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2015, 12(2): 375-392. doi: 10.3934/mbe.2015.12.375

A mixed system modeling two-directional pedestrian flows

Paola Goatin 1, and  Matthias Mimault 2,

1. 

INRIA Sophia Antipolis - Méditerranée, EPI OPALE, 2004, route des Lucioles - BP 93, 06902 Sophia Antipolis Cedex
2. 

INRIA Sophia Antipolis - Méditerranée, 2004, route des Lucioles - BP 93, 06902 Sophia Antipolis Cedex, France

Received  April 2014 Revised  August 2014 Published  December 2014

In this article, we present a simplified model to describe the dynamics of two groups of pedestrians moving in opposite directions in a corridor. The model consists of a $2\times 2$ system of conservation laws of mixed hyperbolic-elliptic type. We study the basic properties of the system to understand why and how bounded oscillations in numerical simulations arise. We show that Lax-Friedrichs scheme ensures the invariance of the domain and we investigate the existence of measure-valued solutions as limit of a subsequence of approximate solutions.
Keywords: macroscopic models for pedestrian flows, Systems of conservation laws, finite volume schemes, mixed hyperbolic-elliptic systems, Young measures..
Mathematics Subject Classification: Primary: 35L65, 35M30; Secondary: 90B2.
Citation: Paola Goatin, Matthias Mimault. A mixed system modeling two-directional pedestrian flows. Mathematical Biosciences & Engineering, 2015, 12 (2) : 375-392. doi: 10.3934/mbe.2015.12.375
References:
[1]

J. B. Bell, J. A. Trangenstein and G. R. Shubin, Conservation laws of mixed type describing three-phase flow in porous media, SIAM J. Appl. Math., 46 (1986), 1000-1017. doi: 10.1137/0146059.  Google Scholar

[2]

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.  Google Scholar

[3]

S. Berres, R. Ruiz-Baier, H. Schwandt and E. M. Tory, An adaptive finite-volume method for a model of two-phase pedestrian flow, Netw. Heterog. Media, 6 (2011), 401-423. doi: 10.3934/nhm.2011.6.401.  Google Scholar

[4]

J. Bick and G. F. Newell, A continuum model for two-directional traffic flow, Q. Appl. Math., XVIII (1960), 191-204. Google Scholar

[5]

A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000.  Google Scholar

[6]

S. Buchmüller and U. Weidmann, Parameters of Pedestrians, Pedestrian Traffic and Walking Facilities, Technical report, ETH Zürich, 2006. Google Scholar

[7]

F. Coquel and P. LeFloch, Convergence of finite difference schemes for conservation laws in several space dimensions: A general theory, SIAM J. Numer. Anal., 30 (1993), 675-700. doi: 10.1137/0730033.  Google Scholar

[8]

R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal., 88 (1985), 223-270. doi: 10.1007/BF00752112.  Google Scholar

[9]

R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, (2000), 713-1020.  Google Scholar

[10]

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

[11]

U. S. Fjordholm, High-order Accurate Entropy Stable Numerical Schemes for Hyperbolic Conservation Laws, Ph.D thesis, ETH Zürich dissertation Nr. 21025, 2013. Google Scholar

[12]

U. S. Fjordholm, R. Käppeli, S. Mishra and E. Tadmor, Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws,, , ().   Google Scholar

[13]

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

[14]

D. Helbing, P. Molnár, I. J. Farkas and K. Bolay, Self-organizing pedestrian movement, Environment and Planning B, 28 (2001), 361-383. doi: 10.1068/b2697.  Google Scholar

[15]

H. Holden, L. Holden and N. H. Risebro, Some qualitative properties of $2\times 2$ systems of conservation laws of mixed type, in Nonlinear Evolution Equations That Change Type, IMA Vol. Math. Appl., 27, Springer, New York, 1990, 67-78. doi: 10.1007/978-1-4613-9049-7_5.  Google Scholar

[16]

E. Isaacson, D. Marchesin, B. Plohr and B. Temple, The Riemann problem near a hyperbolic singularity: The classification of solutions of quadratic Riemann problems. I, SIAM J. Appl. Math., 48 (1988), 1009-1032. doi: 10.1137/0148059.  Google Scholar

[17]

B. L. Keyfitz, A geometric theory of conservation laws which change type, Z. Angew. Math. Mech., 75 (1995), 571-581. doi: 10.1002/zamm.19950750802.  Google Scholar

[18]

B. L. Keyfitz, Singular shocks: Retrospective and prospective, Confluentes Math., 3 (2011), 445-470. doi: 10.1142/S1793744211000424.  Google Scholar

[19]

P. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math., 13 (1960), 217-237. doi: 10.1002/cpa.3160130205.  Google Scholar

[20]

T. P. Liu, The Riemann problem for general $2\times 2$ conservation laws, Trans. Amer. Math. Soc., 199 (1974), 89-112.  Google Scholar

[21]

