Example 3
Use Taylor-series expansion and find a second order finite difference approximation to
x∂φ∂ at the point i and with help of the points i, i+1 and i+2.
i+2
i
i+1
a
b
x
A:
)(
62
4
,
3
33
,
2
22
,
,,1
xO
x
a
x
a
x
a
jiji
ji
jiji
∆+
∂
φ∂
+
∂
φ∂
+
∂
φ∂
+φ=φ
+
B:
)(
62
4
,
3
33
,
2
22
,
,,2
xO
x
b
x
b
x
b
jiji
ji
jiji
∆+
∂
φ∂
+
∂
φ∂
+
∂
φ∂
+φ=φ
+
Derive a second order forward approximation of the first order derivative:
Eliminate the second order derivatives:
⇒−× BA
a
b
2
2
()
()
ji
ji
ji
jijiji
x
bab
x
bb
x
b
a
b
a
b
a
b
,
3
3
32
,
2
2
22
,
2
,
2
2
,2,1
2
2
6
1
2
1
1
∂
φ∂
×−+
+
∂
φ∂
×−+
∂
φ∂
×
−+φ×
−=φ−φ
++
∆∝ab,
)(1
3
,
2
2
,2,1
2
2
,
2
∆+φ
−−φ−φ=
∂
φ∂
×
−
++
O
a
b
a
b
x
b
a
b
jijiji
ji
)(
1
2
2
,
2
2
,2,1
2
2
,
∆+
−
φ
−−φ−φ
=
∂
φ∂
++
O
b
a
b
a
b
a
b
x
jijiji
ji
