Civil Engineering Reference
In-Depth Information
Finally, expanding the Taylor series for a function
y=f
(
x
)
at
x=
(
x
i
−
2
h
)
gives the following equation:
′′
()
−
′′′
()
+
′′′ ′
()
2
3
4
yx hyyh
yhyhy h
i
2
2
2
3
2
(
)
=−′
i
i
i
−
2
2
+
(3.11)
i
i
!
!
4
!
If Equation 3.11 is subtracted from Equation 3.10 and the equation for the
first derivative is substituted into the result, a third derivative relationship
is as follows, with an order of error
h
2
:
y
−
2
y
+
2
y
−
y
i
+
2
i
+
1
i
−
1
i
−
2
y
′′′ =
i
3
2
h
If Equation 3.11 is added to Equation 3.10 and the equation for the second
derivative is substituted into the result, a fourth derivative relationship is
as follows with an order of error
h
2
:
y
−
4
y
+
6
y
−
4
y
+
y
′′′ ′
=
i
+
2
i
+
1
i
i
−
1
i
−
2
y
i
4
h
These are the central difference expressions with error order
h
2
. Higher
order expressions can be derived if we include more terms in each expan-
sion. Forward difference expressions can be derived by using Taylor series
expansion of
x =
(
x
i
+h
)
,
x=
(
x
i
+
2
h
)
,
x=
(
x
i
+
3
h
), and so forth. Backward
difference expressions may also be derived by Taylor series expansion of
x=
(
x
i
-h
)
,
x=
(
x
i
-
2
h
)
,
x=
(
x
i
-
3
h
), and so forth. The following are the cen-
tral, forward, and backward difference expressions of varying error order.
These were compiled from “Applied Numerical Methods for Digital Com-
putations,” by James, Smith, and Wolford (1977). They can also be written
in a reverse graphical form that is sometimes used and compiled from
“Numerical Methods in Engineering,” by Salvadori and Baron (1961).
Central difference expressions with error order
h
2
:
y
−
y
y
′
=
i
+
1
i
−
1
i
2
h
′′=
−+
y
2
y
y
i
+
1
i
i
−
1
y
i
2
h
y
−
2
y
+
2
y
−
y
y
′′′ =
i
+
2
i
+
1
i
−
1
i
−
2
i
3
2
h
y
−
4
y
+
6
y
−
4
y
+
y
y
i
+
2
i
+
1
i
i
−
1
i
−
2
′′ ′′=
i
h
4
