March  2017, 12(1): 93-112. doi: 10.3934/nhm.2017004

A discrete Hughes model for pedestrian flow on graphs

1. 

Dip. di Scienze di Base e Applicate per l'Ingegneria, "Sapienza" Università di Roma, via Scarpa 16, 00161 Roma, Italy

2. 

RICAM, Austrian Academy of Sciences (ÖAW), Altenbergerstr. 69, 4040 Linz, Austria

3. 

Dip. di Matematica, "Sapienza" Università di Roma, P.le Aldo Moro 5, 00185 Roma, Italy

* Corresponding author:Fabio Camilli

Received  April 2016 Revised  December 2016 Published  February 2017

Fund Project: AF is supported the Austrian Academy of Sciences ÖAW via the New Frontiers Group NST-001.

In this paper, we introduce a discrete time-finite state model for pedestrian flow on a graph in the spirit of the Hughes dynamic continuum model. The pedestrians, represented by a density function, move on the graph choosing a route to minimize the instantaneous travel cost to the destination. The density is governed by a conservation law whereas the minimization principle is described by a graph eikonal equation. We show that the discrete model is well-posed and the numerical examples reported confirm the validity of the proposed model and its applicability to describe real situations.

Citation: Fabio Camilli, Adriano Festa, Silvia Tozza. A discrete Hughes model for pedestrian flow on graphs. Networks & Heterogeneous Media, 2017, 12 (1) : 93-112. doi: 10.3934/nhm.2017004
References:
[1]

"Jamarat: Study of Current Conditions and Means of Improvements", Hajj Research Centre, Um Al-Qura University Saudi Arabia, 1984.Google Scholar

[2]

A. AllaM. Falcone and D. Kalise, An efficient policy iteration algorithm for dynamic programming equations, SIAM J. Sci. Comput., 37 (2015), A181-A200. doi: 10.1137/130932284. Google Scholar

[3]

D. Amadori and M.Di Francesco, The one-dimensional Hughes model for pedestrian flow: Riemann-type solutions, Acta Math. Sci., 32 (2012), 259-280. doi: 10.1016/S0252-9602(12)60016-2. Google Scholar

[4]

D. AmadoriP. Goatin and M. D. Rosini, Existence results for Hughes model for pedestrian flows, J. Math. Anal. Appl., 420 (2014), 387-406. doi: 10.1016/j.jmaa.2014.05.072. Google Scholar

[5]

M. Bardi and J. P. Maldonado Lopez, A Dijkstra-type algorithm for dynamic games, Dyn. Games Appl., (2015), 1-4. doi: 10.1007/s13235-015-0156-0. Google Scholar

[6]

N. Bellomo and C. Dogbé, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463. doi: 10.1137/090746677. Google Scholar

[7]

M. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: theoretical study, Netw. Heterog. Media, 9 (2014), 519-552. doi: 10.3934/nhm.2014.9.519. Google Scholar

[8]

F. Camilli and C. Marchi, A comparison among various notions of viscosity solutions for Hamilton-Jacobi equations on networks, J. Math. Anal. Appl., 407 (2013), 112-118. doi: 10.1016/j.jmaa.2013.05.015. Google Scholar

[9]

F. Camilli and C. Marchi, Staionary mean field games systems defined on networks, SIAM J. Cont. Optim., 54 (2016), 1085-1103. doi: 10.1137/15M1022082. Google Scholar

[10]

F. CamilliA. Festa and D. Schieborn, An approximation scheme for a Hamilton-Jacobi equation defined on a network, Appl. Numer. Math., 73 (2013), 33-47. doi: 10.1016/j.apnum.2013.05.003. Google Scholar

[11]

E. CarliniA. FestaF. J. Silva and M. T. Wolfram, A Semi-Lagrangian scheme for a modified version of the Hughes model for pedestrian flow, Dyn. Games Appl., (2016), 1-23. doi: 10.1007/s13235-016-0202-6. Google Scholar

[12]

G. CostesequeJ. P. Lebacque and R. Monneau, A convergent scheme for Hamilton-Jacobi equations on a junction: Application to traffic, Numer. Math., 129 (2015), 405-447. doi: 10.1007/s00211-014-0643-z. Google Scholar

[13]

E. Cristiani and F. S. Priuli, A destination-preserving model for simulating Wardrop equilibria in traffic flow on networks, Netw. Heterog. Media, 10 (2015), 857-876. doi: 10.3934/nhm.2015.10.857. Google Scholar

[14]

