar.fr; Mohammed El Rhabi, INTERACT Research Unit, Institut PolyTechnique UniLaSalle, 3 Rue du Tronquet, 76130 Mont-Saint-Aignan, France, e-mail: mohammed.elrhabi@unilasalle.fr; Abdelilah Hakim, Laboratory of Applied Mathematics and Computer Science, Faculty of Sciences and Technologies of Marrakech, Cadi Ayyad University, BP 549, Av. Abdelkrim El Khattabi, Guéliz, Marrakech, Morocco, e-mail: a.hakim@uca.ma


 
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