September  2011, 6(3): 351-381. doi: 10.3934/nhm.2011.6.351

Two-way multi-lane traffic model for pedestrians in corridors

1. 

1-University Paris-Sud, Laboratory of Theoretical Physics, Batiment 210, F-91405 ORSAY Cedex, France

2. 

1-Université de Toulouse; UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse, F-31062 Toulouse

3. 

5-Department of Mathematics, University of Maryland, College Park, MD 20742-4015, United States

Received  December 2010 Revised  April 2011 Published  August 2011

We extend the Aw-Rascle macroscopic model of car traffic into a two-way multi-lane model of pedestrian traffic. Within this model, we propose a technique for the handling of the congestion constraint, i.e. the fact that the pedestrian density cannot exceed a maximal density corresponding to contact between pedestrians. In a first step, we propose a singularly perturbed pressure relation which models the fact that the pedestrian velocity is considerably reduced, if not blocked, at congestion. In a second step, we carry over the singular limit into the model and show that abrupt transitions between compressible flow (in the uncongested regions) to incompressible flow (in congested regions) occur. We also investigate the hyperbolicity of the two-way models and show that they can lose their hyperbolicity in some cases. We study a diffusive correction of these models and discuss the characteristic time and length scales of the instability.
Citation: Cécile Appert-Rolland, Pierre Degond, Sébastien Motsch. Two-way multi-lane traffic model for pedestrians in corridors. Networks & Heterogeneous Media, 2011, 6 (3) : 351-381. doi: 10.3934/nhm.2011.6.351
References:
[1]

Proceedings of the IEEE ITSC 2006, 2006 IEEE Intelligent Transportation Systems Conference, Toronto, Canada, September, (2006), 17-20. Google Scholar

[2]

SIAM J. Appl. Math., 63 (2002), 259-278. doi: 10.1137/S0036139900380955.  Google Scholar

[3]

SIAM J. Appl. Math., 60 (2000), 916-938. doi: 10.1137/S0036139997332099.  Google Scholar

[4]

Math. Models Methods Appl. Sci., 18 (2008), 1317-1345. doi: 10.1142/S0218202508003054.  Google Scholar

[5]

European J. Appl. Math., 14 (2003), 587-612. doi: 10.1017/S0956792503005266.  Google Scholar

[6]

Arch. Rat. Mech. Anal., 187 (2008), 185-220. doi: 10.1007/s00205-007-0061-9.  Google Scholar

[7]

Math. Models Methods Appl. Sci., 18 (2008), 1269-1298. doi: 10.1142/S0218202508003030.  Google Scholar

[8]

J. Nonlinear Sci., 10 (2000), 639-660. doi: 10.1007/s003320010006.  Google Scholar

[9]

Physica A, 295 (2001), 507-525, arXiv:cond-mat/0102397. doi: 10.1016/S0378-4371(01)00141-8.  Google Scholar

[10]

SIAM J. Sci. Comput., 29 (2007), 539-555. doi: 10.1137/050641211.  Google Scholar

[11]

Math. Methods Appl. Sci., 28 (2005), 1553-1567. doi: 10.1002/mma.624.  Google Scholar

[12]

Transp. Res. B, 29 (1995), 277-286. doi: 10.1016/0191-2615(95)00007-Z.  Google Scholar

[13]

Kinet. Relat. Models, 1 (2008), 279-293. doi: 10.3934/krm.2008.1.279.  Google Scholar

[14]

P. Degond, J. Hua and L. Navoret, Numerical simulations of the Euler system with congestion constraint,, preprint, ().   Google Scholar

[15]

Commun. Comput. Phys., 10 (2011), 1-31, arXiv:0908.1929.  Google Scholar

[16]

Physica A, 387 (2008), 580-586. doi: 10.1016/j.physa.2007.10.001.  Google Scholar

[17]

ACM SIGGRAPH/Eurographics Symposium on Computer Animation (SCA), (2009), 177-187. Google Scholar

[18]

Behavioral Science, 36 (1991), 298-310. doi: 10.1002/bs.3830360405.  Google Scholar

[19]