M. Di FrancescoP. A. MarkowichJ. 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. Google Scholar

[15]

Z. FangQ. LiQ. LiL. D. Han and D. Wang, A proposed pedestrian waiting-time model for improving space-time use efficiency in stadium evacuation scenarios, Build. Environ., 46 (2011), 1774-1784. doi: 10.1016/j.buildenv.2011.02.005. Google Scholar

[16]

M. Garavello and B. Piccoli, "Traffic Flow on Networks" AIMS Series on Applied Mathematics, Vol. 1, American Institute of Mathematical Sciences, 2006. Google Scholar

[17]

L. HuangS. C. WongM. ZhangC. W. Shu and W. H. K. Lam, Revisiting Hughes dynamic continuum model for pedestrian flow and the development of an efficient solution algorithm, Transportat. Res. B-Meth., 43 (2009), 127-141. doi: 10.1016/j.trb.2008.06.003. Google Scholar

[18]

R. L. Hughes, The flow of large crowds of pedestrians, Math. Comput. Simulat., 53 (2000), 367-370. doi: 10.1016/S0378-4754(00)00228-7. Google Scholar

[19]

R. L. Hughes, A continuum theory for the flow of pedestrians, Transport. Res. B-Meth., 36 (2002), 507-535. doi: 10.1016/S0191-2615(01)00015-7. Google Scholar

[20]

R. L. Hughes, The flow of human crowds, Annu. rev. fluid mech., 35 (2003), 169-182. doi: 10.1146/annurev.fluid.35.101101.161136. Google Scholar

[21]

P.-L. Lions and P. E. Souganidis, Viscosity solutions for junctions: Well posedness and stability, Rend. Lincei Mat. Appl., 27 (2016), 535-545. doi: 10.4171/RLM/747. Google Scholar

[22]

J. ManfrediA. Oberman and A. Sviridov, Nonlinear elliptic partial differential equations and p-harmonic functions on graphs, Differ. Integral Equ., 28 (2015), 79-102. Google Scholar

[23]

M. Puterman and S. L. Brumelle, On the convergence of policy iteration in stationary dynamic programming, Math. Oper. Res., 4 (1979), 60-69. doi: 10.1287/moor.4.1.60. Google Scholar

[24]

J. D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38 (2000), 681-698. doi: 10.1137/S0036142999363668. Google Scholar

[25]

A. TreuilleS. Cooper and Z. Popovîc, Continuum crowds, ACM Trans. Graph., 25 (2006), 1160-1168. doi: 10.1145/1179352.1142008. Google Scholar

show all references

References:
[1]

"Jamarat: Study of Current Conditions and Means of Improvements", Hajj Research Centre, Um Al-Qura University Saudi Arabia, 1984.Google Scholar

[2]

A. AllaM. Falcone and D. Kalise, An efficient policy iteration algorithm for dynamic programming equations, SIAM J. Sci. Comput., 37 (2015), A181-A200. doi: 10.1137/130932284. Google Scholar

[3]

D. Amadori and M.Di Francesco, The one-dimensional Hughes model for pedestrian flow: Riemann-type solutions, Acta Math. Sci., 32 (2012), 259-280. doi: 10.1016/S0252-9602(12)60016-2. Google Scholar

[4]

D. AmadoriP. Goatin and M. D. Rosini, Existence results for Hughes model for pedestrian flows, J. Math. Anal. Appl., 420 (2014), 387-406. doi: 10.1016/j.jmaa.2014.05.072. Google Scholar

[5]

M. Bardi and J. P. Maldonado Lopez, A Dijkstra-type algorithm for dynamic games, Dyn. Games Appl., (2015), 1-4. doi: 10.1007/s13235-015-0156-0. Google Scholar

[6]

N. Bellomo and C. Dogbé, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463. doi: 10.1137/090746677. Google Scholar

[7]

M. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: theoretical study, Netw. Heterog. Media, 9 (2014), 519-552. doi: 10.3934/nhm.2014.9.519. Google Scholar

[8]

F. Camilli and C. Marchi, A comparison among various notions of viscosity solutions for Hamilton-Jacobi equations on networks, J. Math. Anal. Appl., 407 (2013), 112-118. doi: 10.1016/j.jmaa.2013.05.015. Google Scholar

[9]

F. Camilli and C. Marchi, Staionary mean field games systems defined on networks, SIAM J. Cont. Optim., 54 (2016), 1085-1103. doi: 10.1137/15M1022082. Google Scholar

