A local version of the Hughes model for pedestrian flow

Roger Hughes proposed a macroscopic model for pedestrian dynamics, in which individuals seek to minimize their travel time but try to avoid regions of high density. One of the basic assumptions is that the overall density of the crowd is known to every agent. In this paper we present a modification of the Hughes model to include local effects, namely limited vision, and a conviction towards decision making. The modified velocity field enables smooth turning and temporary waiting behavior. We discuss the modeling in the micro- and macroscopic setting as well as the efficient numerical simulation of either description. Finally we illustrate the model with various numerical experiments and evaluate the behavior with respect to the evacuation time and the overall performance.

[1]  Sebastien Blandin,et al.  Well-posedness of a conservation law with non-local flux arising in traffic flow modeling , 2016, Numerische Mathematik.

[2]  Rinaldo M. Colombo,et al.  On the Numerical Integration of Scalar Nonlocal Conservation Laws , 2015 .

[3]  G. Theraulaz,et al.  Vision-based macroscopic pedestrian models , 2013, 1307.1953.

[4]  Paola Goatin,et al.  The Wave-Front Tracking Algorithm for Hughes' Model of Pedestrian Motion , 2013, SIAM J. Sci. Comput..

[5]  M. Burger,et al.  Mean field games with nonlinear mobilities in pedestrian dynamics , 2013, 1304.5201.

[6]  P. Degond,et al.  A Hierarchy of Heuristic-Based Models of Crowd Dynamics , 2013, 1304.1927.

[7]  M. Rosini,et al.  On entropy weak solutions of Hughes’ model for pedestrian motion , 2013 .

[8]  Nicola Bellomo,et al.  From the Microscale to Collective Crowd Dynamics , 2013, Multiscale Model. Simul..

[9]  Nicola Bellomo,et al.  Modeling crowd dynamics from a complex system viewpoint , 2012 .

[10]  Cécile Appert-Rolland,et al.  Traffic Instabilities in Self-Organized Pedestrian Crowds , 2012, PLoS Comput. Biol..

[11]  Debora Amadori,et al.  The one-dimensional Hughes model for pedestrian flow: Riemann—type solutions , 2012 .

[12]  Magali Lécureux-Mercier,et al.  Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow , 2011, 1112.4132.

[13]  Marie-Therese Wolfram,et al.  On a mean field game approach modeling congestion and aversion in pedestrian crowds , 2011 .

[14]  R. Colombo,et al.  Nonlocal Crowd Dynamics Models for Several Populations , 2011, 1110.3596.

[15]  Nicola Bellomo,et al.  On the Modeling of Traffic and Crowds: A Survey of Models, Speculations, and Perspectives , 2011, SIAM Rev..

[16]  Cécile Appert-Rolland,et al.  Two-way multi-lane traffic model for pedestrians in corridors , 2011, Networks Heterog. Media.

[17]  R. Colombo,et al.  A CLASS OF NONLOCAL MODELS FOR PEDESTRIAN TRAFFIC , 2011, 1104.2985.

[18]  P. Markowich,et al.  On the Hughes' model for pedestrian flow: The one-dimensional case , 2011 .

[19]  Benedetto Piccoli,et al.  Multiscale Modeling of Granular Flows with Application to Crowd Dynamics , 2010, Multiscale Model. Simul..

[20]  Christian Dogbé,et al.  Modeling crowd dynamics by the mean-field limit approach , 2010, Math. Comput. Model..

[21]  M. J. Castro,et al.  FORCE schemes on unstructured meshes II: Non-conservative hyperbolic systems , 2010 .

[22]  Luca Bruno,et al.  Crowd-structure interaction in lively footbridges under synchronous lateral excitation: A literature review. , 2009, Physics of life reviews.

[23]  Michael Dumbser,et al.  FORCE schemes on unstructured meshes I: Conservative hyperbolic systems , 2009, J. Comput. Phys..

[24]  Chi-Wang Shu,et al.  Revisiting Hughes’ dynamic continuum model for pedestrian flow and the development of an efficient solution algorithm , 2009 .

[25]  Jean-Marie Souriau,et al.  On Geometric Mechanics , 2007 .

[26]  Hongkai Zhao,et al.  Fast Sweeping Methods for Eikonal Equations on Triangular Meshes , 2007, SIAM J. Numer. Anal..

[27]  Adrien Treuille,et al.  Continuum crowds , 2006, ACM Trans. Graph..

[28]  Hongkai Zhao,et al.  A fast sweeping method for Eikonal equations , 2004, Math. Comput..

[29]  Serge P. Hoogendoorn,et al.  Pedestrian route-choice and activity scheduling theory and models , 2004 .

[30]  Ron Kimmel,et al.  Fast Marching Methods , 2004 .

[31]  Roger L. Hughes,et al.  A continuum theory for the flow of pedestrians , 2002 .

[32]  J. Zittartz,et al.  Simulation of pedestrian dynamics using a two dimensional cellular automaton , 2001, cond-mat/0102397.

[33]  Dirk Helbing,et al.  Simulating dynamical features of escape panic , 2000, Nature.

[34]  J A Sethian,et al.  Computing geodesic paths on manifolds. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[35]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[36]  Helbing,et al.  Social force model for pedestrian dynamics. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[37]  P. I. Richards.  Shock Waves on the Highway , 1956 .

[38]  M J Lighthill,et al.  On kinematic waves II. A theory of traffic flow on long crowded roads , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[39]  M J Lighthill,et al.  ON KINEMATIC WAVES.. , 1955 .