Bifurcation under parameter change of Riemann solutions for nonstrictly hyperbolic systems

Abstract

We study the bifurcation of Riemann solutions due to parameter change that alters the type of an umbilic point existing in state space. Solutions with data near generic umbilic points are primarily determined by the local quadratic expansion of flux functions. We observe that near an umbilic point, the bifurcation of the solution is essentially local and its behavior depends solely on the cubic expansion of the flux functions. These phenomena are illustrated for immiscible three-phase flow in porous media, which looses strict hyperbolicity at an isolated point in the interior of the oil–water–gas saturation triangle.