The proposed research involves the numerical solution of various free boundary value problems modelling flows of liquids in Hele-Shaw cells. Such Hele-Shaw flows, as they are called, are important in their own right, and they serve as very useful models of more complicated phenomena involving free boundaries. The proposed research will produce robust numerical schemes capable of handling many different types of nonlinear phenomena whose analytic resolution is currently impossible. The phenomenon of "fingering" of one fluid as it flows into another is a common, easily observed occurrence that is very difficult to describe quantitatively. The proposed research will suggest and evaluate different kinds of methods for studying numerically the set of mathematical equations that govern the evolution in time of an unstable finger. A successful numerical scheme for this particular problem will have much broader applicability to a host of problems in fluid dynamics.