Complex Systems, 6 (1992), 391-415.  Google Scholar

[20]

Physical Review E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282.  Google Scholar

[21]

Proceedings of the International Conference held in Berlin, September 24-28, 1995, Edited by Frank Schweitzer. Gordon and Breach Science Publishers, Amsterdam, 1997.  Google Scholar

[22]

Transportation Research, 8 (1974), 509-515. doi: 10.1016/0041-1647(74)90027-6.  Google Scholar

[23]

Optimal Control Appl. Methods, 24 (2003), 153-172. doi: 10.1002/oca.727.  Google Scholar

[24]

Transportation Research B, 36 (2002), 507-535. doi: 10.1016/S0191-2615(01)00015-7.  Google Scholar

[25]

Ann. Rev. Fluid Mech., 35 (2003), 169-182. doi: 10.1146/annurev.fluid.35.101101.161136.  Google Scholar

[26]

J. Comput. Phys., 160 (2000), 240-282. Google Scholar

[27]

Cambridge Texts in Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[28]

Proc. Roy. Soc., A229 (1955), 281-345. Google Scholar

[29]

Math. Models Methods Appl. Sci., 20 (2010), 1787-1821. doi: 10.1142/S0218202510004799.  Google Scholar

[30]

C. R. Acad. Sci. Paris, 346 (2008), 1245-1250.  Google Scholar

[31]

IEICE Transp. Inf. & Syst., E87-D (2004), 726-732, arXiv:cond-mat/0306262. Google Scholar

[32]

SIGGRAPH '10, 2010. Google Scholar

[33]

Eurographics, 26 (2007), 665-674. Google Scholar

[34]

Pedigree team, Pedestrian flow measurements and analysis in an annular setup,, in preparation., ().   Google Scholar

[35]

SCA '09: Proceedings of the 2009 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, (2009), 189-198. Google Scholar

[36]

Contin. Mech. Thermodyn., 21 (2009), 85-107, arXiv:0812.4390.  Google Scholar

[37]

Arch. Ration. Mech. Anal., 199 (2011), 707-738, arXiv:0811.3383.  Google Scholar

[38]

Proceedings of Game Developers Conference 1999, San Jose, California, (1999), 763-782. Google Scholar

[39]

Phys. Rev. E, 59 (1999), 6328-6339, arXiv:cond-mat/9906430. doi: 10.1103/PhysRevE.59.6328.  Google Scholar

[40]

Int. Journal on Robotics Research, 27 (2008), 1274-1294. doi: 10.1177/0278364908097581.  Google Scholar

[41]

Physica A, 375 (2007), 668-678. doi: 10.1016/j.physa.2006.09.028.  Google Scholar

[42]

Transportation Res. B, 36 (2002), 275-290. doi: 10.1016/S0191-2615(00)00050-3.  Google Scholar

show all references

References:
[1]

Proceedings of the IEEE ITSC 2006, 2006 IEEE Intelligent Transportation Systems Conference, Toronto, Canada, September, (2006), 17-20. Google Scholar

[2]

SIAM J. Appl. Math., 63 (2002), 259-278. doi: 10.1137/S0036139900380955.  Google Scholar

[3]

SIAM J. Appl. Math., 60 (2000), 916-938. doi: 10.1137/S0036139997332099.  Google Scholar

[4]

Math. Models Methods Appl. Sci., 18 (2008), 1317-1345. doi: 10.1142/S0218202508003054.  Google Scholar

[5]

European J. Appl. Math., 14 (2003), 587-612. doi: 10.1017/S0956792503005266.  Google Scholar

[6]

Arch. Rat. Mech. Anal., 187 (2008), 185-220. doi: 10.1007/s00205-007-0061-9.  Google Scholar

[7]

Math. Models Methods Appl. Sci., 18 (2008), 1269-1298. doi: 10.1142/S0218202508003030.  Google Scholar

[8]

J. Nonlinear Sci., 10 (2000), 639-660. doi: 10.1007/s003320010006.  Google Scholar

[9]

Physica A, 295 (2001), 507-525, arXiv:cond-mat/0102397. doi: 10.1016/S0378-4371(01)00141-8.  Google Scholar

