# American Institute of Mathematical Sciences

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 and Heterogeneous Media, 2011, 6 (3) : 351-381. doi: 10.3934/nhm.2011.6.351
##### References:
 [1] S. Al-nasur and P. Kachroo, "A Microscopic-to-Macroscopic Crowd Dynamic Model," Proceedings of the IEEE ITSC 2006, 2006 IEEE Intelligent Transportation Systems Conference, Toronto, Canada, September, (2006), 17-20. [2] A. Aw, A. Klar, A. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278. doi: 10.1137/S0036139900380955. [3] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938. doi: 10.1137/S0036139997332099. [4] N. Bellomo and C. Dogbé, On the modelling crowd dynamics: From scaling to second order hyperbolic macroscopic models, Math. Models Methods Appl. Sci., 18 (2008), 1317-1345. doi: 10.1142/S0218202508003054. [5] S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, European J. Appl. Math., 14 (2003), 587-612. doi: 10.1017/S0956792503005266. [6] F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, Arch. Rat. Mech. Anal., 187 (2008), 185-220. doi: 10.1007/s00205-007-0061-9. [7] F. Berthelin, P. Degond, V. Le Blanc, S. Moutari, J. Royer and M. Rascle, A traffic-flow model with constraints for the modeling of traffic jams, Math. Models Methods Appl. Sci., 18 (2008), 1269-1298. doi: 10.1142/S0218202508003030. [8] F. Bouchut, Y. Brenier, J. Cortes and J. F. Ripoll, A hierachy of models for two-phase flows, J. Nonlinear Sci., 10 (2000), 639-660. doi: 10.1007/s003320010006. [9] C. Burstedde, K. Klauck, A. Schadschneider and J. Zittarz, Simulation of pedestrian dynamics using a 2-dimensional cellular automaton, Physica A, 295 (2001), 507-525, arXiv:cond-mat/0102397. doi: 10.1016/S0378-4371(01)00141-8. [10] C. Chalons, Numerical approximation of a macroscopic model of pedestrian flows, SIAM J. Sci. Comput., 29 (2007), 539-555. doi: 10.1137/050641211. [11] R. M. Colombo and M. D. Rosini, Pedestrian flows and nonclassical shocks, Math. Methods Appl. Sci., 28 (2005), 1553-1567. doi: 10.1002/mma.624. [12] C. Daganzo, Requiem for second order fluid approximations of traffic flow, Transp. Res. B, 29 (1995), 277-286. doi: 10.1016/0191-2615(95)00007-Z. [13] P. Degond and M. Delitala, Modelling and simulation of vehicular traffic jam formation, Kinet. Relat. Models, 1 (2008), 279-293. doi: 10.3934/krm.2008.1.279. [14] P. Degond, J. Hua and L. Navoret, Numerical simulations of the Euler system with congestion constraint, preprint, arXiv:1008.4045. [15] P. Degond and M. Tang, All speed scheme for the low Mach number limit of the isentropic Euler equations, Commun. Comput. Phys., 10 (2011), 1-31, arXiv:0908.1929. [16] R. Y. Guo and H. J. Huang, A mobile lattice gas model for simulating pedestrian evacuation, Physica A, 387 (2008), 580-586. doi: 10.1016/j.physa.2007.10.001. [17] S. J. Guy, J. Chhugani, C. Kim, N. Satish, M. C. Lin, D. Manocha and P. Dubey, "Clearpath: Highly Parallel Collision Avoidance for Multi-Agent Simulation," ACM SIGGRAPH/Eurographics Symposium on Computer Animation (SCA), (2009), 177-187. [18] D. Helbing, A mathematical model for the behavior of pedestrians, Behavioral Science, 36 (1991), 298-310. doi: 10.1002/bs.3830360405. [19] D. Helbing, A fluid-dynamic model for the movement of pedestrians, Complex Systems, 6 (1992), 391-415. [20] D. Helbing and P. Molnàr, Social force model for pedestrian dynamics, Physical Review E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282. [21] D. Helbing