Geoscience Reference
In-Depth Information
Figure 4.6
2-D finite difference grid.
To apply the exponential difference scheme (4.21), one can rearrange Eq. (4.50) as
u
x
∂
u
y
∂
2
f
2
f
f
c
∂
f
c
∂
x
−
ε
+
y
−
ε
=
S
(4.52)
x
2
y
2
∂
∂
∂
∂
Discretizing the convection and diffusion terms with the exponential difference
scheme (4.21) in the
x
- and
y
-directions, respectively, yields
a
P
f
i
,
j
=
a
W
f
i
−
1,
j
+
a
E
f
i
+
1,
j
+
a
S
f
i
,
j
−
1
+
a
N
f
i
,
j
+
1
+
S
i
,
j
(4.53)
where
u
x
2
u
x
a
W
=
x
exp
(
P
x
/
2
)/
sinh
(
P
x
/
2
)
,
a
E
=
x
exp
(
−
P
x
/
2
)/
sinh
(
P
x
/
2
)
,
2
u
y
2
u
y
2
a
S
=
y
exp
(
P
y
/
2
)/
sinh
(
P
y
/
2
)
,
a
N
=
y
exp
(
−
P
y
/
2
)/
sinh
(
P
y
/
2
)
,
(4.54)
a
P
=
a
W
+
a
E
+
a
S
+
a
N
with
P
x
=
/ε
c
.
Scheme (4.53) is also called the five-point hybrid finite analytic scheme (Li and
Yang, 1990; Lu and Si, 1990). Chen and Li (1980) derived the analytic solution for
Eq. (4.50) with constant velocity, diffusivity, and source term at the nine-point cluster
shown in Fig. 4.6 and established a nine-point finite analytic scheme. The nine-point
analytic scheme also has the capability of automatically upwinding and is very stable.
The details can be found in Chen and Li (1980).
u
x
x
/ε
c
and
P
y
=
u
y
y
