Mathematical Biosciences and Engineering (MBE)

A mixed system modeling two-directional pedestrian flows
Pages: 375 - 392, Issue 2, April 2015

doi:10.3934/mbe.2015.12.375      Abstract        References        Full text (832.4K)                  Related Articles

Paola Goatin - INRIA Sophia Antipolis - Méditerranée, EPI OPALE, 2004, route des Lucioles - BP 93, 06902 Sophia Antipolis Cedex, France (email)
Matthias Mimault - INRIA Sophia Antipolis - Méditerranée, 2004, route des Lucioles - BP 93, 06902 Sophia Antipolis Cedex, France (email)

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.       
2 S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, European J. Appl. Math., 14 (2003), 587-612.       
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.       
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.       
8 R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal., 88 (1985), 223-270.       
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,       
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, arXiv:1402.0909.
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.       
14 D. Helbing, P. Molnár, I. J. Farkas and K. Bolay, Self-organizing pedestrian movement, Environment and Planning B, 28 (2001), 361-383.
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.       
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.       
17 B. L. Keyfitz, A geometric theory of conservation laws which change type, Z. Angew. Math. Mech., 75 (1995), 571-581.       
18 B. L. Keyfitz, Singular shocks: Retrospective and prospective, Confluentes Math., 3 (2011), 445-470.       
19 P. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math., 13 (1960), 217-237.       
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.       
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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