American Institute of Mathematical Sciences

December  2018, 11(6): 1333-1358. doi: 10.3934/krm.2018052

A pedestrian flow model with stochastic velocities: Microscopic and macroscopic approaches

 Department of Mathematics, University of Mannheim, 68131 Mannheim, Germany

* Corresponding author: S. Göttlich

Received  March 2017 Revised  December 2017 Published  June 2018

We investigate a stochastic model hierarchy for pedestrian flow. Starting from a microscopic social force model, where the pedestrians switch randomly between the two states stop-or-go, we derive an associated macroscopic model of conservation law type. Therefore we use a kinetic mean-field equation and introduce a new problem-oriented closure function. Numerical experiments are presented to compare the above models and to show their similarities.

Citation: Simone Göttlich, Stephan Knapp, Peter Schillen. A pedestrian flow model with stochastic velocities: Microscopic and macroscopic approaches. Kinetic & Related Models, 2018, 11 (6) : 1333-1358. doi: 10.3934/krm.2018052
References:
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Google Scholar [6] L. Chen, S. Göttlich and Q. Yin, Mean field limit and propagation of chaos for a pedestrian flow model, Journal of Statistical Physics, 166 (2017), 211-229. doi: 10.1007/s10955-016-1679-5. Google Scholar [7] A. Chertock, A. Kurganov, A. Polizzi and I. Timofeyev, Pedestrian flow models with slowdown interactions, Math. Models Methods Appl. Sci., 24 (2014), 249-275. doi: 10.1142/S0218202513400083. Google Scholar [8] E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, vol. 12 of MS&A. Modeling, Simulation and Applications, Springer, Cham, 2014. doi: 10.1007/978-3-319-06620-2. Google Scholar [9] P. Degond, C. Appert-Rolland, M. Moussaïd, J. Pettré and G. Theraulaz, A hierarchy of heuristic-based models of crowd dynamics, J. Stat. Phys., 152 (2013), 1033-1068. doi: 10.1007/s10955-013-0805-x. Google Scholar [10] P. Degond and C. Ringhofer, Stochastic dynamics of long supply chains with random breakdowns, SIAM J. Appl. Math., 68 (2007), 59-79. doi: 10.1137/060674302. Google Scholar [11] P. Degond, C. Appert-Rolland, J. Pettré and G. Theraulaz, Vision-based macroscopic pedestrian models, Kinet. Relat. Models, 6 (2013), 809-839. doi: 10.3934/krm.2013.6.809. Google Scholar [12] G. Dimarco and 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. Google Scholar [13] R. Etikyala, S. Göttlich, A. Klar and S. Tiwari, Particle methods for pedestrian flow models: From microscopic to nonlocal continuum models, Math. Models Methods Appl. Sci., 24 (2014), 2503-2523. doi: 10.1142/S0218202514500274. Google Scholar [14] I. I. Gikhman and A. V. Skorokhod, The Theory of Stochastic Processes. Ⅱ, Classics in Mathematics, Springer-Verlag, Berlin, 2004, Translated from the Russian by S. Kotz, Reprint of the 1975 edition. doi: 10.1007/978-3-642-61921-2. Google Scholar [15] D. Helbing, A fluid dynamic model for the movement of pedestrians, Complex Systems, 6 (1992), 391-415, arXiv: cond-mat/9805213. Google Scholar [16] D. Helbing and P. Molnár, Social force model for pedestrian dynamics, Physical Review E, 51 (1998), 4282-4286, arXiv: cond-mat/9805244. doi: 10.1103/PhysRevE.51.4282. Google Scholar [17] R. L. Hughes, A continuum theory for the flow of pedestrians, Transportation Research Part B: Methodological, 36 (2002), 507-535. doi: 10.1016/S0191-2615(01)00015-7. Google Scholar [18] P.-E. Jabin, Macroscopic limit of Vlasov type equations with friction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 651-672. doi: 10.1016/S0294-1449(00)00118-9. Google Scholar [19] P.-E. Jabin, Various levels of models for aerosols, Math. Models Methods Appl. Sci., 12 (2002), 903-919. doi: 10.1142/S0218202502001957. Google Scholar [20] A. Jüngel, Transport Equations for Semiconductors, vol. 773 of Lecture Notes in Physics, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89526-8. Google Scholar [21] A. Klar, F. Schneider and O. Tse, Approximate models for stochastic dynamic systems with velocities on the sphere and associated fokker-planck equations, Kinetic and Related Models, 7 (2014), 509-529. doi: 10.3934/krm.2014.7.509. Google Scholar [22] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253. Google Scholar [23] B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738. doi: 10.1007/s00205-010-0366-y. Google Scholar [24] B. Piccoli and A. Tosin, Pedestrian flows in bounded domains with obstacles, Contin. Mech. Thermodyn., 21 (2009), 85-107. doi: 10.1007/s00161-009-0100-x. Google Scholar [25] M. Schultz, Stochastic transition model for pedestrian dynamics, in Pedestrian and Evacuation Dynamics 2012, Springer International Publishing, (2013), 971-985, arXiv: 1210.5554. doi: 10.1007/978-3-319-02447-9_81. Google Scholar [26] A. Tordeux and A. Schadschneider, A stochastic optimal velocity model for pedestrian flow, in Parallel Processing and Applied Mathematics, Springer International Publishing, 9574 (2016), 528-538. doi: 10.1007/978-3-319-32152-3_49. Google Scholar [27] A. Tordeux and A. Schadschneider, White and relaxed noises in optimal velocity models for pedestrian flow with stop-and-go waves, Journal of Physics A: Mathematical and Theoretical, 49 (2016), 185101, 16pp. doi: 10.1088/1751-8113/49/18/185101. Google Scholar

