Biomedical Engineering Reference
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the leading truncation error term. The order of an approximation
indicates how fast the error is reduced when the grid is reined; it
does not indicate the absolute magnitude of the error. The error is
thus reduced by a factor of 2, 4, 8, or 16 in the irst-, second-, third- or
fourth-order approximations, respectively. It should be remembered
that this rule is valid only for suficiently small spacing; the deinition
of small enough depends on the proile of the function G ( x ).
5.2.4
Approximation of the Second Derivative
Second derivatives appear in the diffusive terms in Navier-Stokes
equation. To estimate the second derivative at a point, one may use
the approximation for the irst derivative twice. This is the only
possible approach when the luid properties are variable, since we
need the derivative of the product of diffusion coeficient and the irst
derivative. Next, we consider approximation to the second derivative;
application to the diffusive term in the conservation equation will be
discussed later. Geometrically, second derivative is the slope of the
line tangent to the curve representing the irst derivative (see Fig.
5.9. By inserting approximations for the irst derivatives at locations
x i +1 and x i , an approximation for the second derivative is obtained:
u
G
u
G
¥µ ¥µ
¦¶ ¦¶
-
§· §·
u
u
¥ µ
2
x
x
u
G
i
1
i
z
(5.55)
¦ ¶
u § ·
2
x
-
x
x
i
1
i
i
All such approximations involve data from at least three points.
In Eq. (5.55), the outer derivative was estimated by FDS. For inner
derivatives, one may use a different approximation, e.g., BDS; this
results in the following expression:
¥ µ
2
G
G
G
(-
x
x
)
(
x
-)-(
x
x
-
x
)
u
G
i
1
i
i
-1
i
-1
i
1
i
i
i
1
i
-1
(5.56)
¦ ¶
u§ ·
2
2
x
(
x
-) (
x
x
-)
x
i
1
i
i
i
-1
i
One could also use the CDS approach which requires the irst
derivative at x i - 1 and x i +1 . A better choice is to evaluate G /∂x at points
halfway between x i and x i +1 and x i and x i- 1 . The CDS approximations
for these irst derivatives are
GG
-
G
u
G
u
G
¥µ
¥µ
i
1
i
i
i
-1
z
and
z
(5.57)
¦¶
¦¶
§·
u
§·
u
x
x
-
x
x
x
-
x
1
1
i
i
1
i
i
-
i
i
-1
2
2
respectively. The resulting expression for the second derivatives is
 
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