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

Benjamin Seibold; Morris R. Flynn; Aslan R. Kasimov; Rodolfo R. Rosales; Benjamin Seibold; Morris R. Flynn; Aslan R. Kasimov; Rodolfo R. Rosales

doi:10.3934/nhm.2013.8.745

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.
- Second order,
- traffic model,
- Payne-Whitham,
- Aw-Rascle-Zhang,
- jamiton,
- traveling wave,
- fundamental diagram,
- sensor data,
- effective flow rate
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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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.

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