
-
Previous Article
A moment closure based on a projection on the boundary of the realizability domain: 1D case
- KRM Home
- This Issue
-
Next Article
Derivative estimates for screened Vlasov-Poisson system around Penrose-stable equilibria
An anisotropic interaction model with collision avoidance
University of Mannheim, School of Business Informatics and Mathematics, 68159 Mannheim, Germany |
In this article an anisotropic interaction model avoiding collisions is proposed. Starting point is a general isotropic interacting particle system, as used for swarming or follower-leader dynamics. An anisotropy is induced by rotation of the force vector resulting from the interaction of two agents. In this way the anisotropy is leading to a smooth evasion behaviour. In fact, the proposed model generalizes the standard models, and compensates their drawback of not being able to avoid collisions. Moreover, the model allows for formal passage to the limit 'number of particles to infinity', leading to a mesoscopic description in the mean-field sense. Possible applications are autonomous traffic, swarming or pedestrian motion. Here, we focus on the latter, as the model is validated numerically using two scenarios in pedestrian dynamics. The first one investigates the pattern formation in a channel, where two groups of pedestrians are walking in opposite directions. The second experiment considers a crossing with one group walking from left to right and the other one from bottom to top. The well-known pattern of lanes in the channel and travelling waves at the crossing can be reproduced with the help of this anisotropic model at both, the microscopic and the mesoscopic level. In addition, the 'right-before-left' and 'left-before-right' rule appear intrinsically for different anisotropy parameters.
References:
[1] |
G. Albi, M. Bongini, E. Cristiani and D. Kalise,
Invisible control of self-organizing agents leaving unknown environments, SIAM J. Appl. Math., 76 (2016), 1683-1710.
doi: 10.1137/15M1017016. |
[2] |
G. Albi and L. Pareschi,
Modeling self-organized systems interacting with few individuals: From microscopic to macroscopic dynamics, Appl. Math. Letters, 26 (2013), 391-401.
doi: 10.1016/j.aml.2012.10.011. |
[3] |
C. Appert-Rolland, J. Cividini, H. J. Hilhorst and P. Degond,
Pedestrian flows: From individuals to crowds, Transportation Research Procedia, 2 (2014), 468-476.
doi: 10.1016/j.trpro.2014.09.062. |
[4] |
R. Bailo, J. A. Carrillo and P. Degond, Pedestrian models based on rational behaviour, in Crowd Dynamics, Volume 1: Theory, Models, and Safety Problems (eds. L. Gibelli and N. Bellomo), Springer Internat. Publishing, (2018), 259–292. |
[5] |
N. Bellomo, P. Degond and E. Tadmor, Active Particles, Volume 1, Birkhäuser, Basel, 2017.
doi: 10.1007/978-3-319-49996-3. |
[6] |
N. W. F. Bode and E. A. Codling, Statistical models for pedestrian behaviour in front of bottlenecks, in Traffic and Granular Flow'15 (eds. V. Knoop and W. Daamen) Springer, Cham (2016), 81–88.
doi: 10.1007/978-3-319-33482-0_11. |
[7] |
N. W. F. Bode and E. Ronchi,
Statistical model fitting and model selection in pedestrian dynamics research, Collective Dynamics, 4 (2019), 1-32.
doi: 10.17815/CD.2019.20. |
[8] |
R. Borsche and A. Meurer,
Microscopic and macroscopic models for coupled car traffic and pedestiran flow, Journal Computational and Applied Mathematics, 348 (2019), 356-382.
doi: 10.1016/j.cam.2018.08.037. |
[9] |
R. Borsche, A. Klar, S. Kühn and A. Meurer,
Coupling traffic flow networks to pedestrian motion, Math. Mod. Meth. Appl. Sci., 24 (2014), 359-380.
doi: 10.1142/S0218202513400113. |
[10] |
M. Burger, M. Di Francesco, P. A. Markowich and M.-T. Wolfram,
Mean field games with nonlinear mobilities in pedestrian dynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1311-1333.
