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FIGURE 8.2
Notation for derivation of difference expressions.
determined such that
u
=
u k 1
for x
=
x k
h
u
=
u k
for x
=
x k
u
=
u k + 1
for x
=
x k
+
h
Apply these conditions to the polynomial and obtain
u k + (
u k + 1
u k 1 )
x k ) + (
u k 1
2 u k +
u k + 1 )
2
u
=
(
x
(
x
x k )
(8.3a)
2 h
2 h 2
with derivatives
u = (
u k + 1
u k 1
)
+ (
u k 1
2 u k
+
u k + 1
)
(
x
x k
)
(8.3b)
2 h
h 2
u = (
u k 1
2 u k +
u k + 1 )
(8.3c)
h 2
By prescribing a value for x, Eqs. (8.3b and c) provide the first and second derivatives at
any point along the polynomial.
As an example, let x
=
x k to obtain
1
2 h (
u k =
u k 1
+
u k + 1
)
(8.4)
1
h 2 (
u k =
u k 1
2 u k +
u k + 1 )
These are the central difference approximations for u and u . These are referred to as second
order because of the use of the second order polynomial and also because of the truncation
error present. A measure of the errors associated with the finite difference technique is
considered in the following section.
To establish forward difference formulas, set x
=
x k 1 in Eqs. (8.3b and c)
1
2 h (
u k 1 =
+
)
3 u k 1
4 u k
u k + 1
1
h 2 (
u k 1 =
u k 1
2 u k +
u k + 1 )
Or if k
1 is shifted to k, these become
1
2 h (
u k =
3 u k
+
4 u k + 1
u k + 2
)
(8.5)
1
h 2 (
u k =
u k
2 u k + 1
+
u k + 2
)
These are the usual second order forward difference formulas.
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