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The Level Set Formulation: An Initial Value PDE
The central idea in what has become known as level set methods is
take the previous work on
curvature flow, links between
evolving interfaces and hyperbolic conservation laws,
and the
use of numerical shock schemes to track interfaces
which remain a graph,
and use an implicit description of interface to extend these
ideas to evolving fronts which do not remain graphs.
This lies at the core of the OsherSethian level set approach:
it casts the problem in one higher dimension. That is, a curve
propagating in the plane is replaced by the problem of a twodimensional
surface evolving in three dimensions.
More precisely, given a curve gamma(t) propagating in its normal
direction with speed F,
a level set function phi(x,y,t) is introduced such that the zero level set
phi =0 is identified at any time t with the evolving curve gamma(t).
The equation of motion for the evolving level set function then
becomes


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AbstractWe devise new numerical algorithms, called PSC algorithms, for following fronts propagating with curvaturedependent speed. The speed may be an arbitrary function of curvature, and the front can also be passively advected by an underlying flow. These algorithms approximate the equations of motion, which resemble HamiltonJacobi equations with parabolic righthandsides, by using techniques from the hyperbolic conservation laws. Nonoscillatory schemes of various orders of accuracy are used to solve the equations, providing methods that accurately capture the formation of sharp gradients and cusps in the moving fronts. The algorithms handle topological merging and breaking naturally, work in any number of space dimensions, and do not require that the moving surface be written as a function. The methods can be also used for more general HamiltonJacobitype problems. We demonstrate our algorithms by computing the solution to a variety of surface motion problems.
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AbstractIn many physical problems, interfaces move with a speed that depends on the local curvature. Some common examples are flame propagation, crystal growth, and oilwater boundaries. We idealize the front as a closed, nonintersecting, initial hypersurface flowing along its gradient field with a speed that depends on the curvature. Because explicit solutions seldom exist, numerical approximations are often used. In this paper, we review some recent work on algorithms for attacking these problems. We show that algorithms based on direct parameterizations of the moving front face considerable difficulties. This is because such algorithms adhere to local properties of the solution, rather than the global structure. Conversely, the global properties of the motion can be captured by embedding the surface in a higherdimensional function. In this setting, the equations of motion can be solved using numerical techniques borrowed from hyperbolic conservation laws. We apply the algorithms to a variety of complicated shapes, showing corner formation and breaking and merging, and conclude with a study of a dumbbell in #R sup 3# moving under its mean curvature. We follow the collapsing dumbbell as the handle pinches off, a singularity develops, and the front breaks into two separate surfaces.
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