doi: 10.3934/dcdsb.2014.19.1311. |
[11] |
M. Burger, B. Düring, L. M. Kreusser, P. A. Markowich and C.-B. Schönlieb,
Pattern formation of a nonlocal, anisotropic interaction model, Math. Mod. Meth. Appl. Sci., 28 (2018), 409-451.
doi: 10.1142/S0218202518500112. |
[12] |
M. Burger, S. Hittmeir, H. Ranetbauer and M.-T. Wolfram,
Lane formation by side-stepping, SIAM J. Math. Anal., 48 (2016), 981-1005.
doi: 10.1137/15M1033174. |
[13] |
M. Burger, R. Pinnau, C. Totzeck, O. Tse and A. Roth, Instantaneous Control of interacting particle systems in the mean-field limit, J. Comp. Phys., 405 (2020), 109181.
doi: 10.1016/j.jcp.2019.109181. |
[14] |
J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, in Collective dynamics from bacteria to crowds (eds. A. Muntean, F. Toschi), Springer, (2014), 1–46.
doi: 10.1007/978-3-7091-1785-9_1. |
[15] |
J. A. Carrillo, Y.-P. Choi, C. Totzeck and O. Tse,
An analytical framework for consensus-based global optimization method, Math. Mod. Meth. Appl. Sci., 28 (2018), 1037-1066.
doi: 10.1142/S0218202518500276. |
[16] |
J. A. Carrillo, B. Düring, L. M. Kreusser and C.-B. Schönlieb,
Stability analysis of line patterns of an anisotropic interaction model, SIAM J. Appl. Dyn. Sys., 18 (2019), 1798-1845.
doi: 10.1137/18M1181638. |
[17] |
J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences (eds. G. Naldi, L. Pareschi, G. Toscani), Birkhäuser Boston, (2010), 297–336.
doi: 10.1007/978-0-8176-4946-3_12. |
[18] |
J. A. Carrillo, A. Klar and A. Roth,
Single to double mill small noise transition via semi-lagrangian finite volume methods, Comm. Math. Sci., 14 (2016), 1111-1136.
doi: 10.4310/CMS.2016.v14.n4.a12. |
[19] |
J. A. Carrillo, S. Martin and M.-T. Wolfram,
An improved version of the Hughes model for pedestrian flow, Math. Models Meth. Appl. Sci., 26 (2016), 671-697.
doi: 10.1142/S0218202516500147. |
[20] |
J. Cividini, C. Appert-Rolland and H. J. Hilhorst, Diagonal patterns and chevron effect in intersecting traffic flows, EPL, 102 (2013), 20002.
doi: 10.1209/0295-5075/102/20002. |
[21] |
E. Cristiani and D. Peri,
Handling obstacles in pedestrian simulations: Models and optimization, Appl. Math. Mod., 45 (2017), 285-302.
doi: 10.1016/j.apm.2016.12.020. |
[22] |
E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, Springer International Publishing, Switzerland, 2014.
doi: 10.1007/978-3-319-06620-2. |
[23] |
E. Cristiani, F. S. Priuli and A. Tosin,
Modeling rationality to control self-organization of crowds: An environmental approach, SIAM J. Appl. Math., 75 (2015), 605-629.
doi: 10.1137/140962413. |
[24] |
P. Degond, C. Appert-Rolland, M. Moussaid, J. Pettré and G. Theraulaz,
A hierarchy of heuristic-based models of crowd dynamics, J. Statistical Physics, 152 (2013), 1033-1068.
doi: 10.1007/s10955-013-0805-x. |
[25] |
P. Degond, C. Appert-Rolland, J. Pettreé and G. Theraulaz,
Vision-based macroscopic pedestrian models, Kin. Rel. Models, 6 (2013), 809-839.
doi: 10.3934/krm.2013.6.809. |
[26] |
P. Degond, A. Frouvelle and S. Merino-Aceituno,
A new flocking model through body attitude coordination, Math. Mod. Meth. Appl. Sci., 27 (2017), 1005-1049.
