Civil Engineering Reference
In-Depth Information
The first term on the right side in Equation 3.3 is the area or the trapezoid
and the rest is the error as follows:
′′
3
fh
Higherorder termsT
i
E
=−
+
T
12
The exact integral,
I
, can be derived using this error relation from two sep-
arate approximate integrals. The derivation is omitted from this text, but
may be found in “Applied Numerical Methods for Digital Computations,”
by James, Smith, and Wolford (1977). The improved integral is based on
two approximate integrals with a strip where
h
2
< h
1
as follows:
II
I I
≅+
−
−
=+
−
I
I
h
1
h
2
2
h
1
h
2
hh
hI
h
2
2
h
2
2
2
2
h
2
1
−
1
1
h
2
2
h
I
−
I
1
h
h
2
h
1
2
I
≅
2
h
−
1
1
h
2
If the second integration uses a strip one-half that of the first with
h
1
/h
2
=
2,
the equation becomes the following:
()
−
()
−
2
I
2
21
I
h
2
h
1
I
≅
2
This is a defined as a first-order extrapolation. If two first-order extrapo-
lations are performed, then their results can be combined into a second-
order relationship with the following:
()
−
()
−
4
I
2
21
I
h
2
h
1
I
≅
4
The general
n
th order extrapolation would take the following form with
n
being the order of extrapolation:
n
I
4
41
−
−
I
h
2
h
1
I
≅
n
