# American Institute of Mathematical Sciences

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

## A mixed system modeling two-directional pedestrian flows

 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.
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. 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. 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. 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. Google Scholar [5] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem,, Oxford Lecture Series in Mathematics and its Applications, (2000). Google Scholar [6] S. Buchmüller and U. Weidmann, Parameters of Pedestrians, Pedestrian Traffic and Walking Facilities,, Technical report, (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. doi: 10.1137/0730033. Google Scholar [8] R. J. DiPerna, Measure-valued solutions to conservation laws,, Arch. Rational Mech. Anal., 88 (1985), 223. doi: 10.1007/BF00752112. Google Scholar [9] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods,, in Handbook of numerical analysis, (2000), 713. Google Scholar [10] A. D. Fitt, The numerical and analytical solution of ill-posed systems of conservation laws,, Appl. Math. Modelling, 13 (1989), 618. 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, (2102). 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. 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. 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, (1990), 67. 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. 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. doi: 10.1002/zamm.19950750802. Google Scholar [18] B. L. Keyfitz, Singular shocks: Retrospective and prospective,, Confluentes Math., 3 (2011), 445. doi: 10.1142/S1793744211000424. Google Scholar [19] P. Lax and B. Wendroff, Systems of conservation laws,, Comm. Pure Appl. Math., 13 (1960), 217. 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. 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). Google Scholar [22] H. B. Stewart and B. Wendroff, Two-phase flow: Models and methods,, J. Comput. Phys., 56 (1984), 363. 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, (1979), 136. Google Scholar [24] V. Vinod, Structural Stability of Riemann Solutions for a Multiple Kinematic Conservation Law Model that Changes Type,, Ph.D Thesis, (1992). 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. 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. 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. 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. Google Scholar [5] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem,, Oxford Lecture Series in Mathematics and its Applications, (2000). Google Scholar [6] S. Buchmüller and U. Weidmann, Parameters of Pedestrians, Pedestrian Traffic and Walking Facilities,, Technical report, (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. doi: 10.1137/0730033. Google Scholar [8] R. J. DiPerna, Measure-valued solutions to conservation laws,, Arch. Rational Mech. Anal., 88 (1985), 223. doi: 10.1007/BF00752112. Google Scholar [9] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods,, in Handbook of numerical analysis, (2000), 713. Google Scholar [10] A. D. Fitt, The numerical and analytical solution of ill-posed systems of conservation laws,, Appl. Math. Modelling, 13 (1989), 618. 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, (2102). 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. 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. 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, (1990), 67. 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. 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. doi: 10.1002/zamm.19950750802. Google Scholar [18] B. L. Keyfitz, Singular shocks: Retrospective and prospective,, Confluentes Math., 3 (2011), 445. doi: 10.1142/S1793744211000424. Google Scholar [19] P. Lax and B. Wendroff, Systems of conservation laws,, Comm. Pure Appl. Math., 13 (1960), 217. 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. 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). Google Scholar [22] H. B. Stewart and B. Wendroff, Two-phase flow: Models and methods,, J. Comput. Phys., 56 (1984), 363. 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, (1979), 136. Google Scholar [24] V. Vinod, Structural Stability of Riemann Solutions for a Multiple Kinematic Conservation Law Model that Changes Type,, Ph.D Thesis, (1992). Google Scholar
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