# 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 and Heterogeneous Media, 2011, 6 (3) : 401-423. doi: 10.3934/nhm.2011.6.401
##### 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. [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. [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. [4] S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, Eur. J. Appl. Math., 14 (2003), 587-612. [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. [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. [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. [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. [9] J. H. Bick and G. F. Newell, A continuum model for two-directional traffic flow, Quart. Appl. Math., 18 (1960), 191-204. [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. [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. [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. [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. [14] S. Čanić, On the influence of viscosity on Riemann solutions, J. Dyn. Diff. Eqns., 10 (1998), 109-149. [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. [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. [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. [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. [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. [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. [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. [22] A. Harten, Multiresolution representation of data: A general framework, SIAM J. Numer. Anal., 33 (1996), 1205-1256. doi: 10.1137/0733060. [23] D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical features of escape panic, Nature, 407 (2000), 487-490. doi: 10.1038/35035023. [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. [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. [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. [27] S. P. Hoogendoorn and W. Daamen, Self-organization in pedestrian flow, Traff. Granul. Flow, 3 (2005), 373-382. [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. [29] J. Hurley and B. J. Plohr, Some effects of viscous terms on Riemann Problem solutions, Math. Contemp., 8 (1995), 203-224. [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. [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. [32] B. L. Keyfitz, A geometric theory of conservation laws which change type, ZAMM Z. Angew. Math. Mech., 75 (1995), 571-581. [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). [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. [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. [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. [37] D. Marchesin and B. Plohr, Wave structure in WAG recovery, SPE, paper 56480. [38] S. Müller, "Adaptive Multiscale Schemes for Conservation Laws," Lecture Notes in Computational Science and Engineering, 27, Springer-Verlag, Berlin, 2003. [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. [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. [41] P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [42] Th. Slawig, G. Bärwolff and H. Schwandt, "Hybrid macro-Microscopic Simulation of Pedestrian Flow," Technical report, Institut für Mathematik, Technische Universität Berlin, 2007. [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. [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. [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. [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.

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. [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. [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. [4] S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, Eur. J. Appl. Math., 14 (2003), 587-612. [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. [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. [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. [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. [9] J. H. Bick and G. F. Newell, A continuum model for two-directional traffic flow, Quart. Appl. Math., 18 (1960), 191-204. [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. [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. [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. [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. [14] S. Čanić, On the influence of viscosity on Riemann solutions, J. Dyn. Diff. Eqns., 10 (1998), 109-149. [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. [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. [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. [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. [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. [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. [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. [22] A. Harten, Multiresolution representation of data: A general framework, SIAM J. Numer. Anal., 33 (1996), 1205-1256. doi: 10.1137/0733060. [23] D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical features of escape panic, Nature, 407 (2000), 487-490. doi: 10.1038/35035023. [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. [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. [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. [27] S. P. Hoogendoorn and W. Daamen, Self-organization in pedestrian flow, Traff. Granul. Flow, 3 (2005), 373-382. [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. [29] J. Hurley and B. J. Plohr, Some effects of viscous terms on Riemann Problem solutions, Math. Contemp., 8 (1995), 203-224. [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. [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. [32] B. L. Keyfitz, A geometric theory of conservation laws which change type, ZAMM Z. Angew. Math. Mech., 75 (1995), 571-581. [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). [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. [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. [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. [37] D. Marchesin and B. Plohr, Wave structure in WAG recovery, SPE, paper 56480. [38] S. Müller, "Adaptive Multiscale Schemes for Conservation Laws," Lecture Notes in Computational Science and Engineering, 27, Springer-Verlag, Berlin, 2003. [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. [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. [41] P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [42] Th. Slawig, G. Bärwolff and H. Schwandt, "Hybrid macro-Microscopic Simulation of Pedestrian Flow," Technical report, Institut für Mathematik, Technische Universität Berlin, 2007. [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. [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. [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. [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.
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