and P. Molnàr, Self-organization of complex structures. From individual to collective dynamics, Proceedings of the International Conference held in Berlin, September 24-28, 1995, Edited by Frank Schweitzer. Gordon and Breach Science Publishers, Amsterdam, 1997. [22] L. F. Henderson, On the fluid mechanics of human crowd motion, Transportation Research, 8 (1974), 509-515. doi: 10.1016/0041-1647(74)90027-6. [23] S. Hoogendoorn and P. H. L. Bovy, Simulation of pedestrian flows by optimal control and differential games, Optimal Control Appl. Methods, 24 (2003), 153-172. doi: 10.1002/oca.727. [24] R. L. Hughes, A continuum theory for the flow of pedestrians, Transportation Research B, 36 (2002), 507-535. doi: 10.1016/S0191-2615(01)00015-7. [25] R. L. Hughes, The flow of human crowds, Ann. Rev. Fluid Mech., 35 (2003), 169-182. doi: 10.1146/annurev.fluid.35.101101.161136. [26] A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160 (2000), 240-282. [27] R. J. LeVeque, "Finite Volume Methods for Hyperbolic Problems," Cambridge Texts in Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253. [28] M. J. Lighthill and J. B. Whitham, On kinematic waves. I: Flow movement in long rivers. II: A theory of traffic flow on long crowded roads, Proc. Roy. Soc., A229 (1955), 281-345. [29] B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type, Math. Models Methods Appl. Sci., 20 (2010), 1787-1821. doi: 10.1142/S0218202510004799. [30] B. Maury and J. Venel, A mathematical framework for a crowd motion model, C. R. Acad. Sci. Paris, 346 (2008), 1245-1250. [31] K. Nishinari, A. Kirchner, A. Namazi and A. Schadschneider, Extended floor field CA model for evacuation dynamics, IEICE Transp. Inf. & Syst., E87-D (2004), 726-732, arXiv:cond-mat/0306262. [32] J. Ondřej, J. Pettré, A.-H. Olivier and S. Donikian, "A Synthetic-Vision Based Steering Approach for Crowd Simulation," SIGGRAPH '10, 2010. [33] S. Paris, J. Pettré and S. Donikian, Pedestrian reactive navigation for crowd simulation: A predictive approach, Eurographics, 26 (2007), 665-674. [34] Pedigree team, Pedestrian flow measurements and analysis in an annular setup, in preparation. [35] J. Pettré, J. Ondřej, A.-H. Olivier, A. Cretual and S. Donikian, "Experiment-Based Modeling, Simulation and Validation of Interactions Between Virtual Walkers," SCA '09: Proceedings of the 2009 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, (2009), 189-198. [36] B. Piccoli and A. Tosin, Pedestrian flows in bounded domains with obstacles, Contin. Mech. Thermodyn., 21 (2009), 85-107, arXiv:0812.4390. [37] B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738, arXiv:0811.3383. [38] C. W. Reynolds, "Steering Behaviors for Autonomous Characters," Proceedings of Game Developers Conference 1999, San Jose, California, (1999), 763-782. [39] V. Shvetsov and D. Helbing, Macroscopic dynamics of multi-lane traffic, Phys. Rev. E, 59 (1999), 6328-6339, arXiv:cond-mat/9906430. doi: 10.1103/PhysRevE.59.6328. [40] J. van den Berg and H. Overmars, Planning time-minimal safe paths amidst unpredictably moving obstacles, Int. Journal on Robotics Research, 27 (2008), 1274-1294. doi: 10.1177/0278364908097581. [41] W. G. Weng, S. F. Shena, H. Y. Yuana and W. C. Fana, A behavior-based model for pedestrian counter flow, Physica A, 375 (2007), 668-678. doi: 10.1016/j.physa.2006.09.028. [42] M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Res. B, 36 (2002), 275-290. doi: 10.1016/S0191-2615(00)00050-3.

