Environmental Engineering Reference
In-Depth Information
ˆ
ijk
=
ˆ(
x
i
,
y
j
,
z
k
),
u
ijk
=
(
x
i
−
1
/
2
,
y
j
,
z
k
),
v
ijk
=
(
x
i
,
y
j
−
1
/
2
,
z
k
)
u
v
w
ijk
=
w
(
x
i
,
y
j
,
z
k
−
1
/
2
), μ
k
=
μ(
z
k
),
v
ijk
=
v
(
x
i
,
y
j
,
z
k
−
1
/
2
)
The second-order discrete approximation of the operators
A
i
and continuity
Eq. (
2.11
) have the following form (invariable indices
i
,
j
,
k
are omitted)
−
μ
k
ˆ
i
+
1
−
ˆ
i
+
ˆ
i
−
1
A
1
ˆ
u
i
+
1
ˆ
i
+
1
−
u
i
ˆ
i
−
1
2
+
˃ˆ
i
3
ijk
=
(2.40)
2
ʔ
x
(ʔ
x
)
2
−
μ
k
ˆ
j
+
1
−
ˆ
j
+
ˆ
j
−
1
A
2
ˆ
2
v
j
+
1
ˆ
j
+
1
−
v
j
ˆ
j
−
1
+
˃ˆ
j
3
ijk
=
(2.41)
2
2
ʔ
y
(ʔ
y
)
A
3
ˆ
ijk
=
w
k
+
1
ˆ
k
+
1
−
w
k
ˆ
k
−
1
−
μ
k
+
1
(ˆ
k
+
1
−
ˆ
k
)
−
μ
k
(ˆ
k
−
ˆ
k
−
1
)
(ʔ
+
˃ˆ
k
3
(2.42)
2
2
ʔ
z
z
)
(
u
i
+
1
−
u
i
)
+
(
v
j
+
1
−
v
j
)
+
(
w
k
+
1
−
w
k
)
=
0
(2.43)
ʔ
x
ʔ
y
ʔ
z
A
i
)
∗
We immediately obtain the form of adjoint operators
(
if we substitute
u
,
v
,
w
, and
g
, respectively. To show how the
boundary conditions are discretised, we give only one example (see [
41
]formore
details). Let
u
ijk
be a positive value of the
u
-component of the velocity at the left
boundary point
M
w
, and
ˆ
in (
2.40
)-(
2.43
)by
−
u
,
−
v
,
−
0, i.e.,
the point
M
belongs to
S
−
, and conditions (
2.8
) and (
2.22
) are approximated as
=
(
x
1
/
2
,
y
j
,
z
k
)
of the grid domain. Then,
U
n
=−
u
1
jk
<
μ
k
(ˆ
0
jk
−
ˆ
1
jk
)
ʔ
u
1
jk
(ˆ
0
jk
−
ˆ
1
jk
)
2
−
=
0
,
g
0
jk
=
g
1
jk
(2.44)
x
and
A
i
∗
are positive
3), the discrete operators
A
i
Thus, for any
i
(
i
=
1
,
2
,
semidefinite, and they are skew-symmetric if
μ
=
˃
=
0 and
S
is the coast line
(
U
n
=
0 everywhere at
S
).
The problems (
2.4
)-(
2.11
) and (
2.18
)-(
2.24
) are solved in time with the sym-
metrized double-cycle componentwise splitting method by Marchuk [
23
,
41
], i.e.,
within each double time step interval
(
t
n
−
ʔ
t
,
t
n
+
ʔ
t
)
themain and adjoint numerical
schemes have the form
n
n
n
3
−
i
4
−
i
=−
2
A
i
3
−
i
ʦ
−
−
ʦ
−
ʦ
−
3
3
3
n
4
−
i
+
ʦ
−
(
i
=
1
,
2
)
3
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