doi: 10.1142/S0218202517400085. |
[27] |
P. Degond and S. Motsch,
Continuum limit of self-driven particles with orientation interaction, Math. Mod. Meth. Appl. Sci., 18 (2008), 1193-1215.
doi: 10.1142/S0218202508003005. |
[28] |
M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram,
On the Hughes' model for pedestrian flow: The one dimensional case, J. Differential Equations, 250 (2011), 1334-1362.
doi: 10.1016/j.jde.2010.10.015. |
[29] |
M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006), 104302.
doi: 10.1103/PhysRevLett.96.104302. |
[30] |
B. Düring, C. Gottschlich, S. Huckemann, L. M. Kreusser and C. -B.Schönlieb,
An anisotropic interaction model for simulating fingerprints, J. Math. Biol., 78 (2019), 2171-2206.
doi: 10.1007/s00285-019-01338-3. |
[31] |
J. H. M. Evers, R. C. Fetecau and L. Ryzhik, Anisotropic interactions in a first-order aggregation model, Nonlinearity, 28 (2015), 2847.
doi: 10.1088/0951-7715/28/8/2847. |
[32] |
J. H. M. Evers, R. C. Fetecau and W. Sun,
Small intertia regularization of an anisotropic aggregation model, Math. Mod. Meth. Appl. Sci., 27 (2017), 1795-1842.
doi: 10.1142/S0218202517500324. |
[33] |
A. Festa, A. Tosin and M.-T. Wolfram,
Kinetic description of collision avoidance in pedestrian crowds by sidestepping, Kinet. Relat. Models, 11 (2018), 491-520.
doi: 10.3934/krm.2018022. |
[34] |
L. Gibelli and N. Bellomo, Crowd Dynamics, Volume 1, Birkhäuser, Cham, 2018.
doi: 10.1007/978-3-030-05129-7. |
[35] |
F. Golse, The mean-field limit for the dynamics of large particle systems, Journées équations aux dérivées partielles, 9 (2003), 1–47. |
[36] |
S. N. Gomes, A. M. Stuart and M.-T. Wolfram,
Parameter estimation for macroscopic pedestrian dynamics models from microscopic data, SIAM J. Appl. Math., 79 (2019), 1475-1500.
doi: 10.1137/18M1215980. |
[37] |
D. Helbing,
A mathematical model for the behavior of pedestrians, Behav. Sci., 36 (1991), 298-310.
doi: 10.1002/bs.3830360405. |
[38] |
D. Helbing, L. Buzna, A. Johansson and T. Werner,
Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions, Transp. Sci., 39 (2005), 1-24.
doi: 10.1287/trsc.1040.0108. |
[39] |
D. Helbing, I. J. Farkas and T. Vicsek,
Simulating dynamical features of escape panic, Nature, 407 (2000), 487-490.
doi: 10.1038/35035023. |
[40] |
D. Helbing, A. Johansson and H. Z. Al-Abideen, Dynamics of crowd disasters: An empirical study, Phys. Rev. E, 75 (2007), 046109.
doi: 10.1103/PhysRevE.75.046109. |
[41] |
D. Helbing and P. Molnár,
Social force model for pedestiran dynamics, Phys. Rev. E, 51 (1995), 4282-4286.
doi: 10.1103/PhysRevE.51.4282. |
[42] |
S. Hittmeir, H. Ranetbauer, C. Schmeiser and M.-T. Wolfram,
Derivation and analysis of continuum models for crossing pedestrian traffic, Math. Meth. Appl. Sci., 27 (2017), 1301-1325.
doi: 10.1142/S0218202517400164. |
[43] |
P.-E. Jabin,
A review of the mean field limits for Vlasov equations, Kin. Rel. Mod., 7 (2014), 661-711.
doi: 10.3934/krm.2014.7.661. |
[44] |
A. Klar, P. Reuterswärd and M. Seaïd,
A semi-lagrangian method for a Fokker-Planck equation describing fiber dynamics, J. Sci. Comp., 38 (2009), 349-367.
