8 SARAH D. OLSON AND ANITA T. LAYTON
2.2. Immersed interface method. The IB method is a powerful numerical
method but as with almost all numerical methods, it has some drawbacks. IB
methods are typically first-order accurate in the infinity norm. Also, because the
boundary forces are spread out, the jump discontinuity in the pressure solution is
not captured; instead, the computed pressure approximation has a sharp gradient
near the immersed boundary. In fact, the pressure approximation has O(1) Linf
error and error in the pressure gradient translates into inaccuracy in boundary
velocity. As a result, the IB method is known to exhibit “leakage” (i.e., an area
in 2-d or a volume in 3-d enclosed by the immersed boundary or surface tends to
decrease in time), unless corrective procedures are applied (e.g., [32,136]).
As a remedy, LeVeque and Li developed the immersed interface method, which
yields second-order accurate approximates and robustly captures jump discontinu-
ities in the solution and its derivatives. The key idea of the immersed interface
method is to incorporate known jumps in the solution into the finite difference
stencil. This method has been successfully implemented and used for both 2-d and
3-d fluids.
2.2.1. A simple elliptic interface problem. To motivate the immersed interface
method, let’s first consider a simple 1-d elliptic interface problem:
(2.15) (βux)x = f + σδ(x α), 0 x, α 1 ,
where f is smooth but β is discontinuous at x = α. We re-state the problem in
terms of the jump conditions:
(βux)x = f, x (0,α) (α, 1), (2.16)
[u]
u+

u−
= 0, [βux] = σ, [βuxx] = 0 . (2.17)
where

lim
→0+
u(α ± ).
Suppose we discretize Eq. (2.16) using a centered difference scheme. We will
separately consider grid points that are sufficiently far from x = α such that the
associated finite difference stencils do not cross x = α (which will be referred to
as “regular points”), and those whose stencils do cross x = α (“irregular”). For a
regular point xi, the discretized form of Eq. (2.16) is
(2.18)
1
h2
βi+
1
2
(ui+1 ui) βi−
1
2
(ui ui−1) = fi
where h denotes the mesh width.
The finite stencil for an irregular point xi, where xi α xi+1, will need to
be modified in order to attain second-order accuracy. Also, a correction term Ci
will be added:
(2.19) aui−1 + bui + cui+1 = fi + Ci
To determine the coefficients and correction term Ci, we apply Taylor expansion of
ui+1 and ui−1 around α:
u(xi+1)
=u+(α)
+ (xi+1 α)ux
+(α)
+
1
2
(xi+1
α)2uxx(α) +
+
O(h3)
(2.20)
u(xi−1)
=u−(α)
+ (xi α)ux
−(α)
+
1
2
(xi
α)2uxx(α)
+
O(h3)
(2.21)
Now recall that
[βux] = σ
β+ux +

β−ux
= σ
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