Networks and Heterogeneous Media (NHM)

Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models

Pages: 745 - 772, Volume 8, Issue 3, September 2013      doi:10.3934/nhm.2013.8.745

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Benjamin Seibold - Temple University, Department of Mathematics, 1805 North Broad Street Philadelphia, PA 19122, United States (email)
Morris R. Flynn - Department of Mechanical Engineering, University of Alberta, Edmonton, AB, T6G 2G8, Canada (email)
Aslan R. Kasimov - 4700 King Abdullah University of, Science and Technology, Thuwal 23955-6900, Saudi Arabia (email)
Rodolfo R. Rosales - Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, United States (email)

Abstract: Fundamental diagrams of vehicular traffic flow are generally multi-valued in the congested flow regime. We show that such set-valued fundamental diagrams can be constructed systematically from simple second order macroscopic traffic models, such as the classical Payne-Whitham model or the inhomogeneous Aw-Rascle-Zhang model. These second order models possess nonlinear traveling wave solutions, called jamitons, and the multi-valued parts in the fundamental diagram correspond precisely to jamiton-dominated solutions. This study shows that transitions from function-valued to set-valued parts in a fundamental diagram arise naturally in well-known second order models. As a particular consequence, these models intrinsically reproduce traffic phases.

Keywords:  Second order, traffic model, Payne-Whitham, Aw-Rascle-Zhang, jamiton, traveling wave, fundamental diagram, sensor data, effective flow rate.
Mathematics Subject Classification:  Primary: 35B40, 35C07, 35L65, 35Q91; Secondary: 76L05, 91C99.

Received: April 2012;      Revised: June 2013;      Available Online: October 2013.