# American Institute of Mathematical Sciences

September  2011, 6(3): 401-423. doi: 10.3934/nhm.2011.6.401

## An adaptive finite-volume method for a model of two-phase pedestrian flow

 1 Departamento de Ciencias Matemáticas y Físicas, Universidad Católica de Temuco, Temuco, Chile 2 Modeling and Scientific Computing, MATHISCE, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland 3 Institut für Mathematik, Fakultät II Mathematik und Naturwissenschaften, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany 4 Department of Mathematics and Computer Science, Mount Allison University, Sackville, NB E4L 1G6, Canada

Received  December 2010 Revised  June 2011 Published  August 2011

A flow composed of two populations of pedestrians moving in different directions is modeled by a two-dimen\-sional system of convection-diffusion equations. An efficient simulation of the two-dimensional model is obtained by a finite-volume scheme combined with a fully adaptive multiresolution strategy. Numerical tests show the flow behavior in various settings of initial and boundary conditions, where different species move in countercurrent or perpendicular directions. The equations are characterized as hyperbolic-elliptic degenerate, with an elliptic region in the phase space, which in one space dimension is known to produce oscillation waves. When the initial data are chosen inside the elliptic region, a spatial segregation of the populations leads to pattern formation. The entries of the diffusion-matrix determine the stability of the model and the shape of the patterns.
Citation: Stefan Berres, Ricardo Ruiz-Baier, Hartmut Schwandt, Elmer M. Tory. An adaptive finite-volume method for a model of two-phase pedestrian flow. Networks & Heterogeneous Media, 2011, 6 (3) : 401-423. doi: 10.3934/nhm.2011.6.401
##### References:
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Čanić, On the influence of viscosity on Riemann solutions, J. Dyn. Diff. Eqns., 10 (1998), 109-149.  Google Scholar [15] M. Chen, G. Bärwolff and H. Schwandt, A derived grid-based model for simulation of pedestrian flow, J. Zhejiang Univ.: Science A, 10 (2009), 209-220. doi: 10.1631/jzus.A0820049.  Google Scholar [16] E. Cristiani, B. Piccoli and A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics, Multiscale Model. Simul., 9 (2011), 155-182. doi: 10.1137/100797515.  Google Scholar [17] R. R. Clements and R. L. Hughes, Mathematical modelling of a mediaeval battle: The battle of Agincourt, 1415, Math. Comput. Simul., 64 (2004), 259-269. doi: 10.1016/j.matcom.2003.09.019.  Google Scholar [18] W. Daamen, P. H. L. Bovy and S. P. Hoogendoorn, Modelling pedestrians in transfer stations, in "Pedestrian and Evacuation Dynamics" (eds. M. Schreckenberg and S. D. Sharma), Springer-Verlag, Berlin, Heidelberg, (2002), 59-73. Google Scholar [19] J. Esser and M. 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Werner, Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions, Transportation Science, 39 (2005), 1-24. doi: 10.1287/trsc.1040.0108.  Google Scholar [25] D. Helbing, I. Farkas and T. Vicsek, Traffic and related self-driven many-particle systems, Reviews of Modern Physics, 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.  Google Scholar [26] H. Holden, On the Riemann problem for a prototype of a mixed type conservation law, Comm. Pure Appl. Math., 40 (1987), 229-264. doi: 10.1002/cpa.3160400206.  