Equation-Free Analysis of Macroscopic Behavior in Traffic and Pedestrian Flow

Equation-free methods make it possible to analyze the evolution of a few coarse-grained or macroscopic quantities for a detailed and realistic model with a large number of fine-grained or microscopic variables, even though no equations are explicitly given on the macroscopic level. This will facilitate a study of how the model behavior depends on parameter values including an understanding of transitions between different types of qualitative behavior. These methods are introduced and explained for traffic jam formation and emergence of oscillatory pedestrian counter flow in a corridor with a narrow door.

[1]  B. Krauskopf,et al.  Control based bifurcation analysis for experiments , 2008 .

[2]  Neil Fenichel.  Geometric singular perturbation theory for ordinary differential equations , 1979 .

[3]  Ioannis G. Kevrekidis,et al.  Projecting to a Slow Manifold: Singularly Perturbed Systems and Legacy Codes , 2005, SIAM J. Appl. Dyn. Syst..

[4]  Y. Sugiyama,et al.  Traffic jams without bottlenecks—experimental evidence for the physical mechanism of the formation of a jam , 2008 .

[5]  Ioannis G. Kevrekidis,et al.  Equation-free: The computer-aided analysis of complex multiscale systems , 2004 .

[6]  W. Beyn,et al.  Chapter 4 – Numerical Continuation, and Computation of Normal Forms , 2002 .

[7]  A. Schadschneider,et al.  Simulation of pedestrian dynamics using a two dimensional cellular automaton , 2001 .

[8]  Rainer Berkemer,et al.  Implicit Methods for Equation-Free Analysis: Convergence Results and Analysis of Emergent Waves in Microscopic Traffic Models , 2013, SIAM J. Appl. Dyn. Syst..

[9]  Michael Schreckenberg,et al.  Pedestrian and evacuation dynamics , 2002 .

[10]  A. Seyfried,et al.  Basics of Modelling the Pedestrian Flow , 2005, physics/0506189.

[11]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[12]  Equation-Free Analysis of Macroscopic Behavior in Traffic and Pedestrian Flow , 2014 .

[13]  Christophe Vandekerckhove,et al.  Stability and stabilization of the constrained runs schemes for equation-free projection to a slow manifold , 2012 .

[14]  Nakayama,et al.  Dynamical model of traffic congestion and numerical simulation. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[15]  Giovanni Samaey,et al.  Equation-free multiscale computation: algorithms and applications. , 2009, Annual review of physical chemistry.

[16]  J. Starke,et al.  Experimental bifurcation analysis of an impact oscillator—Tuning a non-invasive control scheme , 2013 .

[17]  H. Haken.  Advanced Synergetics: Instability Hierarchies of Self-Organizing Systems and Devices , 1983 .

[18]  Andreas Schadschneider,et al.  Quantitative analysis of pedestrian counterflow in a cellular automaton model. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Mohcine Chraibi,et al.  Efficient and validated simulation of crowds for an evacuation assistant , 2012, Comput. Animat. Virtual Worlds.

[20]  Christophe Vandekerckhove,et al.  A common approach to the computation of coarse-scale steady states and to consistent initialization on a slow manifold , 2011, Comput. Chem. Eng..

[21]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[22]  D. Barton,et al.  Systematic experimental exploration of bifurcations with noninvasive control. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  Hermann Haken,et al.  Synergetics: An Introduction , 1983 .

[24]  Giovanni Samaey,et al.  Equation-free modeling , 2010, Scholarpedia.

[25]  Jens Starke,et al.  Equation-Free Detection and Continuation of a Hopf Bifurcation Point in a Particle Model of Pedestrian Flow , 2012, SIAM J. Appl. Dyn. Syst..

[26]  R. E. Wilson,et al.  Bifurcations and multiple traffic jams in a car-following model with reaction-time delay , 2005 .

[27]  I. G. Kevrekidis,et al.  Esaim: Mathematical Modelling and Numerical Analysis Analysis of the Accuracy and Convergence of Equation-free Projection to a Slow Manifold , 2022 .

[28]  C. W. Gear,et al.  Equation-Free, Coarse-Grained Multiscale Computation: Enabling Mocroscopic Simulators to Perform System-Level Analysis , 2003 .

[29]  H. Haken,et al.  Synergetics , 1988, IEEE Circuits and Devices Magazine.

[30]  Ping Liu,et al.  Coarse-grained particle model for pedestrian flow using diffusion maps. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  I. Gasser,et al.  Bifurcation analysis of a class of ‘car following’ traffic models , 2004 .

[32]  A. Schadschneider,et al.  Ordering in bidirectional pedestrian flows and its influence on the fundamental diagram , 2012 .

[33]  D. Helbing.  Traffic and related self-driven many-particle systems , 2000, cond-mat/0012229.

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