Geoscience Reference
In-Depth Information
n
+
1
n
Numerical stability requires
r
=|
δ
/δ
≤
1
|
, which implies that for the implicit
scheme,
Q
∗
n
+
1
tk
,
i
θ(
1
−
e
k
)
t
+
1
2
A
n
+
1
m
,
i
≥
A
b
,
i
+
1
−
(5.154)
+
1
L
n
+
1
t
,
i
p
m
)
(
1
−
+
1
and for the explicit scheme,
Q
∗
n
+
1
tk
,
i
θ(
−
e
k
)
1
1
t
+
2
A
n
+
1
m
,
i
≤
−
A
b
,
i
+
1
(5.155)
+
1
L
n
+
1
t
,
i
(
1
−
p
m
)
+
1
A
n
+
1
m
,
i
1
, then
p
∗
bk
,
i
+
1
is the bed-material
gradation in the second layer, the influence of which is assumed to be negligible on
the numerical stability of the bed-material gradation in the mixing layer. Thus, the
bed-material gradation error is governed by
A
m
,
i
+
1
<
For erosion, usually
A
b
,
i
+
1
+
+
Q
∗
n
+
1
tk
,
i
n
A
m
,
i
+
1
−
(
f
2
−
f
1
e
k
)(
1
−
θ
)
p
+
1
n
+
1
δ
=
δ
(5.156)
A
n
+
1
m
,
i
f
1
e
k
)θ
p
Q
∗
n
+
1
tk
,
i
+
(
f
2
−
+
1
+
1
from which it is known that the implicit scheme is unconditionally stable, and the
stability condition for the explicit scheme is
Q
∗
n
+
1
tk
,
i
θ(
1
−
e
k
)
t
+
1
A
m
,
i
+
1
+
A
n
+
1
m
,
i
≤
(5.157)
L
n
+
1
t
,
i
+
1
(
−
p
m
)
1
+
1
By definition, the mixing layer should be thicker than the change in bed elevation,
i.e.,
A
n
+
1
m
1, the stability condition (5.154) for the implicit
scheme is automatically satisfied; conditions (5.155) and (5.157) require upper limits
for the time step
≥|
A
b
|
. Because
e
k
<
t
in the explicit scheme. It is evident that the implicit scheme is
much more stable than the explicit scheme.
Requirement of non-negative bed-material gradation
In calculating bed-material gradation, negative values may occur under certain
conditions. Of course, this is a non-physical phenomenon and must be eliminated.
The condition
p
n
+
1
bk
,
i
1
≥
0 for Eq. (5.152) implies that
+
A
m
,
i
+
1
−
(
Q
∗
n
+
1
tk
,
i
p
bk
,
i
+
1
f
1
e
0
k
+
f
0
k
+[
f
2
−
f
1
e
k
)(
1
−
θ
)
]
p
+
1
A
n
+
1
m
,
i
A
m
,
i
+
1
−
p
∗
bk
,
i
+
1
≥
+
(
−
A
b
,
i
+
1
)
0
(5.158)
+
1
