Biomedical Engineering Reference
In-Depth Information
For stability reasons, if
θ<
x
, then Eq. 4.75 is replaced with
u
i
+
1
=
u
I
. Using
Eq. 4.75 in Eq. 4.74 produces the following discretization for the point near the
boundary:
u
I
−
u
i
θ
x
u
i
−
u
i
−
1
x
1
x
=
f
i
.
(4.76)
β
i
+
−
β
i
−
1
2
1
2
In multiple dimensions, this same extrapolation technique is carried out along
each coordinate direction.
The resulting discretization is only first-order accurate near the boundary,
but is second-order accurate overall. This is due to the confinement of the first-
order error to the nodes adjacent to the boundary. On the other hand, the linear
system that comes from this discretization can be solved using faster conjugate
gradient-type algorithms. Increasing the order of the extrapolation to compute
u
i
+
1
can result in a linear system that is more difficult to solve numerically,
because of the non-symmetric stencil, and hence is not preferred.
This method is used primarily for its simplicity, while still yielding second-
order convergence overall. For problems where the accuracy at the boundary
is critical, this is probably not the preferred method, especially if the solution is
difficult to resolve near the boundary. The method has been used in a handful
of applications, for example, see [124].
Comparison of the Elliptic Equation Solvers
The algorithms presented here, for solving elliptic equations in conjunction with
the level set method, vary significantly in sophistication, complexity, and capa-
bility. The X-FEM approach is by far the most difficult to construct, but is also
the most general, and has the greatest potential to solve challenging problems.
In particular, the X-FEM approach provides a much more accurate representa-
tion of the solution near the boundary, a property that is of critical importance
when the velocity of the interface depends on this very value.
The immersed interface method and ghost point method, on the other hand,
are built much more easily, and still produce accurate solutions. The immersed
interface method handles a larger range of equations than does the ghost point
method, which is the most restrictive in this regard. Between these two methods,
the immersed interface method is more accurate at the boundary, but at the
expense of a more difficult system of equations to solve numerically.
