A Simple Example of a Two-Dimensional Model for Traffic: Discussion about Assumptions and Numerical Methods

This article presents a two-dimensional (2D) macroscopic model for traffic flow on a network. We considered a 2D conservation law in which we suggested a new method to estimate the direction of the velocity using only information and properties of the network. We present an algorithm for the computation of the 2D velocity on the network and present numerical simulations. Moreover, the numerical approximation of the 2D equation was carried out using an operator and splitting method and we present the results of some simulations on realistic networks. Lastly, we propose a comparison with the well-known cell transmission model method.

[1]  S. Wong,et al.  A higher-order macroscopic model for pedestrian flows , 2010 .

[2]  Dirk Helbing.  A Fluid-Dynamic Model for the Movement of Pedestrians , 1992, Complex Syst..

[3]  Tibye Saumtally,et al.  Modèles bidimensionnels de trafic , 2012 .

[4]  Nikolaos Geroliminis,et al.  Properties of a well-defined Macroscopic Fundamental Diagram for urban traffic , 2011 .

[5]  Serge P. Hoogendoorn,et al.  Genealogy of traffic flow models , 2015, EURO J. Transp. Logist..

[6]  Yanqun Jiang,et al.  A dynamic traffic assignment model for a continuum transportation system , 2011 .

[7]  M. A. Trapeznikova,et al.  Two-dimensional macroscopic model of traffic flows , 2009 .

[8]  Jean-Patrick Lebacque,et al.  Reactive Dynamic Assignment for a Bi-dimensional Traffic Flow Model , 2016, ICONS 2016.

[9]  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.

[10]  Yan-Qun Jiang,et al.  Macroscopic modeling approach to estimate traffic-related emissions in urban areas , 2015 .

[11]  Anna Kormanová.  A Review on Macroscopic Pedestrian Flow Modelling , 2013, Acta Informatica Pragensia.

[12]  Bart De Schutter,et al.  Optimal dynamic route guidance: A model predictive approach using the macroscopic fundamental diagram , 2013, 16th International IEEE Conference on Intelligent Transportation Systems (ITSC 2013).

[13]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[14]  Alfio Quarteroni,et al.  A distributed model of traffic flows on extended regions , 2010, Networks Heterog. Media.

[15]  Jean-Patrick Lebacque,et al.  First Order Macroscopic Traffic Flow Models for Networks in the Context of Dynamic Assignment , 2002 .

[16]  S. Kružkov.  FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES , 1970 .

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

[18]  H. C. Dickinson,et al.  THE PHOTOGRAPHIC METHOD OF STUDYING TRAFFIC BEHAVIOR , 1934 .

[19]  Ludovic Leclercq,et al.  Macroscopic Traffic Dynamics with Heterogeneous Route Patterns , 2015 .

[20]  Régis Duvigneau,et al.  Comparative Study of Macroscopic Pedestrian Models , 2014 .

[21]  C. Daganzo.  THE CELL TRANSMISSION MODEL.. , 1994 .

[22]  Knut-Andreas Lie.  A Dimensional Splitting Method for Quasilinear Hyperbolic Equations with Variable Coefficients , 1999 .

[23]  M. Beckmann.  A Continuous Model of Transportation , 1952 .

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

[25]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[26]  Mauro Garavello,et al.  Traffic Flow on a Road Network , 2005, SIAM J. Math. Anal..

[27]  Francisco G. Benitez,et al.  Outline of Diffusion Advection in Traffic Flow Modeling , 2008 .

[28]  B. Chetverushkin,et al.  Traffic Flow Simulation by 2 D Macro-and Microscopic Models , 2010 .

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

[30]  N. Geroliminis,et al.  An analytical approximation for the macropscopic fundamental diagram of urban traffic , 2008 .

[31]  Chi-Wang Shu,et al.  Revisiting Jiang’s dynamic continuum model for urban cities , 2013 .

[32]  N. Geroliminis,et al.  Existence of urban-scale macroscopic fundamental diagrams: Some experimental findings - eScholarship , 2007 .

[33]  Laurent Gosse,et al.  A Two-Dimensional Version of the Godunov Scheme for Scalar Balance Laws , 2014, SIAM J. Numer. Anal..

[34]  蒋亚琪.  A dynamic traffic assignment model for a continuum transportation system , 2011 .

[35]  Shing Chung Josh Wong,et al.  A review of the two-dimensional continuum modeling approach to transportation problems , 2006 .