Google Scholar [27] S. P. Hoogendoorn and W. Daamen, Self-organization in pedestrian flow, Traff. Granul. Flow, 3 (2005), 373-382. Google Scholar [28] R. L. Hughes, A continuum theory for the flow of pedestrians, Transp. Res. B, 36 (2002), 507-535. doi: 10.1016/S0191-2615(01)00015-7.  Google Scholar [29] J. Hurley and B. J. Plohr, Some effects of viscous terms on Riemann Problem solutions, Math. Contemp., 8 (1995), 203-224.  Google Scholar [30] Y. Jiang, P. Zhang, S. C. Wong and R. Liu, A higher-order macroscopic model for pedestrian flows, Physica A, 389 (2010), 4623-4635. doi: 10.1016/j.physa.2010.05.003.  Google Scholar [31] B. S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow, Physical Review E, 50 (1994), 54-83. doi: 10.1103/PhysRevE.50.54.  Google Scholar [32] B. L. Keyfitz, A geometric theory of conservation laws which change type, ZAMM Z. Angew. Math. Mech., 75 (1995), 571-581.  Google Scholar [33] B. L. Keyfitz, "Mathematical Properties of Nonhyperbolic Models for Incompressible Two-Phase Flow," Proceedings of the International Conference on Multiphase Flow, New Orleans, May 27-June 1, 2001, (CD-ROM). Google Scholar [34] A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160 (2000), 241-282. doi: 10.1006/jcph.2000.6459.  Google Scholar [35] M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London Ser. A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.  Google Scholar [36] A. Majda and R. L. Pego, Stable viscosity matrices for system of conservation laws, J. Diff. Eqns., 56 (1985), 229-262. doi: 10.1016/0022-0396(85)90107-X.  Google Scholar [37] D. Marchesin and B. Plohr, Wave structure in WAG recovery, SPE, paper 56480. Google Scholar [38] S. Müller, "Adaptive Multiscale Schemes for Conservation Laws," Lecture Notes in Computational Science and Engineering, 27, Springer-Verlag, Berlin, 2003.  Google Scholar [39] A. Nakayama, K. Hasebe and Y. Sugiyama, Instability of pedestrian flow in 2D optimal velocity model with attractive interaction, Comput. Phys. Comm., 177 (2007), 162-163. doi: 10.1016/j.cpc.2007.02.007.  Google Scholar [40] B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738. doi: 10.1007/s00205-010-0366-y.  Google Scholar [41] P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.  Google Scholar [42] Th. Slawig, G. Bärwolff and H. Schwandt, "Hybrid macro-Microscopic Simulation of Pedestrian Flow," Technical report, Institut für Mathematik, Technische Universität Berlin, 2007. Google Scholar [43] Th. Slawig, G. Bärwolff and H. Schwandt, "Simulation of Pedestrian Flows for Traffic Control Systems," in Proceedings of the 7th International Conference on Information and Management Sciences (IMS 2008) (Urumtschi, 12. 8. 08 - 19. 8. 08) (ed. E. Y. Li), California Polytechnic State University, Series on Information and Management Sciences, 7, Pomona/California, 2008, 360-374. Google Scholar [44] E. F. Toro, "Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction," Third edition, Springer-Verlag, Berlin 2009. doi: 10.1007/b79761.  Google Scholar [45] Y. Xia, S. C. Wong, M. P. Zhang, C.-W. Shu and W. H. K. Lam, An efficient discontinuous Galerkin method on triangular meshes for a pedestrian flow model, Int. J. Numer. Meth. Engrg., 76 (2008), 337-350. doi: 10.1002/nme.2329.  Google Scholar [46] P. Zhang, S. C. Wong and C.-W. Shu, A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway, J. Comput. Phys., 212 (2006), 739-756. doi: 10.1016/j.jcp.2005.07.019.  Google Scholar

