Geoscience Reference
In-Depth Information
9.1.3.2 TVD schemes for St. Venant equations
Many investigators, e.g., Garcia-Navapro
et al
. (1992) and Wang
et al
. (2000), have
extended TVD schemes to solve 1-D and 2-D shallow water equations. Here, the
method of Wang
et al
. (2000) is presented.
For the 1-D problem governed by Eq. (9.1), the eigenvalues of
M
in Eq. (9.4) are
gh
,
gh
a
1
a
2
=
q
/
h
−
=
q
/
h
+
(9.40)
and the corresponding right eigenvectors are
1
a
1
,
R
2
1
a
2
R
1
=
=
(9.41)
i
i
, which can be written as
Define
=
−
i
+
1
/
2
+
1
2
l
i
2
R
i
+
1
/
2
=
1
α
(9.42)
i
+
1
/
2
+
1
/
l
=
2
in the coordinate system
R
i
+
1
/
2
.
In analogy to Eq. (9.33), the following intercell flux used in the scheme (9.10) for
solving Eq. (9.1) can be constructed by using Eq. (9.42):
l
i
where
α
2
represents the component of
i
+
1
/
+
1
/
⎧
⎨
⎩
1
2
F
i
+
1
/
2
=
F
i
F
i
+
1
+
⎫
⎬
⎭
2
t
a
i
+
1
/
2
)
2
a
i
+
1
/
2
)
l
i
2
R
i
+
1
/
2
−
x
i
(
ϕ(
r
)
+
(
1
−
ϕ(
r
))ψ(
α
+
1
/
l
=
1
(9.43)
For the 2-D problem, Wang
et al
. (2000) split Eq. (9.5) into two augmented 1-D
systems similar to Eq. (9.25) along the
x
- and
y
-directions. These two 1-D systems are
discretized using a scheme similar to Eq. (9.43) except that the summation is increased
from two to three terms corresponding to the right eigenvectors of matrices
A
and
B
in Eq. (9.8).
9.1.4 WAF schemes
The weighted average flux (WAF) method was first proposed for the Euler equations
by Toro (1989) and then applied to the shallow water equations by Toro (1992)
and others. It is a second-order extension of the Godunov upwind method and
may also be interpreted as a Riemann-problem-based extension of the Lax-Wendroff
method.
