Recent Advances in the Level Set Method - Biomedical Image Analysis - page 238

The ghost point method is probably the fastest, due to its use of faster linear
solvers, but an actual direct comparison has not been done. Both the immersed
interface method and ghost point method will be faster than the X-FEM approach
on the same mesh. However, to obtain the same accuracy near the interface, the
X-FEM will not require as fine a mesh as the others, and hence can make up
the difference in time by using a coarser mesh to obtain comparable results. A
direct comparison of these three methods is the subject of current research.

4.3.4 Particle Level Set Method

Another modification of the level set method, called the particle level set method,
was proposed by Enright et al. in [38]. In the particle level set method, the level
set function is compared with the motion of particles which move along the
characteristics of the same velocity field. For an interface which is passively
advected using the same velocity field, the particles, in theory, should not cross
the interface. By comparing the motion of the particles with the moving interface,
problems with the location of the interface can be identified and corrected.
Suppose the interface velocity is determined by a velocity field v ( x , t). Given
this velocity, the interface speed function, F , in Eq. 4.5 is given by

F = v · n = v · ∇ φ (4.77)

∇ φ .

Substituting this expression for F into Eq. 4.77 gives the passive interface ad-
vection equation

∂φ
∂ t + v ·∇ φ = 0 .

(4.78)

At the same time, the particles themselves are moving with this same velocity,
v . These two evolutions are coupled together when the particles are checked to
see if any has crossed the interface, which in this case indicates that a particle
has moved from a point where φ> 0 to a point where φ< 0, or vice versa. At
that point, the level set function is “corrected.”
In [38], a large number of particles are randomly distributed uniformly in
the neighborhood of the interface φ = 0. Each particle, p , is assigned a sign, s p ,
to indicate whether it is starting where φ> 0or φ< 0, and is also assigned its
distance, r p , to the interface. As the evolution of the interface and the particles