Geoscience Reference
In-Depth Information
A
n
+
1
i
A
i
B
i
δ
=
+
h
i
(5.57)
Q
n
+
1
i
Q
i
=
+
δ
Q
i
(5.58)
where
∗
denotes the estimates at the last iteration step,
δ
h
is the water stage (flow
depth) increment,
δ
Q
is the flow discharge increment, and
B
is the channel width at
the water surface.
Substituting Eqs. (5.57) and (5.58) into Eq. (5.55) yields
ψ
1
−
ψ
+
θ
−
θ
t
B
i
+
1
δ
B
i
δ
h
i
+
1
+
h
i
x
δ
Q
i
+
1
x
δ
Q
i
t
=−
ψ
1
−
ψ
A
i
)
−
θ
A
i
+
1
−
A
i
+
1
)
−
A
i
Q
i
+
1
−
Q
i
)
t
(
t
(
−
x
(
−
θ
1
q
n
+
1
l
,
i
q
n
+
1
l
,
i
Q
i
+
1
−
Q
i
)
+
θ
[
ψ
−
x
(
1
+
(
1
−
ψ)
]
+
q
l
,
i
+
1
+
(
q
l
,
i
]
+
(
1
−
θ)
[
ψ
1
−
ψ)
(5.59)
Eq. (5.59) can be written as
a
i
δ
h
i
+
b
i
δ
Q
i
+
c
i
δ
h
i
+
1
+
d
i
δ
Q
i
+
1
=
p
i
(5.60)
B
i
/
B
i
+
1
/
where
a
i
=
(
1
−
ψ)
t
,
b
i
=−
θ/
x
,
c
i
=
ψ
t
,
d
i
=
θ/
x
, and
=−
ψ
1
−
ψ
A
i
)
−
θ
A
i
+
1
−
A
i
+
1
)
−
A
i
Q
i
+
1
−
Q
i
)
p
i
t
(
t
(
−
x
(
1
−
θ
Q
i
+
1
−
Q
i
)
+
θ
[
ψ
q
n
+
1
l
,
i
q
n
+
1
l
,
i
−
x
(
1
+
(
−
ψ)
]
1
+
q
l
,
i
+
1
+
(
q
l
,
i
]
+
(
1
−
θ)
[
ψ
1
−
ψ)
To linearize the discretized momentum equation, the following relations based on
the first-order Taylor series expansion in terms of
δ
h
and
δ
Q
are used:
z
n
+
1
s
,
i
z
s
,
i
+
δ
=
h
i
(5.61)
Q
n
+
1
i
2
Q
i
)
2
2
Q
i
δ
(
)
=
(
+
Q
i
(5.62)
∂
∗
i
δ
1
1
2
K
=
2
−
h
i
(5.63)
K
i
)
K
i
)
K
n
+
1
i
(
(
3
∂
z
s
2
(
)
∂
∗
i
δ
2
S
f
,
i
K
i
Q
i
|
2
|
K
S
n
+
1
f
,
i
S
f
,
i
+
=
2
δ
Q
i
−
h
i
(5.64)
K
i
)
(
∂
z
s
