IDEAS home Printed from
   My bibliography  Save this paper

Singularities in kinematic wave and variational theories: supershocks, solution properties and some exact solution methods


  • Daganzo, Carlos F


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 PDE's that we call 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 and we call 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.

Suggested Citation

  • Daganzo, Carlos F, 2014. "Singularities in kinematic wave and variational theories: supershocks, solution properties and some exact solution methods," Institute of Transportation Studies, Research Reports, Working Papers, Proceedings qt1rw0p740, Institute of Transportation Studies, UC Berkeley.
  • Handle: RePEc:cdl:itsrrp:qt1rw0p740

    Download full text from publisher

    File URL:;origin=repeccitec
    Download Restriction: no

    More about this item


    Engineering; traffic flow theory; fundamental diagram; kinematic waves; shockwaves; variational theory;

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:cdl:itsrrp:qt1rw0p740. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Lisa Schiff). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.