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##### References:
 [1] S. Al-nasur and P. Kachroo, "A Microscopic-to-Macroscopic Crowd Dynamic Model," Proceedings of the IEEE ITSC 2006, 2006 IEEE Intelligent Transportation Systems Conference, Toronto, Canada, September, (2006), 17-20. [2] A. Aw, A. Klar, A. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278. doi: 10.1137/S0036139900380955. [3] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938. doi: 10.1137/S0036139997332099. [4] N. Bellomo and C. Dogbé, On the modelling crowd dynamics: From scaling to second order hyperbolic macroscopic models, Math. Models Methods Appl. Sci., 18 (2008), 1317-1345. doi: 10.1142/S0218202508003054. [5] S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, European J. Appl. Math., 14 (2003), 587-612. doi: 10.1017/S0956792503005266. [6] F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, Arch. Rat. Mech. Anal., 187 (2008), 185-220. doi: 10.1007/s00205-007-0061-9. [7] F. Berthelin, P. Degond, V. Le Blanc, S. Moutari, J. Royer and M. Rascle, A traffic-flow model with constraints for the modeling of traffic jams, Math. Models Methods Appl. Sci., 18 (2008), 1269-1298. doi: 10.1142/S0218202508003030. [8] F. Bouchut, Y. Brenier, J. Cortes and J. F. Ripoll, A hierachy of models for two-phase flows, J. Nonlinear Sci., 10 (2000), 639-660. doi: 10.1007/s003320010006. [9] C. Burstedde, K. Klauck, A. Schadschneider and J. Zittarz, Simulation of pedestrian dynamics using a 2-dimensional cellular automaton, Physica A, 295 (2001), 507-525, arXiv:cond-mat/0102397. doi: 10.1016/S0378-4371(01)00141-8. [10] C. Chalons, Numerical approximation of a macroscopic model of pedestrian flows, SIAM J. Sci. Comput., 29 (2007), 539-555. doi: 10.1137/050641211. [11] R. M. Colombo and M. D. Rosini, Pedestrian flows and nonclassical shocks, Math. Methods Appl. Sci., 28 (2005), 1553-1567. doi: 10.1002/mma.624. [12] C. Daganzo, Requiem for second order fluid approximations of traffic flow, Transp. Res. B, 29 (1995), 277-286. doi: 10.1016/0191-2615(95)00007-Z. [13] P. Degond and M. Delitala, Modelling and simulation of vehicular traffic jam formation, Kinet. Relat. Models, 1 (2008), 279-293. doi: 10.3934/krm.2008.1.279. [14] P. Degond, J. Hua and L. Navoret, Numerical simulations of the Euler system with congestion constraint, preprint, arXiv:1008.4045. [15] P. Degond and M. Tang, All speed scheme for the low Mach number limit of the isentropic Euler equations, Commun. Comput. Phys., 10 (2011), 1-31, arXiv:0908.1929. [16] R. Y. Guo and H. J. Huang, A mobile lattice gas model for simulating pedestrian evacuation, Physica A, 387 (2008), 580-586. doi: 10.1016/j.physa.2007.10.001. [17] S. J. Guy, J. Chhugani, C. Kim, N. Satish, M. C. Lin, D. Manocha and P. Dubey, "Clearpath: Highly Parallel Collision Avoidance for Multi-Agent Simulation," ACM SIGGRAPH/Eurographics Symposium on Computer Animation (SCA), (2009), 177-187. [18] D. Helbing, A mathematical model for the behavior of pedestrians, Behavioral Science, 36 (1991), 298-310. doi: 10.1002/bs.3830360405. [19] D. Helbing, A fluid-dynamic model for the movement of pedestrians, Complex Systems, 6 (1992), 391-415. [20] D. Helbing and P. Molnàr, Social force model for pedestrian dynamics, Physical Review E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282. [21] D. Helbing and P. Molnàr, Self-organization of complex structures. From individual to collective