[10]

F. CamilliA. Festa and D. Schieborn, An approximation scheme for a Hamilton-Jacobi equation defined on a network, Appl. Numer. Math., 73 (2013), 33-47. doi: 10.1016/j.apnum.2013.05.003. Google Scholar

[11]

E. CarliniA. FestaF. J. Silva and M. T. Wolfram, A Semi-Lagrangian scheme for a modified version of the Hughes model for pedestrian flow, Dyn. Games Appl., (2016), 1-23. doi: 10.1007/s13235-016-0202-6. Google Scholar

[12]

G. CostesequeJ. P. Lebacque and R. Monneau, A convergent scheme for Hamilton-Jacobi equations on a junction: Application to traffic, Numer. Math., 129 (2015), 405-447. doi: 10.1007/s00211-014-0643-z. Google Scholar

[13]

E. Cristiani and F. S. Priuli, A destination-preserving model for simulating Wardrop equilibria in traffic flow on networks, Netw. Heterog. Media, 10 (2015), 857-876. doi: 10.3934/nhm.2015.10.857. Google Scholar

[14]

M. Di FrancescoP. A. MarkowichJ. 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. Google Scholar

[15]

Z. FangQ. LiQ. LiL. D. Han and D. Wang, A proposed pedestrian waiting-time model for improving space-time use efficiency in stadium evacuation scenarios, Build. Environ., 46 (2011), 1774-1784. doi: 10.1016/j.buildenv.2011.02.005. Google Scholar

[16]

M. Garavello and B. Piccoli, "Traffic Flow on Networks" AIMS Series on Applied Mathematics, Vol. 1, American Institute of Mathematical Sciences, 2006. Google Scholar

[17]

L. HuangS. C. WongM. ZhangC. W. Shu and W. H. K. Lam, Revisiting Hughes dynamic continuum model for pedestrian flow and the development of an efficient solution algorithm, Transportat. Res. B-Meth., 43 (2009), 127-141. doi: 10.1016/j.trb.2008.06.003. Google Scholar

[18]

R. L. Hughes, The flow of large crowds of pedestrians, Math. Comput. Simulat., 53 (2000), 367-370. doi: 10.1016/S0378-4754(00)00228-7. Google Scholar

[19]

R. L. Hughes, A continuum theory for the flow of pedestrians, Transport. Res. B-Meth., 36 (2002), 507-535. doi: 10.1016/S0191-2615(01)00015-7. Google Scholar

[20]

R. L. Hughes, The flow of human crowds, Annu. rev. fluid mech., 35 (2003), 169-182. doi: 10.1146/annurev.fluid.35.101101.161136. Google Scholar

[21]

P.-L. Lions and P. E. Souganidis, Viscosity solutions for junctions: Well posedness and stability, Rend. Lincei Mat. Appl., 27 (2016), 535-545. doi: 10.4171/RLM/747. Google Scholar

[22]

J. ManfrediA. Oberman and A. Sviridov, Nonlinear elliptic partial differential equations and p-harmonic functions on graphs, Differ. Integral Equ., 28 (2015), 79-102. Google Scholar

[23]

M. Puterman and S. L. Brumelle, On the convergence of policy iteration in stationary dynamic programming, Math. Oper. Res., 4 (1979), 60-69. doi: 10.1287/moor.4.1.60. Google Scholar

[24]

J. D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38 (2000), 681-698. doi: 10.1137/S0036142999363668. Google Scholar

[25]

A. TreuilleS. Cooper and Z. Popovîc, Continuum crowds, ACM Trans. Graph., 25 (2006), 1160-1168. doi: 10.1145/1179352.1142008. Google Scholar

Figure 1.  Scheme of the network and initial density.
Figure 2.  Test 1 (Dirichlet boundary conditions): density and potential before the first time of interaction.
Figure 3.  Test 1 (Dirichlet boundary conditions): density and potential after the first time of interaction at $(0.2,-0.8)$.
Figure 4.  Test 2 (No-flux boundary conditions): stable configuration obtained for $t>3.5$.
Figure 5.  Test 3 (Dirichlet BCs with diffusion $\epsilon=1$): Density at two different time steps ($t=0.75$ and $t=1.75$).
Figure 6.  The Wuhan Sports Centre (left) and the evacuation network considered in our study (right).
Figure 7.  Initial distribution of density on the graph (up) and drift potential in the initial configuration (down).
Figure 8.  Initial distribution of density on the graph (up) and drift potential in the initial configuration (down).
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