SIMULATING IMMERSED BIOLOGICAL STRUCTURES 9
which, together with
u+
=
u−
and uxx
+
= uxx,

can be used to eliminate from
Eq. (2.20) all u terms with right-side limits to yield
u(xi+1)
=u−(α)
+ (xi+1 α)
β−
β+
ux
−(α)
+
σ
β+
+
1
2
(xi+1
α)2uxx(α)
+
O(h3)
(2.22)
Substituting Eqs. (2.21) and (2.22) into Eq. (2.19) and rearranging, one obtains
aui−1 + bui + cui+1
=(a + b +
c)u−(α)
+ (xi−1 α)a + (xi α)b +
β−
β+
(xi+1 α)c ux
−(α)
+ c(xi+1 α)
σ
β+
+
1
2
(xi−1
α)2a
+ (xi
α)2b
+
β−
β+
(xi+1
α)2c uxx(α)−
=(βux)x = fi + Ci
By matching the coefficients in front of
u−(α),
ux
−(α),
and uxx(α),

we obtain
the following linear system
a + b + c = 0 (2.23)
(xi−1 α)a + (xi α)b +
β−
β+
(xi+1 α)c ux
−(α)
= 0 (2.24)
1
2
(xi−1
α)2a
+ (xi
α)2b
+
β−
β+
(xi+1
α)2c
=
β−
(2.25)
which can be solved for a, b, and c. Then by equating the higher-order terms, we
obtain an expression for the correction terms:
(2.26) Ci = c(xi+1 α)
σ
β+
Ω+
-
Ω
Γ
Figure 2. Model configuration for an immersed interface problem.
2.2.2. Stokes and Navier-Stokes equations. The singular boundary forces in-
duce jump discontinuities in the pressure and normal derivatives of the velocity.
But unlike the elliptic interface problem in the preceeding subsection, those jump
conditions are not given explicitly. Instead, they can be computed from the bound-
ary forces. Let fn and denote the normal and tangential components of the
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