The 2-dimensional Riemann problem for a 2x2 hyperbolic conservation law - I. Isotropic media

摘要

We construct the solutions for a two-dimensional (2-D) Riemann problem for a 2 x 2 hyperbolic nonlinear system based upon the Keyfitz-Kranzer-Isaacson-Temple model. The system is applicable to polymer flooding of an oil reservoir; the parameterization can be adjusted to model either isotropic or anisotropic media. For isotropic media, the solutions are obtained by two methods. The first method utilizes a transformation into a one-dimensional (1-D) Cauchy problem. Such a transformation requires conformity of the x- and y-directional fluxes in the system. The second method involves a 2-D constructive technique which can be used more generally for solving systems. For the isotropic media case, we explicitly construct solutions for the so-called single and four quadrant Riemann problems by both methods and demonstrate the equality of the solutions. This has relevance as a test for the 2-D solution method, as existence and uniqueness results for solutions of systems in 1-D are known, whereas no such results exist for systems in 2-D. [References: 35]