Geoscience Reference
In-Depth Information
κ
(
4
)
.
The HLL scheme (9.18) provides a smooth solution with first-order accuracy. The
TVD scheme (9.43) and the TVD WAF scheme (9.47), which are second-order
accurate, provide better results than the central difference scheme and the HLL scheme.
κ
(
2
)
and
of sharp gradients, which are essentially affected by the coefficients
9.1.5 Upwind flux schemes
Ying
et al
. (2004) proposed an upwind flux scheme to solve the St. Venant equations.
To handle the 1-D dam-break flow in natural rivers, the flow area
A
and discharge
Q
are used as primary variables. The depth gradient and bed slope are combined as
the water surface gradient, which is arranged into the source term. This treatment
can avoid the use of different discretization schemes for the depth gradient and bed
slope terms. The governing equations are still written as Eq. (9.1) with the unknown
variables, fluxes, and source terms:
⎡
⎣
⎤
⎦
Q
Q
2
,
A
Q
,
0
=
F
(
)
=
S
(
)
=
g
n
2
Q
gA
∂
z
s
|
Q
|
/
A
−
x
−
AR
4
/
3
∂
(9.53)
The mesh system is the same as that shown in Fig. 9.1. Integrating Eq. (9.1) over
the
i
th control volume yields the discretized equation (9.10). The intercell flux
F
i
+
1
/
2
is evaluated using the first-order upwind method:
Q
i
+
k
F
i
+
1
/
2
=
(9.54)
Q
i
+
k
)
2
A
i
+
k
(
/
where
k
=
0, if
Q
i
>
0 and
Q
i
+
1
>
0;
k
=
1, if
Q
i
<
0 and
Q
i
+
1
<
0; and
k
=
1
/
2
for others. Here, the subscript
i
+
1
/
2 represents the average of values at grid points
i
and
i
1.
The source term in the momentum equation is evaluated as
+
w
1
z
n
+
1
s
,
i
z
n
+
1
s
,
i
z
n
+
1
s
,
i
z
n
+
1
s
,
i
−
−
g
n
i
Q
i
|
Q
i
|
+
1
−
k
−
k
+
k
−
1
+
k
gA
n
+
1
i
S
i
(
Q
)
=−
x
i
−
k
+
w
2
−
(9.55)
A
i
(
R
i
)
4
/
3
x
i
+
1
−
k
−
x
i
+
k
−
x
i
−
1
+
k
where
w
1
and
w
2
are weighting factors. If
w
1
0.5, Eq. (9.55)
is equivalent to the central difference scheme for water surface gradient. This may
result in non-physical oscillations. In order to improve numerical accuracy, these
weighting factors are related to the Courant number. One of the relations suggested
by Ying
et al
.is
=
0.5 and
w
2
=
−
t
2
(
|
w
1
=
1
U
i
+
1
−
k
|+|
U
i
−
k
|
)/(
x
i
+
1
−
k
−
x
i
−
k
)
=
t
2
(
|
w
2
U
i
+
k
|+|
U
i
−
1
+
k
|
)/(
x
i
+
k
−
x
i
−
1
+
k
)
(9.56)
where
U
is the flow velocity.
