2014, 7(3): 363-377. doi: 10.3934/dcdss.2014.7.363

Interaction of road networks and pedestrian motion at crosswalks

1. 

Department of Mathematics, University of Kaiserslautern, 67663 Kaiserslautern, Germany, Germany

Received  May 2013 Revised  July 2013 Published  January 2014

In the present paper we discuss the coupling of traffic flow with pedestrian motion. First we review the coupling of the Lighthill-Whitham model for road traffic and the Hughes pedestrian model as presented in [5]. In different numerical examples we investigate the mutual interaction of both dynamics. A special focus is given on the possible placement of crosswalks to facilitate the passage for the pedestrians over a crowded street.
Citation: Raul Borsche, Anne Meurer. Interaction of road networks and pedestrian motion at crosswalks. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 363-377. doi: 10.3934/dcdss.2014.7.363
References:
[1]

A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models,, SIAM J. Appl. Math., 63 (2002), 259. doi: 10.1137/S0036139900380955.

[2]

A. Aw and M. Rascle, Resurrection of second order models of traffic flow,, SIAM J. Appl. Math., 60 (2000), 916. doi: 10.1137/S0036139997332099.

[3]

N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives,, SIAM Review, 53 (2011), 409. doi: 10.1137/090746677.

[4]

F. Berthelin, P. Degond, M. Delitla and M. Rascle, A model for the formation and evolution of traffic jams,, Arch. Rat. Mech. Anal., 187 (2008), 185. doi: 10.1007/s00205-007-0061-9.

[5]

R. Borsche, A. Klar, S. Kühn and A. Meurer, Coupling traffic flow networks to pedestrian motion,, Math. Models Methods Appl. Sci., 24 (2014), 213. doi: 10.1142/S0218202513400113.

[6]

G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network,, SIAM J. Math. Anal., 3 (2005), 1862. doi: 10.1137/S0036141004402683.

[7]

R. Colombo, M. Garavello and M. Lecureux-Mercier, A class of non-local models for pedestrian traffic,, MMMAS, 22 (2012). doi: 10.1142/S0218202511500230.

[8]

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. doi: 10.1016/j.jde.2010.10.015.

[9]

P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions,, Math. Comput. Modelling, 44 (2006), 287. doi: 10.1016/j.mcm.2006.01.016.

[10]

J. Greenberg, Extension and amplification of the Aw-Rascle model,, SIAM J. Appl. Math., 62 (2001), 729. doi: 10.1137/S0036139900378657.

[11]

A. Fuegenschuh, S. Goettlich, M. Herty, A. Klar and A. Martin, A discrete optimization approach to large scale supply networks based on partial differential equations,, SIAM Scient. Computing, 30 (2008), 1490. doi: 10.1137/060663799.

[12]

D. Helbing, Traffic and related self-driven many-particle systems,, Rev. Modern Phys., 73 (2001), 1067. doi: 10.1103/RevModPhys.73.1067.

[13]

M. Herty and A. Klar, Modeling, simulation and optimization of traffic flow networks,, SIAM Sci. Comp., 25 (2003), 1066. doi: 10.1137/S106482750241459X.

[14]

H. Holden and N. Risebro, A mathematical model of traffic flow on a network of unidirectional road,, SIAM J. Math. Anal., 4 (1995), 999. doi: 10.1137/S0036141093243289.

[15]

R. L. Hughes, A continuum theory for the flow of pedestrians,, Transp. Res. Part B: Methodological, 36 (2002), 507. doi: 10.1016/S0191-2615(01)00015-7.

[16]

R. L. Hughes, The flow of human crowds,, Annu. Rev. Fluid Mech., 35 (2003), 169. doi: 10.1146/annurev.fluid.35.101101.161136.

[17]

J. Lebacque and M. Khoshyaran, First order macroscopic traffic flow models for networks in the context of dynamic assignment,, in Transportation PlanningState of the Art, (2002). doi: 10.1007/0-306-48220-7_8.

