American Institute of Mathematical Sciences

Advanced
September  2009, 4(3): 605-623. doi: 10.3934/nhm.2009.4.605

Numerical approximation of continuous traffic congestion equilibria

Fethallah Benmansour 1, , Guillaume Carlier 1, , Gabriel Peyré 1, and  Filippo Santambrogio 2,

1. 

CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, Pl. de Lattre de Tassigny, 75775 Paris Cedex 16, France, France, France
2. 

Université Paris Dauphine, Laboratoire CEREMADE, UMR CNRS 7534, Place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16

Received  March 2009 Revised  June 2009 Published  July 2009

Starting from a continuous congested traffic framework recently introduced in [8], we present a consistent numerical scheme to compute equilibrium metrics. We show that equilibrium metric is the solution of a variational problem involving geodesic distances. Our discretization scheme is based on the Fast Marching Method. Convergence is proved via a $\Gamma$-convergence result and numerical results are given.
Keywords: Fast Marching Method., traffic congestion, eikonal equation, subgradient descent, Wardrop equilibria.
Mathematics Subject Classification: Primary: 49M25, 65K10, 90C2.
Citation: Fethallah Benmansour, Guillaume Carlier, Gabriel Peyré, Filippo Santambrogio. Numerical approximation of continuous traffic congestion equilibria. Networks & Heterogeneous Media, 2009, 4 (3) : 605-623. doi: 10.3934/nhm.2009.4.605
[1]

Emiliano Cristiani, Fabio S. Priuli. A destination-preserving model for simulating Wardrop equilibria in traffic flow on networks. Networks & Heterogeneous Media, 2015, 10 (4) : 857-876. doi: 10.3934/nhm.2015.10.857
[2]

Jean-Bernard Baillon, Guillaume Carlier. From discrete to continuous Wardrop equilibria. Networks & Heterogeneous Media, 2012, 7 (2) : 219-241. doi: 10.3934/nhm.2012.7.219
[3]

Mark A. Peletier, Marco Veneroni. Stripe patterns and the Eikonal equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 183-189. doi: 10.3934/dcdss.2012.5.183
[4]

Jing Zhang, Jianquan Lu, Jinde Cao, Wei Huang, Jianhua Guo, Yun Wei. Traffic congestion pricing via network congestion game approach. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020378
[5]

Chadi Nour. Construction of solutions to a global Eikonal equation. Conference Publications, 2007, 2007 (Special) : 779-783. doi: 10.3934/proc.2007.2007.779
[6]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
[7]

Jaime Cruz-Sampedro. Schrödinger-like operators and the eikonal equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 495-510. doi: 10.3934/cpaa.2014.13.495
[8]

Jingmei Zhou, Xiangmo Zhao, Xin Cheng, Zhigang Xu. Visualization analysis of traffic congestion based on floating car data. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1423-1433. doi: 10.3934/dcdss.2015.8.1423
[9]

Jian Gu, Xiantao Xiao, Liwei Zhang. A subgradient-based convex approximations method for DC programming and its applications. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1349-1366. doi: 10.3934/jimo.2016.12.1349
[10]

Herbert Gajewski, Jens A. Griepentrog. A descent method for the free energy of multicomponent systems. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 505-528. doi: 10.3934/dcds.2006.15.505
[11]

Alberto Bressan, Ke Han. Existence of optima and equilibria for traffic flow on networks. Networks & Heterogeneous Media, 2013, 8 (3) : 627-648. doi: 10.3934/nhm.2013.8.627
[12]

Gabriella Bretti, Roberto Natalini, Benedetto Piccoli. Fast algorithms for the approximation of a traffic flow model on networks. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 427-448. doi: 10.3934/dcdsb.2006.6.427
[13]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020184
[14]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020196
[15]

Ting Hu. Kernel-based maximum correntropy criterion with gradient descent method. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4159-4177. doi: 10.3934/cpaa.2020186
[16]

Kai Wang, Lingling Xu, Deren Han. A new parallel splitting descent method for structured variational inequalities. Journal of Industrial & Management Optimization, 2014, 10 (2) : 461-476. doi: 10.3934/jimo.2014.10.461
[17]

Wei-Zhe Gu, Li-Yong Lu. The linear convergence of a derivative-free descent method for nonlinear complementarity problems. Journal of Industrial & Management Optimization, 2017, 13 (2) : 531-548. doi: 10.3934/jimo.2016030
[18]

Alberto Bressan, Khai T. Nguyen. Optima and equilibria for traffic flow on networks with backward propagating queues. Networks & Heterogeneous Media, 2015, 10 (4) : 717-748. doi: 10.3934/nhm.2015.10.717
[19]

Shu-Yu Hsu. Super fast vanishing solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5383-5414. doi: 10.3934/dcds.2020232
[20]

Iryna Egorova, Johanna Michor, Gerald Teschl. Rarefaction waves for the Toda equation via nonlinear steepest descent. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2007-2028. doi: 10.3934/dcds.2018081

2019 Impact Factor: 1.053

Tools

Metrics

  • PDF downloads (28)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences