Biomedical Engineering Reference
In-Depth Information
additional constant,
C
i
, so that the approximation
γ
i
−
1
u
i
−
1
+
γ
i
u
i
+
γ
i
+
1
u
i
+
1
+
κ
i
u
i
=
f
i
+
C
i
(4.62)
is second-order accurate, with jump conditions Eqs. 4.60 and 4.61.
To determine the
γ
i
's, Taylor expansions are taken about the point
x
=
α
to
get
1
2
(
x
i
−
1
−
α
)
2
u
xx
+
O
(
x
3
)
,
u
(
x
i
−
1
)
=
u
−
+
(
x
i
−
1
−
α
)
u
x
+
(4.63)
1
2
(
x
i
−
α
)
2
u
xx
+
O
(
x
3
)
,
u
(
x
i
)
=
u
−
+
(
x
i
−
α
)
u
x
+
(4.64)
1
2
(
x
i
+
1
−
α
)
2
u
xx
+
O
(
x
3
)
.
u
(
x
i
+
1
)
=
u
+
+
(
x
i
+
1
−
α
)
u
x
+
(4.65)
These expansions are inserted into Eq. 4.62, and the
u
+
terms are eliminated
from the equation by using the jump conditions Eqs. 4.60 and 4.61, combined
with the equation
(
β
u
x
)
x
+
κ
u
+
=
(
β
u
x
)
x
+
κ
u
−
,
(4.66)
which comes from the continuity of
f
in Eq. 4.59. The function
f
on the right side
of Eq. 4.62 is replaced with the approximation from the left side,
f
=
(
β
u
x
)
x
+
κ
u
−
. This results in the following equation:
γ
i
−
1
u
−
+
(
x
i
−
1
−
α
)
u
x
+
2
(
x
i
−
1
−
α
)
2
u
xx
1
+
γ
i
u
−
+
(
x
i
−
α
)
u
x
+
2
(
x
i
−
α
)
2
u
xx
+
γ
i
+
1
u
−
+
a
+
(
x
i
+
1
−
α
)(
u
x
+
b
)
+
1
2
(
x
i
+
1
−
α
)
2
u
xx
−
1
b
β
x
−
κ
a
β
2
(
x
i
−
α
)
2
u
xx
=
β
x
u
x
+
β
u
xx
+
κ
u
−
+
C
i
+
O
(
x
3
)
u
−
+
(
x
i
−
α
)
u
x
+
1
+
κ
(4.67)
The coefficients
γ
i
−
1
,
γ
i
,
γ
i
+
1
, and
C
i
are now chosen so that Eq. 4.67 holds up
to second order. This leads to the following equations:
γ
i
−
1
+
γ
i
+
γ
i
+
1
=
0
,
(4.68)
γ
i
−
1
(
x
i
−
1
−
α
)
+
γ
i
(
x
i
−
α
)
+
γ
i
+
1
(
x
i
+
1
−
α
)
+
κ
(
x
i
−
α
)
=
β
x
,
(4.69)
γ
i
−
1
(
x
i
−
1
−
α
)
2
+
γ
i
(
x
i
−
α
)
2
+
γ
i
+
1
(
x
i
+
1
−
α
)
2
+
κ
(
x
i
−
α
)
2
=
2
β,
(4.70)
γ
i
+
1
a
+
b
(
x
i
+
1
−
α
)
−
(
b
β
x
−
κ
a
)(
x
i
+
1
−
α
)
2
β
1
2
=
C
i
.
(4.71)