Geoscience Reference
In-Depth Information
Assuming that
φ
1
and
φ
i
are related by
i
−
φ
=
c
i
φ
+
d
i
(4.213)
i
−
1
i
and substituting Eq. (4.213) into Eq. (4.209) leads to
φ
=
c
i
+
1
φ
+
d
i
+
1
(4.214)
i
i
+
1
where
c
i
+
1
=
a
E
,
i
/(
a
P
,
i
−
c
i
a
W
,
i
)
,
d
i
+
1
=
(
b
i
+
d
i
a
W
,
i
)/(
a
P
,
i
−
c
i
a
W
,
i
)
(4.215)
Comparing the boundary condition (4.210) with Eq. (4.214) at the first point yields
the coefficients
c
2
and
d
2
:
c
2
=
a
E
,1
/
a
P
,1
,
d
2
=
b
1
/
a
P
,1
(4.216)
and then the coefficients
c
i
+
1
and
d
i
+
1
are determined by Eq. (4.215) in the order of
increasing
i
from 2 to
m
1. This is the forward sweep.
At the last grid point, substituting
−
φ
=
c
m
φ
+
d
m
into the boundary condition
m
−
1
m
(4.211) yields
φ
=
(
b
m
+
d
m
a
W
,
m
)/(
a
P
,
m
−
c
m
a
W
,
m
)
(4.217)
m
i
can be obtained using Eq. (4.213) in the order of decreasing
i
from
m
to
2. This is the backward sweep.
The Thomas algorithm is a direct solution method; it is particularly economical and
requires only 5
m
Now all
φ
4 operations (multiplications and divisions) for linear problems.
For non-linear problems, the coefficients and source term in Eq. (4.209) are related to
the solution of
−
φ
, so an iteration procedure is needed. At each iteration step, an initial
guess is given to
φ
at each point, the coefficients and source term are evaluated using
the guessed
φ
, and then the double sweep calculations are performed to obtain the
new value of
at each point. This procedure is repeated until a convergent solution
is reached. However, to obtain the convergent solution, it is necessary that
φ
|
a
P
,
i
|
>
|
a
E
,
i
|+|
a
W
,
i
|
(4.218)
4.5.2 Jacobi and Gauss-Seidel iteration methods
Jacobi and Gauss-Seidel methods solve the algebraic equations point by point in
a certain order. They can be used in the solution of 1-D, 2-D, and 3-D prob-
lems. Consider the following algebraic equation resulting from a 2-D second-order
