Singularities in Kinematic Wave and Variational Theories: Supershocks, Solution Properties and Some Exact Solution Methods

According to the duality theory of traffic flow any well-posed kinematic wave (KW) and/or variational theory (VT) problem can be solved with the same methods either on the time-space plane or the time vs vehicle number plane. To achieve this symmetry, the model parameters and the boundary data need to be expressed in a form appropriate for each plane. It turns out, however, that when boundary data that are bounded in one plane are transformed for the other, singular points with infinite density (jumps in vehicle number) sometimes arise. These singularities require a new form of weak solution to the PDEs called an extended solution. Duality theory indicates that these e-solutions must exist and be unique. The paper characterizes these solutions. It shows that their only added feature is a new type of shock that can contain mass called a supershock. Nothing else is required. The evolution laws of these shocks are described. An exact solution method for e-problems with piecewise linear fundamental diagrams (FDs), not necessarily concave, is given. The paper also addresses the special case where the FD is concave so that VT applies. It is shown that if the FD is piecewise linear then sufficient networks used to solve VT problems with the least cost path method continue to be sufficient in the extended case. Thus, the same solution procedure produces exact results in both the conventional and extended cases.

• Record URL:
  http://www.escholarship.org/uc/item/1rw0p740
• Record URL:
  https://merritt.cdlib.org/d/ark%3A%2F13030%2Fm53f63sm/1/producer%2F887890704.pdf
• 
• Availability:
  • Find a library where document is available. Order URL: http://worldcat.org/oclc/887890704
• Corporate Authors:

  University of California, Berkeley

  Institute of Transportation Studies
  McLaughlin Hall
  Berkeley, CA  United States  94720
• Authors:
  • Daganzo, Carlos F
• Publication Date: 2014-4-10

Language

• English

Media Info

• Media Type: Digital/other
• Features: Appendices; Figures; References;
• Pagination: 18p

Subject/Index Terms

• TRT Terms: Kinematics; Methodology; Partial differential equations; Traffic flow theory; Variational inequalities
• Uncontrolled Terms: Fundamental diagram
• Subject Areas: Data and Information Technology; Highways; I71: Traffic Theory;

Filing Info

• Accession Number: 01539277
• Record Type: Publication
• Source Agency: UC Berkeley Transportation Library
• Report/Paper Numbers: UCB-ITS-WP-2014-2
• Files: BTRIS, TRIS
• Created Date: Sep 26 2014 2:41PM