[18]

J. A. Sethian, Fast marching methods,, SIAM Review, 41 (1999), 199. doi: 10.1137/S0036144598347059.

[19]

E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics,, Springer, (2009). doi: 10.1007/b79761.

[20]

G. Whitham, Linear and Nonlinear Waves,, Wiley, (1974).

[21]

M. Zhang, A non-equilibrium traffic flow model devoid of gas-like behavior,, Transp. Res. B, 36 (2002), 275.

show all references

References:
[1]

A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models,, SIAM J. Appl. Math., 63 (2002), 259. doi: 10.1137/S0036139900380955.

[2]

A. Aw and M. Rascle, Resurrection of second order models of traffic flow,, SIAM J. Appl. Math., 60 (2000), 916. doi: 10.1137/S0036139997332099.

[3]

N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives,, SIAM Review, 53 (2011), 409. doi: 10.1137/090746677.

[4]

F. Berthelin, P. Degond, M. Delitla and M. Rascle, A model for the formation and evolution of traffic jams,, Arch. Rat. Mech. Anal., 187 (2008), 185. doi: 10.1007/s00205-007-0061-9.

[5]

R. Borsche, A. Klar, S. Kühn and A. Meurer, Coupling traffic flow networks to pedestrian motion,, Math. Models Methods Appl. Sci., 24 (2014), 213. doi: 10.1142/S0218202513400113.

[6]

G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network,, SIAM J. Math. Anal., 3 (2005), 1862. doi: 10.1137/S0036141004402683.

[7]

R. Colombo, M. Garavello and M. Lecureux-Mercier, A class of non-local models for pedestrian traffic,, MMMAS, 22 (2012). doi: 10.1142/S0218202511500230.

[8]

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. doi: 10.1016/j.jde.2010.10.015.

[9]

P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions,, Math. Comput. Modelling, 44 (2006), 287. doi: 10.1016/j.mcm.2006.01.016.

[10]

J. Greenberg, Extension and amplification of the Aw-Rascle model,, SIAM J. Appl. Math., 62 (2001), 729. doi: 10.1137/S0036139900378657.

[11]

A. Fuegenschuh, S. Goettlich, M. Herty, A. Klar and A. Martin, A discrete optimization approach to large scale supply networks based on partial differential equations,, SIAM Scient. Computing, 30 (2008), 1490. doi: 10.1137/060663799.

[12]

D. Helbing, Traffic and related self-driven many-particle systems,, Rev. Modern Phys., 73 (2001), 1067. doi: 10.1103/RevModPhys.73.1067.

[13]

M. Herty and A. Klar, Modeling, simulation and optimization of traffic flow networks,, SIAM Sci. Comp., 25 (2003), 1066. doi: 10.1137/S106482750241459X.

[14]

H. Holden and N. Risebro, A mathematical model of traffic flow on a network of unidirectional road,, SIAM J. Math. Anal., 4 (1995), 999. doi: 10.1137/S0036141093243289.

[15]

R. L. Hughes, A continuum theory for the flow of pedestrians,, Transp. Res. Part B: Methodological, 36 (2002), 507. doi: 10.1016/S0191-2615(01)00015-7.

[16]

R. L. Hughes, The flow of human crowds,, Annu. Rev. Fluid Mech., 35 (2003), 169. doi: 10.1146/annurev.fluid.35.101101.161136.

[17]

J. Lebacque and M. Khoshyaran, First order macroscopic traffic flow models for networks in the context of dynamic assignment,, in Transportation PlanningState of the Art, (2002). doi: 10.1007/0-306-48220-7_8.

[18]

J. A. Sethian, Fast marching methods,, SIAM Review, 41 (1999), 199. doi: 10.1137/S0036144598347059.

[19]

E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics,, Springer, (2009). doi: 10.1007/b79761.

[20]

G. Whitham, Linear and Nonlinear Waves,, Wiley, (1974).

[21]

M. Zhang, A non-equilibrium traffic flow model devoid of gas-like behavior,, Transp. Res. B, 36 (2002), 275.

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