June  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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

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

[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. Google Scholar

[12]

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

[13]

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

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

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

[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. Google Scholar

[12]

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

[13]

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

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

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