Supplydemand Diagrams and a New Framework for Analyzing the Inhomogeneous LighthillWhithamRichards Model
 W.L. Jin,
 L. Chen,
 Elbridge Gerry Puckett
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Abstract
Traditionally, the LighthillWhithamRichards (LWR) models for homogeneous and inhomogeneous roads have been analyzed in fluxdensity space with the fundamental diagram of the fluxdensity relation. In this paper, we present a new framework for analyzing the LWR model, especially the Riemann problem at a linear boundary in which the upstream and downstream links are homogeneous and initially carry uniform traffic. We first review the definitions of local supply and demand functions and then introduce the socalled supplydemand diagram, on which a traffic state can be represented by its supply and demand, rather than as density and flux as on a fundamental diagram. It is wellknown that the solutions to the Riemann problem at each link are selfsimilar with a stationary state, and that the wave on the link is determined by the stationary state and the initial state. In our new framework, there can also exist an interior state next to the linear boundary on each link, which takes infinitesimal space, and admissible conditions for the upstream and downstream stationary and interior states can be derived in supplydemand space. With an entropy condition consistent with a local supplydemand method in interior states, we show that the stationary states exist and are unique within the solution framework. We also develop a graphical scheme for solving the Riemann problem, and the results are shown to be consistent with those in the literature. We further discuss asymptotic stationary states on an inhomogeneous ring road with arbitrary initial conditions and demonstrate the existence of interior states with a numerical example. The framework developed in this study is simpler than existing ones and can be extended for analyzing the traffic dynamics in general road networks.
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 Title
 Supplydemand Diagrams and a New Framework for Analyzing the Inhomogeneous LighthillWhithamRichards Model
 Book Title
 Transportation and Traffic Theory 2009: Golden Jubilee
 Book Subtitle
 Papers selected for presentation at ISTTT18, a peer reviewed series since 1959
 Pages
 pp 603635
 Copyright
 2009
 DOI
 10.1007/9781441908209_30
 Print ISBN
 9781441908193
 Online ISBN
 9781441908209
 Publisher
 Springer US
 Copyright Holder
 SpringerVerlag US
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 Editors
 Authors

 W.L. Jin ^{(1)}
 L. Chen ^{(2)}
 Elbridge Gerry Puckett ^{(1)}
 Author Affiliations

 1. University of California, California, U.S.A
 2. University of Science and Technology of China, HongKong, China
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