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References:
 [1] D. Armbruster, S. Martin and A. Thatcher, Elastic and inelastic collisions of swarms, Physica D: Nonlinear Phenomena, 344 (2017), 45-57. doi: 10.1016/j.physd.2016.11.008. Google Scholar [2] D. Armbruster, S. Motsch and A. Thatcher, Swarming in bounded domains, Physica D: Nonlinear Phenomena, 344 (2017), 58-67. doi: 10.1016/j.physd.2016.11.009. Google Scholar [3] H. Bauer, Probability Theory, vol. 23 of De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1996, Translated from the fourth (1991) German edition by Robert B. Burckel and revised by the author. doi: 10.1515/9783110814668. Google Scholar [4] N. Bellomo, C. Bianca and V. Coscia, On the modeling of crowd dynamics: An overview and research perspectives, S$\vec{\rm e}$MA J., 54 (2011), 25-46. Google Scholar [5] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, vol. 106 of Applied Mathematical Sciences, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8. Google Scholar [6] L. Chen, S. Göttlich and Q. Yin, Mean field limit and propagation of chaos for a pedestrian flow model, Journal of Statistical Physics, 166 (2017), 211-229. doi: 10.1007/s10955-016-1679-5. Google Scholar [7] A. Chertock, A. Kurganov, A. Polizzi and I. Timofeyev, Pedestrian flow models with slowdown interactions, Math. Models Methods Appl. Sci., 24 (2014), 249-275. doi: 10.1142/S0218202513400083. Google Scholar [8] E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, vol. 12 of MS&A. Modeling, Simulation and Applications, Springer, Cham, 2014. doi: 10.1007/978-3-319-06620-2. Google Scholar [9] P. Degond, C. Appert-Rolland, M. Moussaïd, J. Pettré and G. Theraulaz, A hierarchy of heuristic-based models of crowd dynamics, J. Stat. Phys., 152 (2013), 1033-1068. doi: 10.1007/s10955-013-0805-x. Google Scholar [10] P. Degond and C. Ringhofer, Stochastic dynamics of long supply chains with random breakdowns, SIAM J. Appl. Math., 68 (2007), 59-79. doi: 10.1137/060674302. Google Scholar [11] P. Degond, C. Appert-Rolland, J. Pettré and G. Theraulaz, Vision-based macroscopic pedestrian models, Kinet. Relat. Models, 6 (2013), 809-839. doi: 10.3934/krm.2013.6.809. Google Scholar [12] G. Dimarco and 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. Google Scholar [13] R. Etikyala, S. Göttlich, A. Klar and S. Tiwari, Particle methods for pedestrian flow models: From microscopic to nonlocal continuum models, Math. Models Methods Appl. Sci., 24 (2014), 2503-2523. doi: 10.1142/S0218202514500274. Google Scholar [14] I. I. Gikhman and A. V. Skorokhod, The Theory of Stochastic Processes. Ⅱ, Classics in Mathematics, Springer-Verlag, Berlin, 2004, Translated from the Russian by S. Kotz, Reprint of the 1975 edition. doi: 10.1007/978-3-642-61921-2. Google Scholar [15] D. Helbing, A fluid dynamic model for the movement of pedestrians, Complex Systems, 6 (1992), 391-415, arXiv: cond-mat/9805213. Google Scholar [16] D. Helbing and P. Molnár, Social force model for pedestrian dynamics, Physical Review E, 51 (1998), 4282-4286, arXiv: cond-mat/9805244. doi: 10.1103/PhysRevE.51.4282. Google Scholar [17] R. L. Hughes, A continuum theory for the flow of pedestrians, Transportation Research Part B: Methodological, 36 (2002), 507-535. doi: 10.1016/S0191-2615(01)00015-7. Google Scholar [18] P.-E. Jabin, Macroscopic limit of Vlasov type equations with friction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 651-672. doi: 10.1016/S0294-1449(00)00118-9. Google Scholar [19] P.