Biomedical Engineering Reference
In-Depth Information
it is not clear what the best choices should be. Nonetheless, the method has
been used successfully in a number of applications, e.g. see the review in [76].
The Ghost Point Method
The ghost point method [50] is another method designed to solve elliptic equa-
tions with irregular and moving boundaries represented by the level set method.
The idea behind this method is similar to the use of what are often called
ghost
points
for discretizing boundary conditions in finite difference methods. In this
context, ghost points are grid points located outside the computational domain,
and are used to enforce boundary conditions.
The method presented in [50] is designed to solve equations of the form
u
∂
=
g
,
∇·
(
β
∇
u
)
=
f
,
(4.72)
in an irregularly shaped domain
, where
β
and
f
are smooth functions defined
on
, and
g
is defined on
∂
, the boundary of
. This is a more restrictive class
of problems than can be handled by the previous two methods described, but
it is a class of problems that often arises. By focusing on this simpler class, a
second-order method with a simple discretization can be employed, which uses
a stencil that has properties which make it easier to solve numerically than the
system created by the previous methods.
To illustrate this method, consider first the one-dimensional problem
(
β
u
x
)
x
=
f
,
(4.73)
with
∂
=
x
I
, and
u
(
x
I
)
=
u
I
. Assume
x
I
lies between the two grid points
x
i
and
x
i
+
1
. For points
x
j
in the interior of the domain, the central difference
discretization, similar to the one used in the immersed interface method, is
used:
β
j
+
u
j
+
1
−
u
j
x
u
j
−
u
j
−
1
x
1
x
=
f
j
.
(4.74)
−
β
j
−
1
2
1
2
At the boundary, the discretization Eq. 4.74 is again employed, but the value
of
u
i
+
1
is not defined because
x
i
+
1
is outside of
. Instead, a ghost value for
u
i
+
1
is computed from the boundary condition using a linear extrapolation:
u
I
+
(
θ
−
1)
u
i
θ
x
I
−
x
i
x
.
u
i
+
1
=
,
where
θ
=
(4.75)
