Geoscience Reference
In-Depth Information
where
A
k
is a WAF limiter function. There are various choices for
A
k
, such as
⎧
⎨
⎩
≤
1
if
r
0
1
−
2
(
1
−|
c
k
|
)
r
if 0
<
r
≤
1
/
2
A
k
(
r
,
|
c
k
|
)
=
|
c
k
|
if 1
/
2
<
r
≤
1
(9.48)
1
−
(
1
−|
c
k
|
)
r
if 1
<
r
≤
2
2
|
c
k
|−
1
if
r
>
2
1
if
r
≤
0
A
k
(
r
,
|
c
k
|
)
=
(9.49)
1
−
2
(
1
−|
c
k
|
)
r
/(
1
+
r
)
if
r
>
0
1
if
r
≤
0
A
k
(
r
,
|
c
k
|
)
=
(9.50)
r
2
1
−
(
1
−|
c
k
|
)
r
(
1
+
r
)/(
1
+
)
if
r
>
0
⎧
⎨
1
if
r
≤
0
A
k
(
r
,
|
c
k
|
)
=
(9.51)
1
−
(
1
−|
c
k
|
)
r
if 0
<
r
≤
1
⎩
|
c
k
|
if
r
>
1
where
r
is the ratio of the upwind change to the local change in a scalar quantity
f
:
⎧
⎨
⎩
f
(
k
)
i
f
(
k
)
i
/
if
c
k
>
0
−
1
/
2
+
1
/
2
r
=
(9.52)
f
(
k
)
i
f
(
k
)
i
/
if
c
k
<
0
+
3
/
2
+
1
/
2
f
(
k
)
i
f
(
k
)
i
f
(
k
)
i
with
=
−
. For the
x
-split two-dimensional shallow water equations,
+
1
/
2
+
1
f
v
, the tangential velocity component, for the
shear wave. For other passive scalars,
f
is set as the corresponding state quantities.
The WAF limiter functions (9.48)-(9.51) are entirely equivalent to the conven-
tional superbee limiter, van Leer's limiter, van Albada's limiter, and minbee limiter,
respectively (Toro, 2001).
The numerical schemes introduced above have been tested extensively in the litera-
ture. For example, the performances of the central difference scheme (9.12), the HLL
scheme (9.18), the TVD scheme (9.43) with van Leer's monotonic limiter, and the
TVDWAF scheme (9.47) with van Albada's limiter are demonstrated in the following
simulation of dam-break flow in a straight rectangular channel with a horizontal bed.
The channel is 1200m long, and a dam is located at 500m from the upstream end. The
water in the reservoir is 10m deep, while the initial downstream water depth is given
as 1 and 0.001m to test the schemes in cases of wet and dry beds. The dam is assumed
to be instantaneously, completely removed. The channel is set to be sufficiently wide
so that the flow is uniform along the transverse direction and an analytical solution in
the frictionless case can be derived (see Graf and Altinakar, 1998). The Manning
n
is
set as 0. The longitudinal grid length is 10m. For the central difference scheme with
artificial diffusion flux, a time step of 0.1 s is used in both wet- and dry-bed cases. For
the HLL, TVD, and TVD WAF schemes, the time step is 0.6 s for the wet-bed case
and 0.3 s for the dry-bed case. Figs. 9.6 and 9.7 compare the calculated water surface
profiles with the analytical solutions at 30 s after dam failure. It can be seen that the
central difference scheme (9.12) with artificial diffusion flux has errors near the vicinity
=
h
for the non-linear waves, and
f
=