doi: 10.1007/s10915-008-9244-2. |
[45] |
R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.
doi: 10.1017/CBO9780511791253.![]() ![]() |
[46] |
N. K. Mahato, A. Klar and S. Tiwari,
A meshfree particle method for a vision-based macroscopic pedestrian model, Int. J. Adv. Eng. Sci. Appl. Math., 10 (2018), 41-53.
doi: 10.1007/s12572-018-0204-2. |
[47] |
S. Motsch and E. Tadmor,
Heterophilious dynamics enhances consensus, SIAM Rev., 56 (2014), 577-621.
doi: 10.1137/120901866. |
[48] |
L. L. Obsu, A. Meurer, S. M. Kassa and A. Klar,
Modelling pedestrians' impact on the performance of a roundabout, Neural Parallel Sci. Comput., 24 (2016), 317-334.
|
[49] |
B. Piccoli and A. Tosin,
Pedestrian flows in bounded domains with obstacles, Continuum Mech. Thermodyn., 21 (2009), 85-107.
doi: 10.1007/s00161-009-0100-x. |
[50] |
R. Pinnau, C. Totzeck, O. Tse and S. Martin,
A consensus-based model for global optimization and its mean-field limit, Math. Mod. Meth. Appl. Sci., 27 (2017), 183-204.
doi: 10.1142/S0218202517400061. |
[51] |
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer-Verlag, Berlin Heidelberg, 1994.
doi: 10.1007/978-3-540-85268-1. |
[52] |
S. Roy, A. Borzí and A. Habbal, Pedestrian motion modelled by Fokker–Planck Nash games, R. Soc. Open Sci., 4 (2017), 170648.
doi: 10.1098/rsos.170648. |
[53] |
A. Sieben, J. Schumann and A. Seyfried, Collective phenomena in crowds-where pedestrian dynamics need social psychology, PLoS one, 12 (2017), e0177328.
doi: 10.1371/journal.pone.0177328. |
[54] |
E. Sonnendrücker, J. Roche, P. Bertrand and A. Ghizzo,
The semi-Lagrangian method for the numerical resolution of the Vlasov equation, J. Comp. Phys., 149 (1999), 201-220.
doi: 10.1006/jcph.1998.6148. |
[55] |
C. Taylor, C. Luzzi and C. Nowzari, On the Effects of Collision Avoidance on Emergent Swarm Behavior, in 2020 American Control Conference (ACC) (2020), 931-936.
doi: 10.23919/ACC45564.2020.9147834. |
[56] |
M. Twarogowska, P. Goatin and R. Duvigneau, Numerical study of macroscopic pedestrian flow models, Appl. Math. Mod., 24 (2014), 5781-5795. Google Scholar |
[57] |
B. Van Leer,
Towards the ultimate conservative difference scheme, J. Comp. Phys., 135 (1997), 227-248.
doi: 10.1006/jcph.1997.5757. |
[58] |
T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen and O. Shochet,
Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.
doi: 10.1103/PhysRevLett.75.1226. |
[59] |
D. Yanagisawa, A. Kimura, A. Tomoeda, R. Nishi, Y. Suma, K. Ohtsuka and K. Nishinari, Introduction of frictional and turning function for pedestrian outflow with an obstacle, Phys. Rev. E, 80 (2009), 036110.
doi: 10.1103/PhysRevE.80.036110. |
show all references
References:
[1] |
G. Albi, M. Bongini, E. Cristiani and D. Kalise,
Invisible control of self-organizing agents leaving unknown environments, SIAM J. Appl. Math., 76 (2016), 1683-1710.
doi: 10.1137/15M1017016. |
[2] |
G. Albi and L. Pareschi,
Modeling self-organized systems interacting with few individuals: From microscopic to macroscopic dynamics, Appl. Math. Letters, 26 (2013), 391-401.
doi: 10.1016/j.aml.2012.10.011. |
[3] |
C. Appert-Rolland, J. Cividini, H. J. Hilhorst and P. Degond,
Pedestrian flows: From individuals to crowds, Transportation Research Procedia, 2 (2014), 468-476.
