THE ONE-DIMENSIONAL HUGHES MODEL FOR EDESTRIAN FLOW: RIEMANN-TYPE SOLUTIONS

doi:10.1016/S0252-9602(12)60016-2

Abstract

Abstract:

This paper deals with a coupled system consisting of a scalar conservation law
and an eikonal equation, called the Hughes model. Introduced in [24], this model attempts
to describe the motion of pedestrians in a densely crowded region, in which they are seen
as a ‘thinking’ (continuum) fluid. The main mathematical diffculty is the discontinuous
gradient of the solution to the eikonal equation appearing in the flux of the conservation
law. On a one dimensional interval with zero Dirichlet conditions (the two edges of the
interval are interpreted as ‘targets’), the model can be decoupled in a way to consider
two classical conservation laws on two sub-domains separated by a turning point at which
the pedestrians change their direction. We shall consider solutions with a possible jump
discontinuity around the turning point. For simplicity, we shall assume they are locally
constant on both sides of the discontinuity. We provide a detailed description of the local-
in-time behavior of the solution in terms of a ‘global’ qualitative property of the pedestrian
density (that we call ‘relative evacuation rate’), which can be interpreted as the attitude
of the pedestrians to direct towards the left or the right target. We complement our result
with explicitly computable examples.

Key words: pedestrian flow, nonlocal conservation law, eikonal equation

CLC Number:

35L65

Debora Amadori, M. Di Francesco. THE ONE-DIMENSIONAL HUGHES MODEL FOR EDESTRIAN FLOW: RIEMANN-TYPE SOLUTIONS[J].Acta mathematica scientia,Series B, 2012, 32(1): 259-280.

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References

[1] Bardos C, Leroux A Y, Nedelec J C. First order quasilinear equations with boundary conditions. Comm
Partial Di?er Equ, 1979, 4: 1017–1034

[2] Bellomo N, Dogb′e C. On the modelling crowd dynamics from scaling to hyperbolic macroscopic models.
Math Models Methods Appl Sci, 2008, 18(suppl): 1317–1345

[3] Betancourt F, B¨urger R, Karlsen K H, Tory E M. On nonlocal conservation laws modelling sedimentation.
Nonlinearity, 2011, 24: 855–885

[4] Bradley G. A proposed mathematical model for computer prediction of crowd movements and their as-
sociated risks//Smith R A, Dickie J F, eds. Engineering for Crowd Safety. Amsterdam: Elsevier, 1993:303–311

[5] Bressan A. Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem. Oxford
University Press, 2000

[6] Bressan A, Colombo R M. P.D.E. models of pedestrian flow. Unpublished, 2007

[7] Burger M, Markowich P A, Pietschmann J -F. Continuous limit of a crowd motion and herding model:
analysis and numerical simulations. Preprint, 2011

[8] Buttazzo G, Jimenez C, Oudet E. An optimization problem for mass transportation with congested dy-
namics. SIAM J Control Optim, 2009, 48: 1961–1976

[9] Cannarsa P, Sinestrari C. Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control.
Vlume 58 of Progress in Nonlinear Di?erential Equations and Their Applications. Birkh¨auser, 2004

[10] Colombo R M, Garavello M, Mercier M. A class of non local models for pedestrian traffc. Preprint, 2011

[11] Colombo R M, Rosini M D. Pedestrian flows and non-classical shocks. Math Methods Appl Sci, 2005,
28(13): 1553–1567

[12] Colombo R M, Rosini M D. Existence of nonclassical solutions in a pedestrian flow model. Nonlinear Anal
Real World Appl, 2009, 10(5): 2716–2728

[13] Cristiani E, Piccoli B, Tosin A. Multiscale modeling of granular flows with applications to crowd dynamics.
Multiscale Modeling and Simulations, 2011, 9: 155–182

[14] Dafermos C M. Polygonal approximations of solutions of the initial value problem for a conservation law.
J Math Anal Appl, 1972, 38: 33–41

[15] Dafermos C M. Hyperbolic Conservation Laws in Continuum Physics. 3rd ed. Springer-Verlag, 2010

[16] Di Francesco M, Markowich P A, Pietschmann J -F, Wolfram M T. On the Hughes’ model for pedestrian
flow: The one-dimensional case. J Differ Equ, 2011, 250: 1334–1362

[17] Dogb′e C. Modeling crowd dynamics by the mean-field limit approach. Math Comp Model, 2010, 52:
1506–1520

[18] Dubois F, LeFloch P. Boundary conditions for nonlinear hyperbolic systems of conservation laws. J Di?er
Equ, 1988, 71(1): 93–122

[19] Fukui M, Ishibashi Y. Self-organized phase transitions in CA-models for pedestrians. J Phys Soc Japan,
1999, 8: 2861–2863

[20] Helbing D. Tra?c and related self-driven many-particle systems. Rev Mod Phys, 2001, 73(4): 1067–1141

[21] Helbing D, Farkas I J, Molnar P, Vicsek T. Simulation of pedestrian crowds in normal and evacuation
situations//Schreckenberg M, Sharma S D, eds. Pedestrian and Evacuation Dynamics. Berlin: Springer,
2002: 21–58

[22] Henderson L F. The statistics of crowd fluids. Nature, 1971, 229: 381–383

[23] Henderson L F. On the fluids mechanics of human crowd motion. Transp Res, 1974, 8: 509–515

[24] Hughes R L. A continuum theory for the flow of pedestrians. Transportation Research Part B: Method-
ological, 2002, 36(6): 507–535

[25] Hughes R L. The flow of human crowds. Annu Rev Fluid Mech, 2003, 35: 169–182

[26] Klausen R A, Risebro N H. Stability of conservation laws with discontinuous coe?cients. J Differ Equ,
1999, 157: 41–60

[27] Kruˇzkov S N. First order quasilinear equations in several independent variables. Math USSR Sb, 1970,
10: 217–243

[28] Kruˇzkov S N. Generalized solutions of the Hamilton-Jacobi equations of eikonal type. i. formulation of
the problems; existence, uniqueness and stability theorems; some properties of the solutions. Math USSR
Sb, 1975, 27(3): 406–446

[29] Lax P D. Hyperbolic systems of conservation laws and the mathematical theory of shock waves. CBMS
Regional Conference Series in Applied Mathematics, No 11. Philadelphia: SIAM, 1973

[30] Lighthill M J, Whitham G B. On kinematic waves. ii. a theory of traffc flow on long crowded roads.
Royal Society of London Proceedings Series A, 1955, 229: 317–345

[31] Maury B, Roudneff-Chupin A, Santambrogio F. A macroscopic crowd motion model of the gradient-flow
type. Math Model Meth Appl Sci, 2010, 20: 1787–1821

[32] Muramatsu M, Nagatani T. Jamming transition in two-dimensional pedestrian traffc. Physica A, 2000,
275: 281–291

[33] Oleininik O A. Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear
equation. Uspehi Mat Nauk, 1959, 14(2(86)): 165–170

[34] Piccoli B, Tosin A. Pedestrian flows in bounded domains with obstacles. Contin Mech Thermodyn, 2009,
21(2): 85–107

[35] Schadschneider A, Klingsch W, Kluepfel H, Kretz T, Rogsch C, Seyfried A. Encyclopedia of Complexity
and System Science. Meyers R A, ed. Vol 3, Chapter Evacuation Dynamics: Empirical Results, Modeling
and Applications. Springer, 2009

[36] Whitham G. Linear and Nonlinear Waves. New York: Wiley & Sons, 1974