HISTORY OF THE METHODS/FLOW CHART
ABOUT THE AUTHOR/CV
In contrast, Level Set Methods and Fast Marching Methods share a common view of how to solve these problems. Rather than focus on the moving boundaries themselves, these techniques exploit a strong link between moving interfaces and equations from computational fluid equations. The idea of building numerical methods for tracking interfaces by exploting this link was discussed in
"Most algorithms place marker particles along the front and advance the positions of the particles in accordance with a set of finite difference approximations to the equations of motion. Such schemes usually go unstable and blow up as the curvature builds around a cusp, since small errors in the position produce large errors in the determination of the curvature. One alternative is to consider the reformulation of the equations of motion as a conservation law with viscocity, and solve these equations with the techniques developed for gas dynamics. These techniques, based on high-order upwind formulations, are particularly attractive, since they are highly stable, accurate, and preserve monotonicity...."
This idea forms one cornerstone of partial differential equations based numerical methods for tracking evolving fronts. It has contributed to two different, yet complementary techniques for tracking evolving fronts: (1) An extremely efficient Fast Marching Method for certain specialized front problems. (2) A more general, but slower all-purpose time-dependent Level Set Method.
Both methods are designed to handle problems in which the separating interfaces develop sharp corners and cusps, change topology, and become truly intricate. Their virtues are most on display in problems in three dimensions (or more) in which interface motion is intricately bound up with complex physics and geometry.