show all references

##### References:
 [1] B. Andreianov, M. Bendahmane and R. Ruiz-Baier, Analysis of a finite volume method for a cross-diffusion model in population dynamics, Math. Models Meth. Appl. Sci., 21 (2011), 307-344. doi: 10.1142/S0218202511005064.  Google Scholar [2] A. V. Azevedo, D. Marchesin, B. Plohr and K. Zumbrun, Capillary instability in models for three-phase flow, Z. Angew. Math. Phys., 53 (2002), 713-746. doi: 10.1007/s00033-002-8180-5.  Google Scholar [3] J. B. Bell, J. A. Trangenstein and G. R. Shubin, Conservation laws of mixed type describing three-phase flows in porous media, SIAM J. Appl. Math., 46 (1986), 1000-1017. doi: 10.1137/0146059.  Google Scholar [4] S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, Eur. J. Appl. Math., 14 (2003), 587-612.  Google Scholar [5] S. Berres, R. Bürger and K. H. Karlsen, Central schemes and systems of conservation laws with discontinuous coefficients modeling gravity separation of polydisperse suspensions, J. Comput. Appl. Math., 164/165 (2004), 53-80. doi: 10.1016/S0377-0427(03)00496-5.  Google Scholar [6] S. Berres, R. Bürger and A. Kozakevicius, Numerical approximation of oscillatory solutions of hyperbolic-elliptic systems of conservation laws by multiresolution schemes, Adv. Appl. Math. Mech., 1 (2009), 581-614.  Google Scholar [7] S. Berres and R. Ruiz-Baier, A fully adaptive numerical approximation for a two-dimensional epidemic model with non-linear cross-diffusion, Nonlin. Anal. Real World Appl., 12 (2011), 2888-2903. doi: 10.1016/j.nonrwa.2011.04.014.  Google Scholar [8] S. Berres, R. Ruiz-Baier, H. Schwandt and E. M. Tory, Two-dimensional models of pedestrian flow, in "Series in Contemporary Applied Mathematics" (Proceedings of HYP 2010) (eds. P. G. Ciarlet and Ta-Tsien Li), Higher Education Press, Beijing, World Scientific, Singapore, 2011, to appear. Google Scholar [9] J. H. Bick and G. F. Newell, A continuum model for two-directional traffic flow, Quart. Appl. Math., 18 (1960), 191-204. Google Scholar [10] L. Bruno, A. Tosin, P. Tricerri and F. Venuti, Non-local first-order modelling of crowd dynamics: A multidimensional framework with applications, Appl. Math. Model., 35 (2011), 426-445. doi: 10.1016/j.apm.2010.07.007.  Google Scholar [11] R. Bürger, K. H. Karlsen, E. M. Tory and W. L. Wendland, Model equations and instability regions for the sedimentation of polydisperse suspensions of spheres, ZAMM Z. Angew. Math. Mech., 82 (2002), 699-722.  Google Scholar [12] R. Bürger, R. Ruiz-Baier and K. Schneider, Adaptive multiresolution methods for the simulation of waves in excitable media, J. Sci. Comput., 43 (2010), 261-290. doi: 10.1007/s10915-010-9356-3.  Google Scholar [13] C. Burstedde, K. Klauck, A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A: Stat. Mech. Appl., 295 (2001), 507-525. doi: 10.1016/S0378-4371(01)00141-8.  Google Scholar [14] S. Čanić, On the influence of viscosity on Riemann solutions, J. Dyn. Diff. Eqns., 10 (1998), 109-149.  Google Scholar [15] M. Chen, G. Bärwolff and H. Schwandt, A derived grid-based model for simulation of pedestrian flow, J. Zhejiang Univ.: Science A, 10 (2009), 209-220. doi: 10.1631/jzus.A0820049.  Google Scholar [16] E. Cristiani, B. Piccoli and A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics, Multiscale Model. Simul., 9 (2011), 155-182. doi: 10.1137/100797515.  Google Scholar [17] R. R. Clements and R. L. Hughes, Mathematical modelling of a mediaeval battle: The battle of Agincourt, 1415, Math. Comput. Simul., 64 (2004), 259-269. doi: 10.1016/j.matcom.2003.09.019.  Google Scholar [18] W. Daamen, P. H. L. Bovy and S. P. Hoogendoorn, Modelling pedestrians in transfer stations, in "Pedestrian and Evacuation Dynamics" (eds. M. Schreckenberg and S. D. Sharma), Springer-Verlag, Berlin, Heidelberg, (2002), 59-73. Google Scholar [19] J. Esser and M. Schreckenberg, Microscopic simulation of urban traffic based on cellular automata, Int. J. Mod. Phys. C, 8 (1997), 1025-1036. doi: 10.1142/S0129183197000904.  Google Scholar [20] A. D. Fitt, The numerical and analytical solution of ill-posed systems of conservation laws, Appl. Math. Modelling, 13 (1989), 618-631. doi: 10.1016/0307-904X(89)90171-6.  Google Scholar [21] H. Frid and I.-S. Liu, Oscillation waves in Riemann problems inside elliptic regions for conservation laws of mixed type, Z. Angew. Math. Phys., 46 (1995), 913-931. doi: 10.1007/BF00917877.  