Biomedical Engineering Reference
In-Depth Information
One could attempt to alleviate this by reducing the problem's size by using large
spatial and temporal scales and offsetting the loss of accuracy incurred by employing
higher-order (in terms of accuracy) methods. However, unless the DE is extremely
stable, using too high an order can introduce spurious solutions and for second order
initial value problems, such as the diffusion equation, Press et al. [37] recommend
that one should go no higher than second order in space and time. Thus for multi-
dimensional DEs one must resign oneself to long running times, high memory re-
quirements and a certain loss of accuracy. If this is not practical a lower dimensional
model can in some circumstances be used to approximate higher dimensions [45].
4.2.3.2
Finite difference schemes used
For the numerical solutions given here, we used the alternating direction implicit
(ADI) method in two and three space dimensions [1]. This is recommended for
diffusive problems as it is fast, second-order accurate in space and time, uncondi-
tionally stable and unlike simpler schemes, allows for examination of the solution at
all time-steps [1, 30, 37]. The equation to be approximated is:
C
2
C
t
−
D
=
P
(
x
,
t
)
−
S
(
x
)
C
−
λ
C
(4.42)
where:
Q
for points inside the source during synthesis
P
(
x
,
t
)=
(4.43)
0
else
and:
η
for points inside sinks
S
(
x
)=
(4.44)
0
else
where a sink is a local high concentration of an NO-binding moiety such as a heme-
protein. Thus in two dimensions we have [1]:
D
y
u
n
u
n
+
1
/
2
−
u
n
x
u
n
+
1
/
2
+
δ
=
+
P
(
i
,
j
,
n
)
1
2
∆
t
λ
+
S
(
i
,
j
)
2
u
n
+
1
/
2
+
(4.45)
u
n
−
D
y
u
n
+
1
P
i
2
u
n
+
1
−
u
n
+
1
/
2
x
u
n
+
1
/
2
+
δ
1
=
+
,
j
,
n
+
1
2
∆
t
λ
+
S
(
i
,
j
)
2
u
n
+
1
(4.46)
u
n
+
1
/
2
−
+
Extending these equations to three space variables leads to a method that is unsta-
ble for any useful spatial and temporal scales [1] and so the following variant is used.
Instead of taking three third-steps one generates three subsequent approximations for
the solution at the advanced time-step, the third one being used as the actual solution.
We obtain the first approximation
u
n
+
1
at time-step
n
+
1 in the following way [1]:
D
2
δ
x
u
n
+
1
+
u
n
+
δ
z
u
n
u
n
+
1
−
u
n
y
u
n
+
δ
=
)
−
λ
2
u
n
+
1
+
u
n
(4.47)
t
+
P
(
n
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