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

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

doi:10.3934/nhm.2013.8.745

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

Benjamin Seibold, Morris R. Flynn, Aslan R. Kasimov, Rodolfo Rubén Rosales

Research output: Contribution to journal › Article › peer-review

31 Scopus citations

Abstract

Fundamental diagrams of vehicular traiic ow are generally multivalued in the congested ow regime. We show that such set-valued fundamental diagrams can be constructed systematically from simple second order macroscopic traiic 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 traiic phases. © American Institute of Mathematical Sciences.

Original language	English (US)
Pages (from-to)	745-772
Number of pages	28
Journal	Networks and Heterogeneous Media
Volume	8
Issue number	3
DOIs	https://doi.org/10.3934/nhm.2013.8.745
State	Published - Oct 4 2013

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: The authors would like to thank J.-C. Nave for helpful discussions. The authors would like to acknowledge the support by the National Science Foundation. R. R. Rosales and B. Seibold were supported through grants DMS-1007899 and DMS-1007967, respectively. In addition, R. R. Rosales wishes to acknowledge partial support by the NSF through grants DMS-0813648, DMS-1115278, and DMS-0907955, B. Seibold through grants DMS-0813648 and DMS-1115269, and A. R. Kasimov through grant DMS-0907955. M. R. Flynn wishes to acknowledge support by the NSERC Discovery Grant Program.

ASJC Scopus subject areas

Applied Mathematics
General Engineering
Statistics and Probability
Computer Science Applications

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10.3934/nhm.2013.8.745

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title = "Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models",

abstract = "Fundamental diagrams of vehicular traiic ow are generally multivalued in the congested ow regime. We show that such set-valued fundamental diagrams can be constructed systematically from simple second order macroscopic traiic 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 traiic phases. {\textcopyright} American Institute of Mathematical Sciences.",

author = "Benjamin Seibold and Flynn, {Morris R.} and Kasimov, {Aslan R.} and Rosales, {Rodolfo Rub{\'e}n}",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: The authors would like to thank J.-C. Nave for helpful discussions. The authors would like to acknowledge the support by the National Science Foundation. R. R. Rosales and B. Seibold were supported through grants DMS-1007899 and DMS-1007967, respectively. In addition, R. R. Rosales wishes to acknowledge partial support by the NSF through grants DMS-0813648, DMS-1115278, and DMS-0907955, B. Seibold through grants DMS-0813648 and DMS-1115269, and A. R. Kasimov through grant DMS-0907955. M. R. Flynn wishes to acknowledge support by the NSERC Discovery Grant Program.",

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doi = "10.3934/nhm.2013.8.745",

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T1 - Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models

AU - Seibold, Benjamin

AU - Flynn, Morris R.

AU - Kasimov, Aslan R.

AU - Rosales, Rodolfo Rubén

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: The authors would like to thank J.-C. Nave for helpful discussions. The authors would like to acknowledge the support by the National Science Foundation. R. R. Rosales and B. Seibold were supported through grants DMS-1007899 and DMS-1007967, respectively. In addition, R. R. Rosales wishes to acknowledge partial support by the NSF through grants DMS-0813648, DMS-1115278, and DMS-0907955, B. Seibold through grants DMS-0813648 and DMS-1115269, and A. R. Kasimov through grant DMS-0907955. M. R. Flynn wishes to acknowledge support by the NSERC Discovery Grant Program.

PY - 2013/10/4

Y1 - 2013/10/4

N2 - Fundamental diagrams of vehicular traiic ow are generally multivalued in the congested ow regime. We show that such set-valued fundamental diagrams can be constructed systematically from simple second order macroscopic traiic 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 traiic phases. © American Institute of Mathematical Sciences.

AB - Fundamental diagrams of vehicular traiic ow are generally multivalued in the congested ow regime. We show that such set-valued fundamental diagrams can be constructed systematically from simple second order macroscopic traiic 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 traiic phases. © American Institute of Mathematical Sciences.

UR - http://hdl.handle.net/10754/562949

UR - http://arxiv.org/abs/arXiv:1204.5510v2

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M3 - Article

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VL - 8

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JO - Networks and Heterogeneous Media

JF - Networks and Heterogeneous Media

IS - 3

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

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