[10]

SIAM J. Sci. Comput., 29 (2007), 539-555. doi: 10.1137/050641211.  Google Scholar

[11]

Math. Methods Appl. Sci., 28 (2005), 1553-1567. doi: 10.1002/mma.624.  Google Scholar

[12]

Transp. Res. B, 29 (1995), 277-286. doi: 10.1016/0191-2615(95)00007-Z.  Google Scholar

[13]

Kinet. Relat. Models, 1 (2008), 279-293. doi: 10.3934/krm.2008.1.279.  Google Scholar

[14]

P. Degond, J. Hua and L. Navoret, Numerical simulations of the Euler system with congestion constraint,, preprint, ().   Google Scholar

[15]

Commun. Comput. Phys., 10 (2011), 1-31, arXiv:0908.1929.  Google Scholar

[16]

Physica A, 387 (2008), 580-586. doi: 10.1016/j.physa.2007.10.001.  Google Scholar

[17]

ACM SIGGRAPH/Eurographics Symposium on Computer Animation (SCA), (2009), 177-187. Google Scholar

[18]

Behavioral Science, 36 (1991), 298-310. doi: 10.1002/bs.3830360405.  Google Scholar

[19]

Complex Systems, 6 (1992), 391-415.  Google Scholar

[20]

Physical Review E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282.  Google Scholar

[21]

Proceedings of the International Conference held in Berlin, September 24-28, 1995, Edited by Frank Schweitzer. Gordon and Breach Science Publishers, Amsterdam, 1997.  Google Scholar

[22]

Transportation Research, 8 (1974), 509-515. doi: 10.1016/0041-1647(74)90027-6.  Google Scholar

[23]

Optimal Control Appl. Methods, 24 (2003), 153-172. doi: 10.1002/oca.727.  Google Scholar

[24]

Transportation Research B, 36 (2002), 507-535. doi: 10.1016/S0191-2615(01)00015-7.  Google Scholar

[25]

Ann. Rev. Fluid Mech., 35 (2003), 169-182. doi: 10.1146/annurev.fluid.35.101101.161136.  Google Scholar

[26]

J. Comput. Phys., 160 (2000), 240-282. Google Scholar

[27]

Cambridge Texts in Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[28]

Proc. Roy. Soc., A229 (1955), 281-345. Google Scholar

[29]

Math. Models Methods Appl. Sci., 20 (2010), 1787-1821. doi: 10.1142/S0218202510004799.  Google Scholar

[30]

C. R. Acad. Sci. Paris, 346 (2008), 1245-1250.  Google Scholar

[31]

IEICE Transp. Inf. & Syst., E87-D (2004), 726-732, arXiv:cond-mat/0306262. Google Scholar

[32]

SIGGRAPH '10, 2010. Google Scholar

[33]

Eurographics, 26 (2007), 665-674. Google Scholar

[34]

Pedigree team, Pedestrian flow measurements and analysis in an annular setup,, in preparation., ().   Google Scholar

[35]

SCA '09: Proceedings of the 2009 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, (2009), 189-198. Google Scholar

[36]

Contin. Mech. Thermodyn., 21 (2009), 85-107, arXiv:0812.4390.  Google Scholar

[37]

Arch. Ration. Mech. Anal., 199 (2011), 707-738, arXiv:0811.3383.  Google Scholar

[38]

Proceedings of Game Developers Conference 1999, San Jose, California, (1999), 763-782. Google Scholar

[39]

Phys. Rev. E, 59 (1999), 6328-6339, arXiv:cond-mat/9906430. doi: 10.1103/PhysRevE.59.6328.  Google Scholar

[40]

Int. Journal on Robotics Research, 27 (2008), 1274-1294. doi: 10.1177/0278364908097581.  Google Scholar

[41]

Physica A, 375 (2007), 668-678. doi: 10.1016/j.physa.2006.09.028.  Google Scholar

[42]

Transportation Res. B, 36 (2002), 275-290. doi: 10.1016/S0191-2615(00)00050-3.  Google Scholar

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