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Since from Eq. (8.8) this expression is equal to zero (plus
R
), the coefficients of the first three
terms may be set equal to zero, i.e.,
a
+
a
0
+
a
1
=
0
−
1
−
a
+
1
+
a
1
=
0
or
a
=
1
/
2
,
0
=
0
,
1
=−
1
/
2
−
1
−
1
a
+
a
1
=
0
−
1
The remaining terms provide a measure of the discretization error
R
. Then, Eq. (8.8) becomes
h
3
3!
u
1
2
u
−
1
1
2
u
1
hu
0
+
+
0
−
=−
+···
0
or
h
2
6
1
2
h
(
−
u
0
=
u
0
u
−
1
+
0
+
u
1
)
−
+···
Thus, the truncation error is
h
2
6
u
0
R
=−
+···
(8.9)
h
2
This is often expressed as
R
,
where the symbol
O
means terms of a certain
order
or higher. The lowest power of
h
in
R
gives the
order
of the approximation or of the
error. Thus, we say that the first order derivative is of second order accuracy, and this finite
difference formula is a second order approximation.
The higher the power of h, the better the
approximation and the faster the convergence of the approximation to the true value.
=
Error
=
O
(
)
8.1.3
Computational Molecules
A convenient notation for expressing difference relations is referred to as
computational
molecules
or
difference stars.
For example, the first central difference derivative of
u
(Eq. 8.1),
with respect to
x,
can be written as
1
2
h
x
−
u
k
=
1
0
1
u
+
R
(8.10)
|−−◦−−|
k
−
1
k
k
+
1
where indicates the location of
k,
the
reference point.
Also,
h
has been replaced by
h
x
in
order to make clear that the derivative with respect to
x
is being approximated. In the
y
direction,
−−
k
+
1
1
|
|
◦
1
2
h
y
u
k
=
u
+
R
k
(8.11)
0
|
|
−−
−
1
k
−
1
Similar molecules hold for higher derivatives.
In the central difference formulas of Eqs. (8.10) and (8.11), the reference point is located
at the center of the node (grid) points included in the difference expression. The central
difference formulas (or central difference stars) are not symmetric for odd derivatives, e.g.,
1
2
h
−
u
k
=
1
0
1
u
+
R
(8.12)
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