doi: 10.1016/j.trpro.2014.09.062. |
[4] |
R. Bailo, J. A. Carrillo and P. Degond, Pedestrian models based on rational behaviour, in Crowd Dynamics, Volume 1: Theory, Models, and Safety Problems (eds. L. Gibelli and N. Bellomo), Springer Internat. Publishing, (2018), 259–292. |
[5] |
N. Bellomo, P. Degond and E. Tadmor, Active Particles, Volume 1, Birkhäuser, Basel, 2017.
doi: 10.1007/978-3-319-49996-3. |
[6] |
N. W. F. Bode and E. A. Codling, Statistical models for pedestrian behaviour in front of bottlenecks, in Traffic and Granular Flow'15 (eds. V. Knoop and W. Daamen) Springer, Cham (2016), 81–88.
doi: 10.1007/978-3-319-33482-0_11. |
[7] |
N. W. F. Bode and E. Ronchi,
Statistical model fitting and model selection in pedestrian dynamics research, Collective Dynamics, 4 (2019), 1-32.
doi: 10.17815/CD.2019.20. |
[8] |
R. Borsche and A. Meurer,
Microscopic and macroscopic models for coupled car traffic and pedestiran flow, Journal Computational and Applied Mathematics, 348 (2019), 356-382.
doi: 10.1016/j.cam.2018.08.037. |
[9] |
R. Borsche, A. Klar, S. Kühn and A. Meurer,
Coupling traffic flow networks to pedestrian motion, Math. Mod. Meth. Appl. Sci., 24 (2014), 359-380.
doi: 10.1142/S0218202513400113. |
[10] |
M. Burger, M. Di Francesco, P. A. Markowich and M.-T. Wolfram,
Mean field games with nonlinear mobilities in pedestrian dynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1311-1333.
doi: 10.3934/dcdsb.2014.19.1311. |
[11] |
M. Burger, B. Düring, L. M. Kreusser, P. A. Markowich and C.-B. Schönlieb,
Pattern formation of a nonlocal, anisotropic interaction model, Math. Mod. Meth. Appl. Sci., 28 (2018), 409-451.
doi: 10.1142/S0218202518500112. |
[12] |
M. Burger, S. Hittmeir, H. Ranetbauer and M.-T. Wolfram,
Lane formation by side-stepping, SIAM J. Math. Anal., 48 (2016), 981-1005.
doi: 10.1137/15M1033174. |
[13] |
M. Burger, R. Pinnau, C. Totzeck, O. Tse and A. Roth, Instantaneous Control of interacting particle systems in the mean-field limit, J. Comp. Phys., 405 (2020), 109181.
doi: 10.1016/j.jcp.2019.109181. |
[14] |
J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, in Collective dynamics from bacteria to crowds (eds. A. Muntean, F. Toschi), Springer, (2014), 1–46.
doi: 10.1007/978-3-7091-1785-9_1. |
[15] |
J. A. Carrillo, Y.-P. Choi, C. Totzeck and O. Tse,
An analytical framework for consensus-based global optimization method, Math. Mod. Meth. Appl. Sci., 28 (2018), 1037-1066.
doi: 10.1142/S0218202518500276. |
[16] |
J. A. Carrillo, B. Düring, L. M. Kreusser and C.-B. Schönlieb,
Stability analysis of line patterns of an anisotropic interaction model, SIAM J. Appl. Dyn. Sys., 18 (2019), 1798-1845.
doi: 10.1137/18M1181638. |
[17] |
J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences (eds. G. Naldi, L. Pareschi, G. Toscani), Birkhäuser Boston, (2010), 297–336.
doi: 10.1007/978-0-8176-4946-3_12. |
[18] |
J. A. Carrillo, A. Klar and A. Roth,
Single to double mill small noise transition via semi-lagrangian finite volume methods, Comm. Math. Sci., 14 (2016), 1111-1136.
