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Networks and Heterogeneous Media

2013, Volume 8, Issue 3: 745-772. doi: 10.3934/nhm.2013.8.745
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Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models

  • 1.
    Temple University, Department of Mathematics, 1805 North Broad Street Philadelphia, PA 19122
  • 2.
    Department of Mechanical Engineering, University of Alberta, Edmonton, AB, T6G 2G8
  • 3.
    4700 King Abdullah University of, Science and Technology, Thuwal 23955-6900
  • 4.
    Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139
  • Received: 01 April 2012 Revised: 01 June 2013

  • Primary: 35B40, 35C07, 35L65, 35Q91; Secondary: 76L05, 91C99.

  • 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.

    Citation: Benjamin Seibold, Morris R. Flynn, Aslan R. Kasimov, Rodolfo R. Rosales. Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models[J]. Networks and Heterogeneous Media, 2013, 8(3): 745-772. doi: 10.3934/nhm.2013.8.745

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  • 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.


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