dynamics, Proceedings of the International Conference held in Berlin, September 24-28, 1995, Edited by Frank Schweitzer. Gordon and Breach Science Publishers, Amsterdam, 1997. [22] L. F. Henderson, On the fluid mechanics of human crowd motion, Transportation Research, 8 (1974), 509-515. doi: 10.1016/0041-1647(74)90027-6. [23] S. Hoogendoorn and P. H. L. Bovy, Simulation of pedestrian flows by optimal control and differential games, Optimal Control Appl. Methods, 24 (2003), 153-172. doi: 10.1002/oca.727. [24] R. L. Hughes, A continuum theory for the flow of pedestrians, Transportation Research B, 36 (2002), 507-535. doi: 10.1016/S0191-2615(01)00015-7. [25] R. L. Hughes, The flow of human crowds, Ann. Rev. Fluid Mech., 35 (2003), 169-182. doi: 10.1146/annurev.fluid.35.101101.161136. [26] A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160 (2000), 240-282. [27] R. J. LeVeque, "Finite Volume Methods for Hyperbolic Problems," Cambridge Texts in Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253. [28] M. J. Lighthill and J. B. Whitham, On kinematic waves. I: Flow movement in long rivers. II: A theory of traffic flow on long crowded roads, Proc. Roy. Soc., A229 (1955), 281-345. [29] B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type, Math. Models Methods Appl. Sci., 20 (2010), 1787-1821. doi: 10.1142/S0218202510004799. [30] B. Maury and J. Venel, A mathematical framework for a crowd motion model, C. R. Acad. Sci. Paris, 346 (2008), 1245-1250. [31] K. Nishinari, A. Kirchner, A. Namazi and A. Schadschneider, Extended floor field CA model for evacuation dynamics, IEICE Transp. Inf. & Syst., E87-D (2004), 726-732, arXiv:cond-mat/0306262. [32] J. Ondřej, J. Pettré, A.-H. Olivier and S. Donikian, "A Synthetic-Vision Based Steering Approach for Crowd Simulation," SIGGRAPH '10, 2010. [33] S. Paris, J. Pettré and S. Donikian, Pedestrian reactive navigation for crowd simulation: A predictive approach, Eurographics, 26 (2007), 665-674. [34] Pedigree team, Pedestrian flow measurements and analysis in an annular setup, in preparation. [35] J. Pettré, J. Ondřej, A.-H. Olivier, A. Cretual and S. Donikian, "Experiment-Based Modeling, Simulation and Validation of Interactions Between Virtual Walkers," SCA '09: Proceedings of the 2009 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, (2009), 189-198. [36] B. Piccoli and A. Tosin, Pedestrian flows in bounded domains with obstacles, Contin. Mech. Thermodyn., 21 (2009), 85-107, arXiv:0812.4390. [37] B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738, arXiv:0811.3383. [38] C. W. Reynolds, "Steering Behaviors for Autonomous Characters," Proceedings of Game Developers Conference 1999, San Jose, California, (1999), 763-782. [39] V. Shvetsov and D. Helbing, Macroscopic dynamics of multi-lane traffic, Phys. Rev. E, 59 (1999), 6328-6339, arXiv:cond-mat/9906430. doi: 10.1103/PhysRevE.59.6328. [40] J. van den Berg and H. Overmars, Planning time-minimal safe paths amidst unpredictably moving obstacles, Int. Journal on Robotics Research, 27 (2008), 1274-1294. doi: 10.1177/0278364908097581. [41] W. G. Weng, S. F. Shena, H. Y. Yuana and W. C. Fana, A behavior-based model for pedestrian counter flow, Physica A, 375 (2007), 668-678. doi: 10.1016/j.physa.2006.09.028. [42] M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Res. B, 36 (2002), 275-290. doi: 10.1016/S0191-2615(00)00050-3.
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