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TABLE 8.2
Multiple Position Difference Expressions
Reference
Point 0 and
Order of the
Configuration
∗
Difference Relation
Approximation
h
4
1
u
=
180
u
k
+···
1
/
6 1
4
1
/
2
h
−
1
0
1
u
−
h
6
/
1
u
=
/
−
−
−
2100
u
v
ii
+···
1
5 1
3
1
60
h
1
28
0
28
1
u
k
h
4
/
1
u
=
/
h
2
−
+
240
u
v
i
+···
1
12 1
10
1
1
2
1
u
k
23
h
6
1
/
15 2
11
2
u
=
1
/
20
h
2
116
−
34
16
1
u
−
75600
u
v
iii
+···
k
h
4
1
/
4 1
2
1
u
=
1
/
2
h
3
−
12
0
−
21
u
−
240
u
v
ii
+···
k
h
4
720
u
v
iii
1
/
6 1
4
1
u
i
v
=
1
/
h
4
1
−
4
6
−
41
u
+
+···
k
3
h
2
u
k
+
11
hu
k
+
1
/
6
85
−
108
27
−
4
u
=
0
+
R
(
h
5
)
h
2
7
−
1
−
11
u
+
30
hu
k
+
43
−
60
21
−
4
u
=
0
+
R
(
h
6
)
12
hu
k
−
1
+
3
h
4
u
i
k
+
2 11
−
18
9
−
2
u
=
0
+
R
(
h
5
)
3
h
4
u
i
k
−
12
hu
k
+
1
+
2
−
29
−
18
11
u
=
0
+
R
(
h
5
)
12
h
2
u
k
−
1
−
11
h
4
u
i
k
+
12
−
2
5
−
41
u
=
0
+
R
(
h
5
)
−
11
h
4
u
i
k
+
12
h
2
u
k
+
1
+
12 1
−
4
5
−
2
u
=
0
+
R
(
h
5
)
∗
The symbol
+
indicates the points whose derivatives are used in the formulas.
of derivative boundary conditions can cause the system of finite difference equations to
become unsymmetric, which can be important if efficiency is of concern.
8.1.7 Convergence
In Section 8.1.6, the problem of solving differential equations is reduced to an algebraic
problem in which a set of simultaneous equations is solved. Often, this will require a
computer solution. The error of the finite differences approximation decreases as the mesh
size decreases. More precisely, the error is proportional to a power of the mesh size. Hence,
if an exact solution is not available for comparison,
convergence
of a finite difference solution
is ascertained by adjusting the mesh size. The solution accuracy can also be improved by
switching to a higher order difference scheme.
Consult a textbook on the finite difference method for more formal and complete discus-
sions on convergence characteristics of finite difference approximate solutions.
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