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6 Answers to Supplementary Problems
3.8 Centered difference approximation of the fourth order derivative
u i+1 =u i +
d
1 u
dX 1
i
X
1!
+
d
2 u
dX 2
i
X
2
2!
+
d
3 u
dX 3
i
X
3
3!
+
+
d
4 u
dX 4
i
X
4
4!
+
d
5 u
dX 5
i
X
5
5!
+
d
6 u
dX 6
i
X
6
6!
+ · · · ,
(6.63)
u i−1 =u i −
d
1 u
dX 1
i
X
1!
+
d
2 u
dX 2
i
X
2
2!
−
d
3 u
dX 3
i
X
3
3!
+
+
d
4 u
dX 4
i
X
4
4!
−
d
5 u
dX 5
i
X
5
5!
+
d
6 u
dX 6
i
X
6
6!
− · · · .
(6.64)
Summing up the expressions (6.63) and (6.64) and rearranging for the second order
derivative gives:
d
2 u
dX 2
i
=
u i+1 − 2u 1 + u i−1
X 2
−
d
4 u
dX 4
i
X
2
12
−
d
6 u
dX 6
i
X
4
360
− · · · . (6.65)
u i+2 = u i +
d
1 u
dX 1
i
(2X )
1!
+
d
2 u
dX 2
i
(2X )
2
2!
+
d
3 u
dX 3
i
(2X )
3
3!
+
+
d
4 u
dX 4
i
(2X )
4
4!
+
d
5 u
dX 5
i
(2X )
5
5!
+
d
6 u
dX 6
i
(2X )
6
6!
+ · · · , (6.66)
u i−2 = u i −
d
1 u
dX 1
i
(2X )
1!
+
d
2 u
dX 2
i
(2X )
2
2!
−
d
3 u
dX 3
i
(2X )
3
3!
+
+
d
4 u
dX 4
i
(2X )
4
4!
−
d
5 u
dX 5
i
(2X )
5
5!
+
d
6 u
dX 6
i
(2X )
6
6!
− · · · . (6.67)
Summing up the expressions (6.66) and (6.67) and considering the expression for
the second order derivative given in Eq. (6.65) allows to express the fourth order
derivative finally as:
d
4 u
dX 4
i
=
u i+2 − 4u i+1 + 6u i − 4u i−1 + u i−2
((X ) 4
−
d
6 u
dX 6
i
((X )
2
6
O((X 2 )
.
(6.68)