Google Scholar [22] A. Harten, Multiresolution representation of data: A general framework, SIAM J. Numer. Anal., 33 (1996), 1205-1256. doi: 10.1137/0733060.  Google Scholar [23] D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical features of escape panic, Nature, 407 (2000), 487-490. doi: 10.1038/35035023.  Google Scholar [24] D. Helbing, L. Buzna, A. Johansson and T. Werner, Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions, Transportation Science, 39 (2005), 1-24. doi: 10.1287/trsc.1040.0108.  Google Scholar [25] D. Helbing, I. Farkas and T. Vicsek, Traffic and related self-driven many-particle systems, Reviews of Modern Physics, 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.  Google Scholar [26] H. Holden, On the Riemann problem for a prototype of a mixed type conservation law, Comm. Pure Appl. Math., 40 (1987), 229-264. doi: 10.1002/cpa.3160400206.  Google Scholar [27] S. P. Hoogendoorn and W. Daamen, Self-organization in pedestrian flow, Traff. Granul. Flow, 3 (2005), 373-382. Google Scholar [28] R. L. Hughes, A continuum theory for the flow of pedestrians, Transp. Res. B, 36 (2002), 507-535. doi: 10.1016/S0191-2615(01)00015-7.  Google Scholar [29] J. Hurley and B. J. Plohr, Some effects of viscous terms on Riemann Problem solutions, Math. Contemp., 8 (1995), 203-224.  Google Scholar [30] Y. Jiang, P. Zhang, S. C. Wong and R. Liu, A higher-order macroscopic model for pedestrian flows, Physica A, 389 (2010), 4623-4635. doi: 10.1016/j.physa.2010.05.003.  Google Scholar [31] B. S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow, Physical Review E, 50 (1994), 54-83. doi: 10.1103/PhysRevE.50.54.  Google Scholar [32] B. L. Keyfitz, A geometric theory of conservation laws which change type, ZAMM Z. Angew. Math. Mech., 75 (1995), 571-581.  Google Scholar [33] B. L. Keyfitz, "Mathematical Properties of Nonhyperbolic Models for Incompressible Two-Phase Flow," Proceedings of the International Conference on Multiphase Flow, New Orleans, May 27-June 1, 2001, (CD-ROM). Google Scholar [34] A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160 (2000), 241-282. doi: 10.1006/jcph.2000.6459.  Google Scholar [35] M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London Ser. A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.  Google Scholar [36] A. Majda and R. L. Pego, Stable viscosity matrices for system of conservation laws, J. Diff. Eqns., 56 (1985), 229-262. doi: 10.1016/0022-0396(85)90107-X.  Google Scholar [37] D. Marchesin and B. Plohr, Wave structure in WAG recovery, SPE, paper 56480. Google Scholar [38] S. Müller, "Adaptive Multiscale Schemes for Conservation Laws," Lecture Notes in Computational Science and Engineering, 27, Springer-Verlag, Berlin, 2003.  Google Scholar [39] A. Nakayama, K. Hasebe and Y. Sugiyama, Instability of pedestrian flow in 2D optimal velocity model with attractive interaction, Comput. Phys. Comm., 177 (2007), 162-163. doi: 10.1016/j.cpc.2007.02.007.  Google Scholar [40] B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738. doi: 10.1007/s00205-010-0366-y.  Google Scholar [41] P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.  Google Scholar [42] Th. Slawig, G. Bärwolff and H. Schwandt, "Hybrid macro-Microscopic Simulation of Pedestrian Flow," Technical report, Institut für Mathematik, Technische Universität Berlin, 2007. Google Scholar [43] Th. Slawig, G. Bärwolff and H. Schwandt, "Simulation of Pedestrian Flows for Traffic Control Systems," in Proceedings of the 7th International Conference on Information and Management Sciences (IMS 2008) (Urumtschi, 12. 8. 08 - 19. 8. 08) (ed. E. Y. Li), California Polytechnic State University, Series on Information and Management Sciences, 7, Pomona/California, 2008, 360-374. Google Scholar [44] E. F. Toro, "Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction," Third edition, Springer-Verlag, Berlin 2009. doi: 10.1007/b79761.  Google Scholar [45] Y. Xia, S. C. Wong, M. P. Zhang, C.-W. Shu and W. H. K. Lam, An efficient discontinuous Galerkin method on triangular meshes for a pedestrian flow model, Int. J. Numer. Meth. Engrg., 76 (2008), 337-350. doi: 10.1002/nme.2329.  Google Scholar [46] P. Zhang, S. C. Wong and C.-W. Shu, A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway, J. Comput. Phys., 212 (2006), 739-756. doi: 10.1016/j.jcp.2005.07.019.  Google Scholar
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