METHODS FOR PROPAGATING INTERFACES

Adaptivity provides a way to construct optimal algorithms for tracking moving interfaces which arise in a wide collection of physical applications. Here, we summarize the development and interconnection between Narrow Band Level Set Methods and Fast Marc hing Methods, which provide ecient techniques for tracking fronts. We end with a small collection of examples to demonstrate the applicability of the techniques. Over the past ten years, a collection of numerical techniques have been de- veloped to track propagating interfaces that arise in physical phenomena. These techniques allow for evolution under complex speed laws, including the eects of curvature and anisotropy , easily couple to the underlying physics, allow for natural topological change in the evolving interface, including splitting and merging, and are unchanged in three or more space dimensions. They take on a partial dier- ential equations approach to the interface problem, casting the motion as either an initial value or boundary value partial dierential equation, and rely on finite dierence approximations to provide convergence, consistent numerical techniques of high order. Because of this reliance on finite dierence schemes, the error of the solution is known at the start, and can be rigorously controlled. At their core, these techniques hinge on the "viscosity solutions" view of the underlying equations, in which the correct weak solution is chosen which inter- prets the propagating front as a physical boundary between two regions. Two such techniques are Level Set techniques, in troduced by Osher and Sethian (6), and Fast Marching Methods, introduced by Sethian in (12). Both grew out of the theory of curve and surface evolution developed in (9), (10), (11), which develops the notion of w eak solutions and entropy limits for evolving interfaces, and links upwind numerical methodology for hyperbolic conservation laws to fron t