doi: 10.4310/CMS.2016.v14.n4.a12. |
[19] |
J. A. Carrillo, S. Martin and M.-T. Wolfram,
An improved version of the Hughes model for pedestrian flow, Math. Models Meth. Appl. Sci., 26 (2016), 671-697.
doi: 10.1142/S0218202516500147. |
[20] |
J. Cividini, C. Appert-Rolland and H. J. Hilhorst, Diagonal patterns and chevron effect in intersecting traffic flows, EPL, 102 (2013), 20002.
doi: 10.1209/0295-5075/102/20002. |
[21] |
E. Cristiani and D. Peri,
Handling obstacles in pedestrian simulations: Models and optimization, Appl. Math. Mod., 45 (2017), 285-302.
doi: 10.1016/j.apm.2016.12.020. |
[22] |
E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, Springer International Publishing, Switzerland, 2014.
doi: 10.1007/978-3-319-06620-2. |
[23] |
E. Cristiani, F. S. Priuli and A. Tosin,
Modeling rationality to control self-organization of crowds: An environmental approach, SIAM J. Appl. Math., 75 (2015), 605-629.
doi: 10.1137/140962413. |
[24] |
P. Degond, C. Appert-Rolland, M. Moussaid, J. Pettré and G. Theraulaz,
A hierarchy of heuristic-based models of crowd dynamics, J. Statistical Physics, 152 (2013), 1033-1068.
doi: 10.1007/s10955-013-0805-x. |
[25] |
P. Degond, C. Appert-Rolland, J. Pettreé and G. Theraulaz,
Vision-based macroscopic pedestrian models, Kin. Rel. Models, 6 (2013), 809-839.
doi: 10.3934/krm.2013.6.809. |
[26] |
P. Degond, A. Frouvelle and S. Merino-Aceituno,
A new flocking model through body attitude coordination, Math. Mod. Meth. Appl. Sci., 27 (2017), 1005-1049.
doi: 10.1142/S0218202517400085. |
[27] |
P. Degond and S. Motsch,
Continuum limit of self-driven particles with orientation interaction, Math. Mod. Meth. Appl. Sci., 18 (2008), 1193-1215.
doi: 10.1142/S0218202508003005. |
[28] |
M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram,
On the Hughes' model for pedestrian flow: The one dimensional case, J. Differential Equations, 250 (2011), 1334-1362.
doi: 10.1016/j.jde.2010.10.015. |
[29] |
M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006), 104302.
doi: 10.1103/PhysRevLett.96.104302. |
[30] |
B. Düring, C. Gottschlich, S. Huckemann, L. M. Kreusser and C. -B.Schönlieb,
An anisotropic interaction model for simulating fingerprints, J. Math. Biol., 78 (2019), 2171-2206.
doi: 10.1007/s00285-019-01338-3. |
[31] |
J. H. M. Evers, R. C. Fetecau and L. Ryzhik, Anisotropic interactions in a first-order aggregation model, Nonlinearity, 28 (2015), 2847.
doi: 10.1088/0951-7715/28/8/2847. |
[32] |
J. H. M. Evers, R. C. Fetecau and W. Sun,
Small intertia regularization of an anisotropic aggregation model, Math. Mod. Meth. Appl. Sci., 27 (2017), 1795-1842.
doi: 10.1142/S0218202517500324. |
[33] |
A. Festa, A. Tosin and M.-T. Wolfram,
Kinetic description of collision avoidance in pedestrian crowds by sidestepping, Kinet. Relat. Models, 11 (2018), 491-520.
doi: 10.3934/krm.2018022. |
[34] |
L. Gibelli and N. Bellomo, Crowd Dynamics, Volume 1, Birkhäuser, Cham, 2018.
doi: 10.1007/978-3-030-05129-7. |
[35] |
F. Golse, The mean-field limit for the dynamics of large particle systems, Journées équations aux dérivées partielles, 9 (2003), 1–47. |
[36] |
S. N. Gomes, A. M. Stuart and M.-T. Wolfram,
Parameter estimation for macroscopic pedestrian dynamics models from microscopic data, SIAM J. Appl. Math., 79 (2019), 1475-1500.
