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 rightside 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 higherorder 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 NavierStokes 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 fτ denote the normal and tangential components of the