Biomedical Engineering Reference
In-Depth Information
FIGURE 13.1
Y
Two-dimensional grid for the finite-difference
method, where
i
denotes grid spaces in the x-direction and
j
denotes
grid spaces in the y-direction. Note that the spacing does not have to
be uniform across the entire geometry. By using either the forward
difference method or the backward difference method, parameters of
interest can be obtained from known boundary conditions and
known initial conditions.
j
+
1
Δ
x
Δ
y
j
j
−
1
i
−
1
i
i
+
1
X
where
x
0
and
y
0
are the initial points within the mesh. For any given function (
f
) that can
be described at each of these grid points, the Taylor expansion states that the function at
neighboring points can be defined using the spatial partial derivative of that function, pro-
vided that the function is continuous. In one dimension, the Taylor expansion for a contin-
uous function is defined by
i
Δx1
@x
2
i
Δx
2
@x
3
i
Δx
3
f
i1
1
5 f
i
1
@
f
@x
2
@
2
f
6
@
3
f
1
1
1
1
...
i
Δx1
@x
2
i
Δx
2
@x
3
i
Δx
3
f
i2
1
5 f
i
2
@
f
@x
1
2
@
2
f
1
6
@
3
f
2
12
...
ð
13
:
4
Þ
Using
Equation 13.4
, the first spatial partial derivative of
f
at
i
can be solved using a for-
ward difference (
Equation 13.5
) or a backward difference (
Equation 13.6
):
i
5
@f
@x
f
i1
1
2 f
i
Δx
1H:O:T:
ð
13
:
5
Þ
i
5
@f
@x
f
i
2
f
i
2
1
Δx
1H:O:T:
ð
13
:
6
Þ
Δx
2
or more. These are typically
neglected, which gives a truncation error in the finite-difference method of the first degree.
The H.O.T. can be ignored because they will be relatively small as compared to the
Δx
term (if and only if
H.O.T. denotes all of the higher-order terms of order
Δx
approaches zero). If
Equations 13.5
and
13.6
are added together
and averaged, a formulation for the central difference is obtained, which has a second-
order accuracy (if the H.O.T. are ignored).
i
5
@f
@x
f
i1
1
2 f
i2
1
Δx
1H:O:T:
ð
13
:
7
Þ
2
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