doi: 10.1137/18M1215980. |
[37] |
D. Helbing,
A mathematical model for the behavior of pedestrians, Behav. Sci., 36 (1991), 298-310.
doi: 10.1002/bs.3830360405. |
[38] |
D. Helbing, L. Buzna, A. Johansson and T. Werner,
Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions, Transp. Sci., 39 (2005), 1-24.
doi: 10.1287/trsc.1040.0108. |
[39] |
D. Helbing, I. J. Farkas and T. Vicsek,
Simulating dynamical features of escape panic, Nature, 407 (2000), 487-490.
doi: 10.1038/35035023. |
[40] |
D. Helbing, A. Johansson and H. Z. Al-Abideen, Dynamics of crowd disasters: An empirical study, Phys. Rev. E, 75 (2007), 046109.
doi: 10.1103/PhysRevE.75.046109. |
[41] |
D. Helbing and P. Molnár,
Social force model for pedestiran dynamics, Phys. Rev. E, 51 (1995), 4282-4286.
doi: 10.1103/PhysRevE.51.4282. |
[42] |
S. Hittmeir, H. Ranetbauer, C. Schmeiser and M.-T. Wolfram,
Derivation and analysis of continuum models for crossing pedestrian traffic, Math. Meth. Appl. Sci., 27 (2017), 1301-1325.
doi: 10.1142/S0218202517400164. |
[43] |
P.-E. Jabin,
A review of the mean field limits for Vlasov equations, Kin. Rel. Mod., 7 (2014), 661-711.
doi: 10.3934/krm.2014.7.661. |
[44] |
A. Klar, P. Reuterswärd and M. Seaïd,
A semi-lagrangian method for a Fokker-Planck equation describing fiber dynamics, J. Sci. Comp., 38 (2009), 349-367.
doi: 10.1007/s10915-008-9244-2. |
[45] |
R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.
doi: 10.1017/CBO9780511791253.![]() ![]() |
[46] |
N. K. Mahato, A. Klar and S. Tiwari,
A meshfree particle method for a vision-based macroscopic pedestrian model, Int. J. Adv. Eng. Sci. Appl. Math., 10 (2018), 41-53.
doi: 10.1007/s12572-018-0204-2. |
[47] |
S. Motsch and E. Tadmor,
Heterophilious dynamics enhances consensus, SIAM Rev., 56 (2014), 577-621.
doi: 10.1137/120901866. |
[48] |
L. L. Obsu, A. Meurer, S. M. Kassa and A. Klar,
Modelling pedestrians' impact on the performance of a roundabout, Neural Parallel Sci. Comput., 24 (2016), 317-334.
|
[49] |
B. Piccoli and A. Tosin,
Pedestrian flows in bounded domains with obstacles, Continuum Mech. Thermodyn., 21 (2009), 85-107.
doi: 10.1007/s00161-009-0100-x. |
[50] |
R. Pinnau, C. Totzeck, O. Tse and S. Martin,
A consensus-based model for global optimization and its mean-field limit, Math. Mod. Meth. Appl. Sci., 27 (2017), 183-204.
doi: 10.1142/S0218202517400061. |
[51] |
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer-Verlag, Berlin Heidelberg, 1994.
doi: 10.1007/978-3-540-85268-1. |
[52] |
S. Roy, A. Borzí and A. Habbal, Pedestrian motion modelled by Fokker–Planck Nash games, R. Soc. Open Sci., 4 (2017), 170648.
doi: 10.1098/rsos.170648. |
[53] |
A. Sieben, J. Schumann and A. Seyfried, Collective phenomena in crowds-where pedestrian dynamics need social psychology, PLoS one, 12 (2017), e0177328.
doi: 10.1371/journal.pone.0177328. |
[54] |
E. Sonnendrücker, J. Roche, P. Bertrand and A. Ghizzo,
The semi-Lagrangian method for the numerical resolution of the Vlasov equation, J. Comp. Phys., 149 (1999), 201-220.
