A kinetic theory approach to the dynamics of crowd evacuation from bounded domains. (English) Zbl 1309.35176
Summary: A mathematical model of the evacuation of a crowd from bounded domains is derived by a hybrid approach with kinetic and macro-features. Interactions at the micro-scale, which modify the velocity direction, are modeled by using tools of game theory and are transferred to the dynamics of collective behaviors. The velocity modulus is assumed to depend on the local density. The modeling approach considers dynamics caused by interactions of pedestrians not only with all the other pedestrians, but also with the geometry of the domain, such as walls and exits. Interactions with the boundary of the domain are non-local and described by games. Numerical simulations are developed to study evacuation time depending on the size of the exit zone, on the initial distribution of the crowd and on a parameter which weighs the unconscious attraction of the stream and the search for less crowded walking directions.
MSC:
| 35Q91 | PDEs in connection with game theory, economics, social and behavioral sciences |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 90C90 | Applications of mathematical programming |
| 91A80 | Applications of game theory |
References:
| [1] | Appert-Rolland, C.; Degond, P.; Motsch, S., Netw. Heterog. Media, 6, 351, 2011 · Zbl 1260.90051 |
| [2] | Bellomo, N.; Bellouquid, A., Netw. Heterog. Media, 6, 383, 2011 · Zbl 1260.90052 |
| [3] | Bellomo, N.; Bellouquid, A.; Knopoff, D., SIAM Multiscale Model. Simulat., 11, 943, 2013 · Zbl 1280.90019 |
| [4] | Bellomo, N.; Knopoff, D.; Soler, J., Math. Models Methods Appl. Sci., 23, 1861, 2013 · Zbl 1315.35137 |
| [5] | Bellomo, N.; Piccoli, B.; Tosin, A., Math. Models Methods Appl. Sci., 22, 29, 2012 · Zbl 1252.35192 |
| [6] | Bellomo, N.; Soler, J., Math. Models Methods Appl. Sci., 22, 29, 2012 · Zbl 1242.92065 |
| [7] | Bellouquid, A.; De Angelis, E.; Fermo, L., Math. Models Methods Appl. Sci., 22, 35, 2012 · Zbl 1243.35157 |
| [8] | Borsche, R., Math. Models Methods Appl. Sci., 24, 359, 2014 · Zbl 1279.90035 |
| [9] | Bruno, L., Appl. Math. Model., 35, 426, 2011 |
| [10] | S. Buchmueller and U. Weidman, Parameters of pedestrians, pedestrian traffic and walking facilities, ETH Report No. 132 (2006). |
| [11] | Cercignani, C.; Illner, R.; Pulvirenti, M., The Mathematical Theory of Dilute Gases, 1994, Springer · Zbl 0813.76001 |
| [12] | Chertock, A., Math. Models Methods Appl. Sci., 24, 249, 2014 · Zbl 1282.90041 |
| [13] | Colombo, R. M.; Garavello, M.; Lécureux-Mercier, M., Math. Models Methods Appl. Sci., 22, 34, 2012 · Zbl 1248.35213 |
| [14] | Coscia, V.; Canavesio, C., Math. Models Methods Appl. Sci., 18, 1217, 2008 · Zbl 1171.91018 |
| [15] | Cristiani, E.; Piccoli, B.; Tosin, A., Multiscale Model. Simulat., 9, 155, 2011 · Zbl 1221.35232 |
| [16] | De Angelis, E., Math. Comput. Model., 29, 83, 1999 · Zbl 0987.90012 |
| [17] | Degond, P., Kinet. Relat. Models, 6, 809, 2013 · Zbl 1426.76051 |
| [18] | Degond, P., J. Stat. Phys., 152, 1033, 2013 |
| [19] | Degond, P.; Dimarco, G.; Mac, T. B. N., Math. Models Methods Appl. Sci., 24, 277, 2014 · Zbl 1285.35120 |
| [20] | Delitala, M.; Tosin, A., Math. Models Methods Appl. Sci., 17, 901, 2007 · Zbl 1117.35320 |
| [21] | Eftimie, R., J. Math. Biol., 65, 35, 2012 · Zbl 1252.92012 |
| [22] | Fermo, L.; Tosin, A., SIAM J. Appl. Math., 75, 1533, 2013 |
| [23] | Ha, S.-Y.; Kang, M.-J.; Kwon, B., Math. Models Methods Appl. Sci., 24, 2311, 2014 · Zbl 1300.35073 |
| [24] | Hartmann, D.; Von Sivers, I., Netw. Heterog. Media, 8, 985, 2013 · Zbl 1284.35279 |
| [25] | Helbing, D.; Johansson, A.; Al-Abideen, H. Z., Phys. Rev. E, 75, 046109, 2007 |
| [26] | Helbing, D.; Molnár, P., Phys. Rev. E, 51, 4282, 1995 |
| [27] | Helbing, D., Environ. Plan. B, Plan. Design, 28, 361, 2001 |
| [28] | Holden, H., Splitting Methods for Partial Differential Equations with Rough Solutions, 2010, European Mathematical Society (EMS) · Zbl 1191.35005 |
| [29] | Hughes, R. L., Transp. Res. B, Methodol., 36, 507, 2002 |
| [30] | Knopoff, D., Math. Models Methods Appl. Sci., 23, 541, 2013 · Zbl 1357.91035 |
| [31] | Lorenz, T.; Surulescu, C., Math. Models Methods Appl. Sci., 24, 2383, 2014 · Zbl 1300.92016 |
| [32] | Moussaïd, M., Proc. Roy. Soc. B, 276, 2755, 2009 |
| [33] | Piccoli, B.; Tosin, A., Contin. Mech. Thermodyn., 21, 85, 2009 · Zbl 1170.90351 |
| [34] | Piccoli, B.; Tosin, A., Arch. Rational Mech. Anal., 199, 707, 2011 · Zbl 1237.90057 |
| [35] | Roggen, D., Netw. Heterog. Media, 6, 521, 2011 |
| [36] | Santos, F. C.; Pacheco, J. M.; Lenaerts, T., Proc. Natl. Acad. Sci. USA, 103, 3490, 2006 |
| [37] | Santos, F. C., Math. Models Methods Appl. Sci., 22, 17, 2012 · Zbl 1318.91150 |
| [38] | Schadschneider, A.; Seyfried, A., Netw. Heterog. Media, 6, 545, 2011 · Zbl 1260.90057 |
| [39] | M. Twarogowska, P. Goatin and R. Duvigneau, Macroscopic modeling and simulations of room evacuation, preprint (2013), [math.NA], arXiv:1308.1770. |
| [40] | Verbeni, M., Phys. Life Rev., 10, 457, 2013 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
