Geoscience Reference
In-Depth Information
The discretized sediment equations are then directly solved in the following
sequence:
(1) Compute
A
b
,
i
+
1
using Eq. (5.146);
(2) Calculate
A
bk
,
i
+
1
using Eq. (5.145);
(3) Compute
p
n
+
1
bk
,
i
1
using Eq. (5.135);
+
(4) Calculate
Q
n
+
1
t
1
using Eq. (5.141);
∗
k
,
i
+
(5) Compute
Q
n
+
1
tk
,
i
1
using Eq. (5.142), and
(6) Update the cross-section topography using Eq. (5.134), and calculate the bed-
material gradations in the subsurface layers.
+
However, the coupled sediment calculation is still decoupled from the flow
calculation so that the entire flow and sediment calculation procedure is in a
semi-coupled form.
The above direct solution method can also be used in the bed-load and suspended-
load transport model in Section 5.3.2, by writing Eqs. (5.130)-(5.132) as Eqs. (5.142)
and (5.143) with
C
k
,
i
+
1
,
C
1
,
Q
bk
,
i
+
1
, and
Q
b
∗
k
,
i
+
1
as unknowns and deriv-
ing an equation similar to Eq. (5.146) to compute the total bed change
∗
k
,
i
+
A
b
,
i
+
1
directly.
5.3.3.3 Stability of Preissmann scheme for sediment
transport equation
Neglecting the influence of the source term, the error in the sediment transport rate
determined using Eq. (5.138) is governed by
n
+
1
n
+
1
n
i
n
i
c
1
δ
=
c
2
δ
+
c
3
δ
+
c
4
δ
(5.147)
i
+
1
i
+
1
n
i
where
δ
is the Fourier component of the error at point
i
and time level
n
, defined
n
i
V
n
e
i
σ
x
i
, with
V
n
and
as
being its amplitude and wave number, respectively.
Inserting this definition expression into Eq. (5.147) yields the growth factor:
δ
=
σ
V
n
+
1
V
n
c
3
e
i
σ
x
+
c
4
r
=
=
(5.148)
c
1
e
i
σ
x
−
c
2
The coefficients of Eq. (5.138) satisfy that
c
1
≥
0 and
c
2
+
c
3
+
c
4
≤
c
1
at the locally
uniform state. Supposing
c
2
,
c
3
, and
c
4
≥
0 yields
=
≤
c
3
e
i
σ
x
c
4
e
−
i
σ
x
+
c
4
c
3
+
c
3
+
c
4
|
r
|=
c
2
≤
1
(5.149)
c
1
e
i
σ
x
c
2
e
−
i
σ
x
−
c
2
c
1
−
c
1
−
which means that the von Neumann stability condition is satisfied. At the locally
uniform state, in which
U
and
L
t
are constant in each element, the constraints
c
2
,
c
3
,
and
c
4
≥
0 imply