M. Moussaïd, E. G. Guillot, M. Moreau, J. Fehrenbach, O. Chabiron, S. Lemercier, J. Pettré, C. Appert-Rolland, P. Degond and G. Theraulaz, Traffic instabilities in self-organized pedestrian crowds, PLoS Comput. Biol., 8 (2012), e1002442. Google Scholar

[22]

H. B. Stewart and B. Wendroff, Two-phase flow: Models and methods, J. Comput. Phys., 56 (1984), 363-409. doi: 10.1016/0021-9991(84)90103-7.  Google Scholar

[23]

L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., 39, Pitman, Boston, Mass.-London, 1979, 136-212.  Google Scholar

[24]

V. Vinod, Structural Stability of Riemann Solutions for a Multiple Kinematic Conservation Law Model that Changes Type, Ph.D Thesis, University of Houston, Houston, 1992. 68 pp.  Google Scholar

show all references

References:
[1]

J. B. Bell, J. A. Trangenstein and G. R. Shubin, Conservation laws of mixed type describing three-phase flow in porous media, SIAM J. Appl. Math., 46 (1986), 1000-1017. doi: 10.1137/0146059.  Google Scholar

[2]

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.  Google Scholar

[3]

S. Berres, R. Ruiz-Baier, H. Schwandt and E. M. Tory, An adaptive finite-volume method for a model of two-phase pedestrian flow, Netw. Heterog. Media, 6 (2011), 401-423. doi: 10.3934/nhm.2011.6.401.  Google Scholar

[4]

J. Bick and G. F. Newell, A continuum model for two-directional traffic flow, Q. Appl. Math., XVIII (1960), 191-204. Google Scholar

[5]

A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000.  Google Scholar

[6]

S. Buchmüller and U. Weidmann, Parameters of Pedestrians, Pedestrian Traffic and Walking Facilities, Technical report, ETH Zürich, 2006. Google Scholar

[7]

F. Coquel and P. LeFloch, Convergence of finite difference schemes for conservation laws in several space dimensions: A general theory, SIAM J. Numer. Anal., 30 (1993), 675-700. doi: 10.1137/0730033.  Google Scholar

[8]

R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal., 88 (1985), 223-270. doi: 10.1007/BF00752112.  Google Scholar

[9]

R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, (2000), 713-1020.  Google Scholar

[10]

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

[11]

U. S. Fjordholm, High-order Accurate Entropy Stable Numerical Schemes for Hyperbolic Conservation Laws, Ph.D thesis, ETH Zürich dissertation Nr. 21025, 2013. Google Scholar

[12]

U. S. Fjordholm, R. Käppeli, S. Mishra and E. Tadmor, Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws,, , ().   Google Scholar

[13]

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

[14]

D. Helbing, P. Molnár, I. J. Farkas and K. Bolay, Self-organizing pedestrian movement, Environment and Planning B, 28 (2001), 361-383. doi: 10.1068/b2697.  Google Scholar

[15]

H. Holden, L. Holden and N. H. Risebro, Some qualitative properties of $2\times 2$ systems of conservation laws of mixed type, in Nonlinear Evolution Equations That Change Type, IMA Vol. Math. Appl., 27, Springer, New York, 1990, 67-78. doi: 10.1007/978-1-4613-9049-7_5.  Google Scholar

[16]

E. Isaacson, D. Marchesin, B. Plohr and B. Temple, The Riemann problem near a hyperbolic singularity: The classification of solutions of quadratic Riemann problems. I, SIAM J. Appl. Math., 48 (1988), 1009-1032. doi: 10.1137/0148059.  Google Scholar

[17]

B. L. Keyfitz, A geometric theory of conservation laws which change type, Z. Angew. Math. Mech., 75 (1995), 571-581. doi: 10.1002/zamm.19950750802.  Google Scholar

[18]

B. L. Keyfitz, Singular shocks: Retrospective and prospective, Confluentes Math., 3 (2011), 445-470. doi: 10.1142/S1793744211000424.  Google Scholar

[19]

P. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math., 13 (1960), 217-237. doi: 10.1002/cpa.3160130205.  Google Scholar

[20]

T. P. Liu, The Riemann problem for general $2\times 2$ conservation laws, Trans. Amer. Math. Soc., 199 (1974), 89-112.  Google Scholar

[21]

M. Moussaïd, E. G. Guillot, M. Moreau, J. Fehrenbach, O. Chabiron, S. Lemercier, J. Pettré, C. Appert-Rolland, P. Degond and G. Theraulaz, Traffic instabilities in self-organized pedestrian crowds, PLoS Comput. Biol., 8 (2012), e1002442. Google Scholar

[22]

H. B. Stewart and B. Wendroff, Two-phase flow: Models and methods, J. Comput. Phys., 56 (1984), 363-409. doi: 10.1016/0021-9991(84)90103-7.  Google Scholar

[23]

L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., 39, Pitman, Boston, Mass.-London, 1979, 136-212.  Google Scholar

[24]

V. Vinod, Structural Stability of Riemann Solutions for a Multiple Kinematic Conservation Law Model that Changes Type, Ph.D Thesis, University of Houston, Houston, 1992. 68 pp.  Google Scholar

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