Geoscience Reference
In-Depth Information
where
y
are the spatial difference operators of
L
x
and
L
y
, respectively. In
the first step, the operator
L
x
is approximated implicitly, while
L
y
is approximated
explicitly. In the second step,
L
x
is treated explicitly, while
L
y
is treated implicitly.
For example, the ADI scheme for the 2-D diffusion equation (4.55) is
x
and
⎧
⎨
f
n
+
1
/
2
i
,
j
f
i
,
j
−
xx
f
n
+
1
/
2
i
,
j
S
n
+
1
/
2
i
,
j
yy
f
i
,
j
)
+
=
ε
(δ
+
δ
c
t
/
2
(4.59)
⎩
f
n
+
1
i
,
j
f
n
+
1
/
2
i
,
j
−
xx
f
n
+
1
/
2
i
,
j
yy
f
n
+
1
i
,
j
S
n
+
1
i
,
j
=
ε
(δ
+
δ
)
+
c
t
/
2
where
δ
xx
and
δ
yy
are the central difference operators corresponding to the second
y
2
, respectively.
The ADI scheme (4.59) is unconditionally stable. Because only a 1-D difference
equation with three unknowns needs to be solved at each step, it is simpler than the
full-domain implicit difference equation (4.56).
2
x
2
and
2
derivatives
∂
/∂
∂
/∂
Operator splitting method
The operator splitting method was proposed by Yanenko (1971) and others. It
splits the differential equation into several operators and then treats each operator
separately.
Consider a differential equation:
∂
f
=
L
1
f
+
L
2
f
(4.60)
∂
t
where
L
1
and
L
2
are spatial operators. The corresponding difference equation is
written as
f
n
+
1
f
n
−
1
f
n
+
1
2
f
n
+
1
=
+
(4.61)
t
where
2
are the difference operators of
L
1
and
L
2
, respectively.
Eq. (4.61) can be split as
1
and
⎨
f
n
+
1
/
2
f
n
−
1
f
n
+
1
/
2
=
t
(4.62)
⎩
f
n
+
1
f
n
+
1
/
2
−
2
f
n
+
1
=
t
Note that the operator splitting method can be used for 1-D, 2-D, and 3-D
problems. In other words, operators
L
1
and
L
2
in Eq. (4.60) can be one-, two-,
or three-dimensional.
The consistency of the operator splitting method for linear differential equations
has been proven, but not yet for nonlinear differential equations. However, extensive
