Geoscience Reference
In-Depth Information
are evaluated using the latest estimates of primary variables. To
determine the matrices
[
L
i
]
,
[
R
i
]
, and
{
S
i
}
, the derivatives of each function (
F
1
,
F
2
,
F
3
k
)
with respect to each primary dependent variable (
A
,
Q
,
Q
tk
) at two ends of each reach
are required. The derivatives with respect to auxiliary variables are transformed into
the primary-variable derivatives through chain-rule expansion. The dependence of the
Manning
n
and flow density on primary variables should be considered.
The equation system (5.189) for all reaches and the associated boundary conditions
are solved using a block-bidiagonal algorithm. Suppose that there is a relation of the
form:
[
L
i
]
and
[
R
i
]
[
E
i
]{
δ
W
1
}+[
H
i
]{
δ
W
i
}+{
G
i
}=
0
(5.191)
Then deriving
{
δ
W
i
}
from Eq. (5.189) and substituting it into Eq. (5.191) yields
[
E
i
+
1
]{
δ
W
1
}+[
H
i
+
1
]{
δ
W
i
+
1
}+{
G
i
+
1
}=
0
(5.192)
where the coefficient matrices are
[
E
i
+
1
]=[
E
i
]
(5.193)
]
−
1
[
H
i
+
1
]=−[
H
i
][
L
i
[
R
i
]
(5.194)
]
−
1
{
G
i
+
1
}=−[
H
i
][
L
i
{
S
i
}+{
G
i
}
(5.195)
Comparing Eqs. (5.189) and (5.192) at
i
=
1 results in
[
E
2
]=[
L
1
]
,
[
H
2
]=[
R
1
]
, and
{
G
2
}={
S
1
}
. The forward sweep can then be carried out using Eqs. (5.193)-(5.195)
=
...
from
i
2, 4,
,
I
. At the end of the forward sweep, the following relation is obtained:
[
E
I
]{
δ
W
1
}+[
H
I
]{
δ
W
I
}+{
G
I
}=
0
(5.196)
The boundary conditions at upstream and downstream points can be linearized
locally and written in the vector form:
[
α
]{
δ
W
1
}+[
β
]{
δ
W
I
}+{
γ
}=
0
(5.197)
Eliminating
{
δ
W
1
}
from Eqs. (5.196) and (5.197) yields
E
I
]
−
1
H
I
]+[
β
]]
−
1
E
I
]
−
1
{
δ
W
I
} = [−[
α
][
[
{[
α
][
{
G
I
}−{
γ
}}
(5.198)
{
δ
W
I
}
is determined, the remaining unknown vectors can be obtained by the
“return-sweep” inversion of Eq. (5.189):
Once
]
−
1
]
−
1
{
δ
W
i
}=−[
L
i
[
R
i
]{
δ
W
i
+
1
}−[
L
i
{
S
i
}
(5.199)
It can be seen from Eqs. (5.193)-(5.195), (5.198), and (5.199) that this algorithm
requires (
I
−
1) inversions of a
(
2
+
N
)
×
(
2
+
N
)
matrix for each iteration step.
