A mixed system modeling two-directional pedestrian flows

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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

Abstract
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Under a Creative Commons license

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.
[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.
[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.
[4]	J. Bick and G. F. Newell, A continuum model for two-directional traffic flow, Q. Appl. Math., XVIII (1960), 191-204.
[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.
[6]	S. Buchmüller and U. Weidmann, Parameters of Pedestrians, Pedestrian Traffic and Walking Facilities, Technical report, ETH Zürich, 2006.
[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.
[8]	R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal., 88 (1985), 223-270. doi: 10.1007/BF00752112.
[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.
[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.
[11]	U. S. Fjordholm, High-order Accurate Entropy Stable Numerical Schemes for Hyperbolic Conservation Laws, Ph.D thesis, ETH Zürich dissertation Nr. 21025, 2013.
[12]	U. S. Fjordholm, R. Käppeli, S. Mishra and E. Tadmor, Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws,, , ().
[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.
[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.
[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.
[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.
[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.
[18]	B. L. Keyfitz, Singular shocks: Retrospective and prospective, Confluentes Math., 3 (2011), 445-470. doi: 10.1142/S1793744211000424.
[19]	P. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math., 13 (1960), 217-237. doi: 10.1002/cpa.3160130205.
[20]	T. P. Liu, The Riemann problem for general $2\times 2$ conservation laws, Trans. Amer. Math. Soc., 199 (1974), 89-112.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[4]	J. Bick and G. F. Newell, A continuum model for two-directional traffic flow, Q. Appl. Math., XVIII (1960), 191-204.
[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.
[6]	S. Buchmüller and U. Weidmann, Parameters of Pedestrians, Pedestrian Traffic and Walking Facilities, Technical report, ETH Zürich, 2006.
[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.
[8]	R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal., 88 (1985), 223-270. doi: 10.1007/BF00752112.
[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.
[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.
[11]	U. S. Fjordholm, High-order Accurate Entropy Stable Numerical Schemes for Hyperbolic Conservation Laws, Ph.D thesis, ETH Zürich dissertation Nr. 21025, 2013.
[12]	U. S. Fjordholm, R. Käppeli, S. Mishra and E. Tadmor, Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws,, , ().
[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.
[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.
[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.
[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.
[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.
[18]	B. L. Keyfitz, Singular shocks: Retrospective and prospective, Confluentes Math., 3 (2011), 445-470. doi: 10.1142/S1793744211000424.
[19]	P. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math., 13 (1960), 217-237. doi: 10.1002/cpa.3160130205.
[20]	T. P. Liu, The Riemann problem for general $2\times 2$ conservation laws, Trans. Amer. Math. Soc., 199 (1974), 89-112.
[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.
[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.
[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.
[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.

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