Computational Fluid Dynamics
Lecture 3
Discretization Continued….
A fourth order approximation to
f
x
∂
∂
can be found using Taylor Series.
()
(
)
(
)
( ) ( ) ( )
00
0
0
0
0
0
0
0
0
2
2
0:
22
1
:
44
0
:
88
0
:
16
16
v
a
fx
x b
x c
d
x e
x
f x
abcde
abd e
abd
e
0
′
+∆ +
+
−∆ +
−∆ =
++++=
′
+−− =
′′
+++ =
′′′
′
+++
=
4
th
order solution:
( )
0
0
41
32
3 4
x
x
x
x
xx
+∆
−
−∆
+ ∆
−
− ∆
−
∆∆
Another Example:
1
st
derivative near a boundary
1
st
grid point
3
rd
order one sided Dirichlet B.C.using Taylor Series expansions at grid points 2, 3, and 4 with
hx
= ∆
.
[]
11
2
3
4
23
4
21
4
31
4
1
1
2
1
1
26
4
3
99
3
14
3
9
3
2
fA
f
B
f
C
f
D
f
h
hh
f
f
hf
f
f
h
f
f
hf
h f
h f
h
f
f
hf
h f
h f
h
f
Bf
Bhf
B
f
B
f
Af
Cf
C hf
C h f
C
h f
h
Df
D hf
D
h
σ
′
=+
+
+
′
=+ +
+
+
′
+
+
+
′
+
+
+
′
′
′
+ + +
+
++ +
′
++
+
3
9
2
fD
h
f
+
1
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1
11
2
1
0
23
9
20
2
49
0
63 2
ABCD f
h
fBCD
f
BC D
h
f
BC
D
hf
+++
=
′′
=++
++
=
′′′
=
_____________________________________________________________________________
12
1
1111
0
0123
1
19
02
0
22
149
0
0
632
Matrix inversion gives:
11
1.8333
6
18
3.0
6
39
1.5
26
36
which agrees exactly with page 61 of ATP.
1
11
18
6
A
B
C
D
A
B
C
D
f
ff
zz
=
=−
==
∂
=−+
∂∆
[]
34
92
−+
Consistency:
The Finite difference approximation is consistent if the truncation error for the whole equation goes to
zero as the mesh size goes to zero.
Accuracy is rate that F.D. approximation approaches zero as
0
x
∆
→
.
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 Spring '07
 Slinn
 finite difference, Finite difference method, Finite differences, Ak eikj∆x

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