Geoscience Reference
In-Depth Information
4.2.2.2 Discretization of multidimensional unsteady
problems
The important issue for discretizing 2-D and 3-D unsteady problems is how to arrange
spatial difference operators in different directions or fractional steps. Widely used
approaches include the full-domain implicit (or explicit) method, alternating direction
implicit method, and operator splitting method.
Full-domain implicit (or explicit) method
The full-domain implicit (or explicit) method discretizes all spatial derivatives in 2-D
and 3-D differential equations at the same time level. For example, extending Eq.
(4.33) to the 2-D diffusion equation
t
=
ε
c
2
f
2
f
∂
f
∂
x
2
+
∂
+
S
(4.55)
∂
y
2
∂
∂
yields
f
n
+
1
i
f
n
+
1
i
,
j
f
i
,
j
2
f
n
+
1
i
,
j
f
n
+
1
i
f
n
+
1
i
,
j
2
f
n
+
1
i
,
j
f
n
+
1
i
,
j
−
−
+
−
+
−
1,
j
+
1,
j
−
1
+
1
=
ε
c
θ
+
x
2
y
2
t
f
i
−
1,
j
−
2
f
i
,
j
+
f
i
+
1,
j
f
i
,
j
−
1
−
2
f
i
,
j
+
f
i
,
j
+
1
S
n
+
θ
i
,
j
+
ε
(
1
−
θ)
+
+
c
x
2
y
2
(4.56)
0, Eq. (4.56) is explicit in both
x
- and
y
-directions; its sufficient and
necessary stability condition is
r
When
θ
=
h
2
1,
Eq. (4.56) is implicit in both
x
- and
y
-directions and unconditionally stable; however,
the discretized equation at each grid point involves five unknowns and usually needs
to be solved by an iteration method.
=
ε
t
/
≤
1
/
4, if
x
=
y
=
h
. When
θ
=
c
Alternating direction implicit method
The alternating direction implicit (ADI) method was proposed by Peaceman and
Rachford (1955). It usually divides the computation into two or three steps and
discretizes the spatial derivatives implicitly in only one direction at each step.
Consider a 2-D partial differential equation:
∂
f
=
L
x
f
+
L
y
f
(4.57)
∂
t
where
L
x
and
L
y
are differential operators in the
x
- and
y
-directions. The correspond-
ing two-step ADI difference equations can be written as
⎨
f
n
+
1
/
2
f
n
−
x
f
n
+
1
/
2
y
f
n
=
+
t
/
2
(4.58)
⎩
f
n
+
1
f
n
+
1
/
2
−
x
f
n
+
1
/
2
y
f
n
+
1
=
+
t
/
2
