Biomedical Engineering Reference
In-Depth Information
where
h
must be small enough to provide a reasonable approximation to represent
the actual continuous function. This is the basic idea behind the FD method which
will be demonstrated.
The Taylor series is defined as:
f
(
a
)
1
f
(
a
)
2
f
(3)
(
a
)
3
a
)
2
a
)
3
f
(
x
)
≈
f
(
a
)
+
(
x
−
a
)
+
(
x
−
+
(
x
−
+···
(7.3)
!
!
!
which can be rewritten in sigma notation as
∞
f
(
n
)
(
a
)
n
a
)
n
f
(
x
)
≈
(
x
−
!
n
=
0
Referring to Fig.
7.2
, at the point (
i
), we let there be a generic flow field variable
φ
.
Following the Taylor series, we substitute
f
(
a
)as
φ
i
and
f(x)
as
φ
i
+
1
. This gives the
variable at point (
i
+ 1) expanded about the point (
i
)as
dφ
dx
d
2
φ
dx
2
d
3
φ
dx
3
x
2
2
x
2
6
φ
i
+
1
=
φ
i
+
x
+
+
+···
(7.4)
i
i
i
Rearranging to obtain the derivative we get
dφ
dx
d
2
φ
dx
2
d
3
φ
dx
3
x
2
6
φ
i
+
1
−
φ
i
x
2
+
i
=
+
+···
(7.5)
x
i
i
If we only take the first term on the right hand side as the approximation for the
partial derivative of the left hand side, we have
∂φ
∂x
φ
i
+
1
−
φ
i
i
≈
(7.6)
x
which is in the same form as that presented in Eq. (7.2). The terms that are omitted
constitute the truncation error, which we can write as
∂φ
∂x
φ
i
+
1
−
φ
i
i
=
+
O(
x
)
Truncation error
x
∂
2
φ
∂x
2
∂
3
φ
∂x
3
x
2
6
x
2
+
where
O(
x
)
=
+···
(7.7)
i
i
The term O(
x
) signifies the truncation error of the FD approximation, which mea-
sures the accuracy of the approximation and determines the rate at which the error
decreases based on the lowest-order term in the truncated terms. In Eq. (7.7), this
value is
x
which is of first order and hence the FD is
first-order-accurate
. Addition-
ally since the Eq. (7.7) describes the function from a point (
i
) to a position in front
of it at (
i
1), the FD formulation is called a
forward difference
as it is formulated
and therefore biased from information to the right or in front of the origin.
+
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