The Classification of Solutions of Quadratic Riemann Problems. I.
Abstract
We are interested in classifying the solutions of Riemann problems for the 2 x 2 conservation laws which have homogeneous quadratic flux functions. Such flux functions approximate an arbitrary 2 x 2 system in a neighborhood of an isolated point where strict hyperbolicity fails. This problem was motivated by Marchesin and Paes-Leme who discovered such a singularity in a system of equations arising in oil reservoir simulation. Schaeffer, Shearer, Marchesin and Paes-Leme solved the Riemann problem for this system in a neighborhood of the singular point. Isaacson and Temple outlined a program for classifying such singularities by means of locating normal forms for the equivalence classes of equations generated by linear changes in dependent variables. A 2-parameter family of such normal forms were found by Plohr. In the important work of Schaeffer and Shearer a new normal form was found which reduced the classification of integral curves to a theorem of Darboux on the classification of umbilic points for homogeneous cubic equations. The integral curves fall into four isomorphism classes, called Regions I-IV. In this paper we give the solution of the Riemann problem for the systems in Region IV which exhibit up-down symmetry. A presentation of the solutions of the corresponding systems in Regions II and III will follow.
Document Details
- Document Type
- Technical Report
- Publication Date
- Dec 01, 1985
- Accession Number
- ADA163706
Entities
People
- B. Plohr
- B. Temple
- D. Marchesin
- E. Isaacson
Organizations
- University of Wisconsin–Madison