-E. Jabin, Various levels of models for aerosols, Math. Models Methods Appl. Sci., 12 (2002), 903-919. doi: 10.1142/S0218202502001957. Google Scholar [20] A. Jüngel, Transport Equations for Semiconductors, vol. 773 of Lecture Notes in Physics, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89526-8. Google Scholar [21] A. Klar, F. Schneider and O. Tse, Approximate models for stochastic dynamic systems with velocities on the sphere and associated fokker-planck equations, Kinetic and Related Models, 7 (2014), 509-529. doi: 10.3934/krm.2014.7.509. Google Scholar [22] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253. Google Scholar [23] B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738. doi: 10.1007/s00205-010-0366-y. Google Scholar [24] B. Piccoli and A. Tosin, Pedestrian flows in bounded domains with obstacles, Contin. Mech. Thermodyn., 21 (2009), 85-107. doi: 10.1007/s00161-009-0100-x. Google Scholar [25] M. Schultz, Stochastic transition model for pedestrian dynamics, in Pedestrian and Evacuation Dynamics 2012, Springer International Publishing, (2013), 971-985, arXiv: 1210.5554. doi: 10.1007/978-3-319-02447-9_81. Google Scholar [26] A. Tordeux and A. Schadschneider, A stochastic optimal velocity model for pedestrian flow, in Parallel Processing and Applied Mathematics, Springer International Publishing, 9574 (2016), 528-538. doi: 10.1007/978-3-319-32152-3_49. Google Scholar [27] A. Tordeux and A. Schadschneider, White and relaxed noises in optimal velocity models for pedestrian flow with stop-and-go waves, Journal of Physics A: Mathematical and Theoretical, 49 (2016), 185101, 16pp. doi: 10.1088/1751-8113/49/18/185101. Google Scholar
Velocity vector at the boundary
Overview of the deterministic and stochastic model hierarchy equations
Densities at different times: $u_{ij}^{\text{Mic}, n}$ for the microscopic and $u_{ij}^{\text{Mac}, n}$ for the macroscopic model
Mass balances at $x = -1$ and $x = 0$
$L^1$ and $L^2$ error
Densities for $\lambda_1$ at different times: $u_{ij}^{\text{Mic}, n}$ for the microscopic and $u_{ij}^{\text{Mac}, n}$ for the macroscopic model
Densities for $\lambda_2$ at different times: $u_{ij}^{\text{Mic}, n}$ for the microscopic and $u_{ij}^{\text{Mac}, n}$ for the macroscopic model
Mass balances at $x = 1$
$L^1$ and $L^2$ errors
Numerical error and EOOC for the first example
 $\mathsf{err}$ EOOC $\Delta x = {}^{1}\!\!\diagup\!\!{}_{5}\;$ 0.3251 - $\Delta x = {}^{1}\!\!\diagup\!\!{}_{10}\;$ 0.1755 0.8897 $\Delta x = {}^{1}\!\!\diagup\!\!{}_{20}\;$ 0.0717 1.2919
 $\mathsf{err}$ EOOC $\Delta x = {}^{1}\!\!\diagup\!\!{}_{5}\;$ 0.3251 - $\Delta x = {}^{1}\!\!\diagup\!\!{}_{10}\;$ 0.1755 0.8897 $\Delta x = {}^{1}\!\!\diagup\!\!{}_{20}\;$ 0.0717 1.2919
Numerical error and EOOC for the second example with rate function $\lambda_1$
 $\mathsf{err}$ EOOC $\displaystyle \Delta x = {}^{1}\!\!\diagup\!\!{}_{5}\;$ 0.4457 - $\displaystyle \Delta x = {}^{1}\!\!\diagup\!\!{}_{10}\;$ 0.2215 1.0085 $\displaystyle \Delta x = {}^{1}\!\!\diagup\!\!{}_{20}\;$ 0.0889 1.3170
 $\mathsf{err}$ EOOC $\displaystyle \Delta x = {}^{1}\!\!\diagup\!\!{}_{5}\;$ 0.4457 - $\displaystyle \Delta x = {}^{1}\!\!\diagup\!\!{}_{10}\;$ 0.2215 1.0085 $\displaystyle \Delta x = {}^{1}\!\!\diagup\!\!{}_{20}\;$ 0.0889 1.3170
Numerical error and EOOC for the second example with rate function $\lambda_2$
 $\mathsf{err}$ EOOC $\displaystyle \Delta x = {}^{1}\!\!\diagup\!\!{}_{5}\;$ 0.5203 - $\displaystyle \Delta x = {}^{1}\!\!\diagup\!\!{}_{10}\;$ 0.2873 0.8567 $\displaystyle \Delta x = {}^{1}\!\!\diagup\!\!{}_{20}\;$ 0.1153 1.3176
 $\mathsf{err}$ EOOC $\displaystyle \Delta x = {}^{1}\!\!\diagup\!\!{}_{5}\;$ 0.5203 - $\displaystyle \Delta x = {}^{1}\!\!\diagup\!\!{}_{10}\;$ 0.2873 0.8567 $\displaystyle \Delta x = {}^{1}\!\!\diagup\!\!{}_{20}\;$ 0.1153 1.3176
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