Recent Advances in the Level Set Method - Biomedical Image Analysis - page 233

important when having to interpolate the data to determine the front speed on
the boundary contour. Of the three methods, this is the only one that has this
capability.

The Immersed Interface Method

The immersed interface method, introduced by LeVeque and Li [74], has also
been coupled to the level set method [76, 78]. Like the X-FEM described above,
the immersed interface method is designed to solve elliptic equations which arise
in a variety of physical applications. The advantage of the immersed interface
method is that it is second-order accurate, even near the interface where jump
conditions may appear.
The immersed interface method is designed to solve equations of the form

∇· ( β ( x ) ∇ u ( x )) + κ ( x ) u ( x ) = f ( x ) , (4.58)

where the coefficient functions β , κ , and f may have discontinuities across an
interface . The function f may also have a delta function singularity, which
often arises, for example, from surface tension in multiphase flow.
The key idea in the immersed interface method is to modify the discretization
of Eq. 4.58 in such a way that the jump discontinuities and singularities are
accounted for, leading to a fully second-order method. At points away from
the interface, where the coefficient functions and the solution are smooth, the
standard central difference approximation is used. However, for grid points
which are near the interface, an additional grid point is added to the usual
central difference stencil to account for a second-order Taylor approximation
around a point on the interface.
To illustrate how this method works, consider the one-dimensional problem

( β u x ) x + κ u = f , x ∈ [0 , 1] \ α, (4.59)

u + − u − = a , at x = α, (4.60)

u x − u x = b ,

at x = α, (4.61)

where u − is the value of u on the interval [0 ,α ], and u + is the value of u on the
interval [ α, 1]. Suppose that the point α is located between the uniformly spaced
grid points x i and x i + 1 . The idea is to calculate coefficients γ i − 1 , γ i , γ i + 1 , and an