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

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

Simone Göttlich , , Stephan Knapp and  Peter Schillen

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.

Keywords: Interacting particle system, stochastic processes, mean field equations, hydrodynamic limit, macroscopic pedestrian model, numerical simulations.
Mathematics Subject Classification: Primary: 90B20, 65Cxx, 35L60.
Citation: Simone Göttlich, Stephan Knapp, Peter Schillen. A pedestrian flow model with stochastic velocities: Microscopic and macroscopic approaches. Kinetic and Related Models, 2018, 11 (6) : 1333-1358. doi: 10.3934/krm.2018052
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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

D. Helbing, A fluid dynamic model for the movement of pedestrians, Complex Systems, 6 (1992), 391-415, arXiv: cond-mat/9805213.

[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.

[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.

[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.

[19]

P.-E. Jabin, Various levels of models for aerosols, Math. Models Methods Appl. Sci., 12 (2002), 903-919.  doi: 10.1142/S0218202502001957.

[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.

[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.

[22]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253.

[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.

[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.

[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.

[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.

[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.

show all references

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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

D. Helbing, A fluid dynamic model for the movement of pedestrians, Complex Systems, 6 (1992), 391-415, arXiv: cond-mat/9805213.

[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.

[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.

[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.

[19]

P.-E. Jabin, Various levels of models for aerosols, Math. Models Methods Appl. Sci., 12 (2002), 903-919.  doi: 10.1142/S0218202502001957.

[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.

[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.

[22]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253.

[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.

[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.

[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.

[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.

[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.

Figure 1.  Velocity vector at the boundary
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Figure 2.  Overview of the deterministic and stochastic model hierarchy equations
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Figure 3.  Densities at different times: $u_{ij}^{\text{Mic}, n}$ for the microscopic and $u_{ij}^{\text{Mac}, n}$ for the macroscopic model
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Figure 4.  Mass balances at $x = -1$ and $x = 0$
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Figure 5.  $L^1$ and $L^2$ error
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Figure 6.  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
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Figure 7.  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
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Figure 8.  Mass balances at $x = 1$
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Figure 9.  $L^1$ and $L^2$ errors
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Table 1.  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
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$ \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
Table 2.  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
Table Options

Download as excel

$\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
Table 3.  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
Table Options

Download as excel

$ \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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