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

u±

≡ 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