Geoscience Reference
In-Depth Information
Note that the momentum equation (5.2) in the dynamic wave model and its
simplification Eq. (5.4) in the diffusion wave model can also be discretized using
the Preissmann implicit scheme and linearized locally using Eqs. (5.61)-(5.67), and
the resulting equations can be written as Eq. (5.68) with different coefficients. The
detailed derivation is left to interested readers.
5.2.2.3 Solution of discretized unsteady flow equations
Algorithm for a single channel
−
As shown in Fig. 5.7, a single channel is segmented to
I
1 reaches with
I
cross-
sections. The pentadiagonal matrix of Eqs. (5.60) and (5.68) is solved by successively
applying a double sweep algorithm, which is often called the Thomas algorithm.
A linear relationship between the unknowns
δ
δ
h
i
and
Q
i
is assumed to be of the
type:
δ
Q
i
=
S
i
δ
h
i
+
T
i
(5.69)
Substituting Eq. (5.69) into Eqs. (5.60) and (5.68) and eliminating
δ
h
i
yields
δ
Q
i
+
1
=
S
i
+
1
δ
h
i
+
1
+
T
i
+
1
(5.70)
where
S
i
+
1
and
T
i
+
1
are recurrence coefficients:
c
i
−
(
a
i
+
b
i
S
i
S
i
+
1
=−
(
a
i
+
b
i
S
i
)
)
c
i
(5.71)
d
i
−
(
a
i
+
b
i
S
i
(
a
i
+
b
i
S
i
)
)
d
i
p
i
−
b
i
T
i
a
i
+
b
i
S
i
=
(
a
i
+
b
i
S
i
)(
)
−
(
)(
p
i
−
b
i
T
i
)
T
i
+
1
(5.72)
d
i
−
(
a
i
+
b
i
S
i
(
a
i
+
b
i
S
i
)
)
d
i
In the first (forward) sweep, Eqs. (5.71) and (5.72) are applied recursively, with
i
varying from 1 to
I
−
1. To perform this sweep,
S
1
and
T
1
at cross-section 1 (inlet) are
derived from the upstream boundary condition. For simplicity, the case of subcritical
flow is considered here. Therefore,
Q
n
+
1
is known by the given discharge hydrograph
at the inlet, and the recurrence coefficients
S
1
and
T
1
read
Q
n
+
1
1
Q
1
S
1
=
0,
T
1
=
−
(5.73)
Substituting Eq. (5.69) into Eq. (5.60) yields
=
(
p
i
−
b
i
T
i
)
−
(
c
i
δ
h
i
+
1
+
d
i
δ
Q
i
+
1
)
δ
h
i
(5.74)
a
i
+
b
i
S
i
Therefore,
in the second (return) sweep,
δ
h
i
and
δ
Q
i
can be calculated using
Eqs. (5.74) and (5.69) recursively, with
i
from
I
−
1to1.