doi: 10.1006/jcph.1998.6148. |
[55] |
C. Taylor, C. Luzzi and C. Nowzari, On the Effects of Collision Avoidance on Emergent Swarm Behavior, in 2020 American Control Conference (ACC) (2020), 931-936.
doi: 10.23919/ACC45564.2020.9147834. |
[56] |
M. Twarogowska, P. Goatin and R. Duvigneau, Numerical study of macroscopic pedestrian flow models, Appl. Math. Mod., 24 (2014), 5781-5795. Google Scholar |
[57] |
B. Van Leer,
Towards the ultimate conservative difference scheme, J. Comp. Phys., 135 (1997), 227-248.
doi: 10.1006/jcph.1997.5757. |
[58] |
T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen and O. Shochet,
Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.
doi: 10.1103/PhysRevLett.75.1226. |
[59] |
D. Yanagisawa, A. Kimura, A. Tomoeda, R. Nishi, Y. Suma, K. Ohtsuka and K. Nishinari, Introduction of frictional and turning function for pedestrian outflow with an obstacle, Phys. Rev. E, 80 (2009), 036110.
doi: 10.1103/PhysRevE.80.036110. |














[1] |
Illés Horváth, Kristóf Attila Horváth, Péter Kovács, Miklós Telek. Mean-field analysis of a scaling MAC radio protocol. Journal of Industrial & Management Optimization, 2021, 17 (1) : 279-297. doi: 10.3934/jimo.2019111 |
[2] |
Theresa Lange, Wilhelm Stannat. Mean field limit of ensemble square root filters - discrete and continuous time. Foundations of Data Science, 2021 doi: 10.3934/fods.2021003 |
[3] |
Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026 |
[4] |
Josselin Garnier, Knut Sølna. Enhanced Backscattering of a partially coherent field from an anisotropic random lossy medium. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1171-1195. doi: 10.3934/dcdsb.2020158 |
[5] |
Lei Yang, Lianzhang Bao. Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1083-1109. doi: 10.3934/dcdsb.2020154 |
[6] |
Jean-Paul Chehab. Damping, stabilization, and numerical filtering for the modeling and the simulation of time dependent PDEs. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021002 |
[7] |
Shigui Ruan. Nonlinear dynamics in tumor-immune system interaction models with delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 541-602. doi: 10.3934/dcdsb.2020282 |
[8] |
Laura Aquilanti, Simone Cacace, Fabio Camilli, Raul De Maio. A Mean Field Games model for finite mixtures of Bernoulli and categorical distributions. Journal of Dynamics & Games, 2020 doi: 10.3934/jdg.2020033 |
[9] |
Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391 |
[10] |
Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405 |
[11] |
Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039 |
[12] |
Alain Bensoussan, Xinwei Feng, Jianhui Huang. Linear-quadratic-Gaussian mean-field-game with partial observation and common noise. Mathematical Control & Related Fields, 2021, 11 (1) : 23-46. doi: 10.3934/mcrf.2020025 |
[13] |
Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106 |
[14] |
Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257 |
[15] |
Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020406 |
[16] |
Vincent Ducrot, Pascal Frey, Alexandra Claisse. Levelsets and anisotropic mesh adaptation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 165-183. doi: 10.3934/dcds.2009.23.165 |
[17] |
Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020368 |
[18] |
Chang-Yeol Jung, Roger Temam. Interaction of boundary layers and corner singularities. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 315-339. doi: 10.3934/dcds.2009.23.315 |
[19] |
Jie Li, Xiangdong Ye, Tao Yu. Mean equicontinuity, complexity and applications. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 359-393. doi: 10.3934/dcds.2020167 |
[20] |
Xin Guo, Lexin Li, Qiang Wu. Modeling interactive components by coordinate kernel polynomial models. Mathematical Foundations of Computing, 2020, 3 (4) : 263-277. doi: 10.3934/mfc.2020010 |
2019 Impact Factor: 1.311
Tools
Metrics
Other articles
by authors
[Back to Top]