Biomedical Engineering Reference
In-Depth Information
Fig. 5.21  A representation of a one-dimensional and two-dimensional uniformly distributed Car-
tesian grid for the finite difference method (full symbols denote boundary nodes and open symbols
denote computational nodes)
at which the error decreases based on the lowest-order term in the truncated terms.
Since the equation describes the function from point (  i ) to a position in front of it at
(  i + 1 ), the finite difference formulation is called a forward difference and is influ-
enced by information to the right or in front of the origin.
Similarly the Taylor series expansion for the variable at point (  i −1) with respect
to point (  i ) gives the backward difference approximation as
=
φ
φφ
(
)
i
i
1
+
O
x
x
x
i
Truncation error
(5.38)
2
3
2
∂ ∆∂ ∆
∆=
φ
x
φ
x
(
)
where
O
x
+
2
3
2
6
x
x
i
i
Combining the forward and backward difference equations we get the central dif-
ference approximation
3
2
φ
φφ
∂ ∆
φ
x
= + ∆
i
+
1
i
1
2
2
O(
x
)
where
O(
∆= +
x
)
(5.39)
3
x
2
x
3
x
i
i
Truncation error
For second order derivatives, such as the diffusion term in the momentum equation,
we sum the Taylor series expansions from Eqs. (5.37) and (5.38), which gives
2
−+
φ
φ φφ
2
i
+
1
i
i
1
2
=
+ ∆
O(
x
)
Central difference
(5.40)
2
2
x
()
x
i
Truncation error
This equation represents the central finite difference for the second order derivative
with respect to x evaluated at the point (  i ).
Visual Representation Figure 5.22 shows the grid points that are involved and their
contribution in terms of being added or subtracted towards the finite difference
approximation for the first and second order derivatives.
 
Search WWH ::




Custom Search