Biomedical Engineering Reference
In-Depth Information
coefficients
g
(
G
σ,
k
g
(
G
σ,
k
2
4
i
,
j
)
i
,
j
)
1
2
1
2
n
i
,
j
=
τ
+
(
G
i
,
j
)
2
,
i
,
j
=
τ
+
(
G
i
,
j
)
2
,
2
ε
2
ε
k
=
1
k
=
3
g
(
G
σ,
k
g
(
G
σ,
k
6
i
,
j
)
8
i
,
j
)
1
2
1
2
s
i
,
j
=
τ
+
(
G
i
,
j
)
2
,
e
i
,
j
=
τ
+
(
G
i
,
j
)
2
ε
2
ε
2
k
=
5
k
=
7
and we use either (cf. (11.17))
1
m
i
,
j
=
8
k
=
1
G
i
,
j
2
8
ε
2
+
or (cf. (11.18))
8
1
8
1
m
i
,
j
=
2
+
(
G
i
,
j
)
2
ε
k
=
1
to define diagonal coefficients
c
i
,
j
=
n
i
,
j
+
w
i
,
j
+
s
i
,
j
+
e
i
,
j
+
m
i
,
j
h
2
.
If we define right-hand sides at the
n
th discrete time step by
r
i
,
j
=
m
i
,
j
h
2
u
n
−
1
,
i
,
j
then for DF node corresponding to couple (
i
,
j
) we get the equation
c
i
,
j
u
n
i
,
j
−
n
i
,
j
u
n
i
+
1
,
j
−
w
i
,
j
u
n
i
,
j
−
1
−
s
i
,
j
u
n
i
−
1
,
j
−
e
i
,
j
u
n
i
,
j
+
1
=
r
i
,
j
.
(11.28)
Collecting these equations for all DF nodes and taking into account Dirichlet
boundary conditions, we get the linear system to be solved.
We solve this system by the so-called SOR (successive over relaxation) it-
erative method, which is a modification of the basic Gauss-Seidel algorithm
(see e.g. [46]). At the
n
th discrete time step we start the iterations by setting
u
n
(0)
i
,
j
=
u
n
−
1
i
,
j
,
i
=
1
,...,
m
1
,
j
=
1
,...,
m
2
. Then in every iteration
l
=
1
,...
and
for every
i
=
1
,...,
m
1
,
j
=
1
,...,
m
2
, we use the following two-step procedure:
Y
=
(
s
i
,
j
u
n
(
l
)
i
−
1
,
j
+
w
i
,
j
u
n
(
l
)
i
,
j
−
1
+
e
i
,
j
u
n
(
l
−
1)
i
,
j
+
1
+
n
i
,
j
u
n
(
l
−
1)
i
+
1
,
j
+
r
i
,
j
)
/
c
i
,
j
u
n
(
l
)
i
,
j
=
u
n
(
l
−
1)
i
,
j
+
ω
(
Y
−
u
n
(
l
−
1)
i
,
j
)
.
