Geoscience Reference
In-Depth Information
The computation procedure consists of two sweeps. In the first sweep, Eq. (9.26a) is
solved along the
i
-direction to obtain
n
+
1
/
2
n
ij
; in the second sweep, Eq. (9.26b)
from
ij
n
+
1
/
2
n
+
1
is solved along the
j
-direction to obtain
from
. Both sweeps have a
ij
ij
common time step,
t
. In each sweep, Eq. (9.22) can be applied to evaluate the
intercell fluxes
F
and
G
.
The aforementioned splitting scheme is first-order accurate in time. More first-
and second-order accurate splitting schemes can be found in Toro (2001) and other
references.
9.1.3 TVD schemes
9.1.3.1 TVD schemes for scalar problems
Consider a one-dimensional function
φ
in the domain shown in Fig. 9.1. The total
variation at time level
t
n
is defined as
−
I
1
n
n
i
n
i
TV
(φ
)
=
1
|
φ
−
φ
|
(9.27)
+
1
i
=
n
+
1
n
i
n
i
n
i
A numerical scheme
φ
=
H
(φ
k
,
...
,
φ
,
...
,
φ
)
is said to be Total Variation
i
−
+
l
Diminishing (TVD) if
n
+
1
n
TV
(φ
)
≤
TV
(φ
)
∀
n
(9.28)
A TVD scheme can preserve the monotonicity of the solution. This means that if
the data
n
+
1
n
i
{
φ
}
is monotone, the solution
{
φ
}
by a TVD scheme is monotone in the
i
same sense (Harten, 1983).
Suppose that numerical schemes written as
n
+
1
n
i
n
i
n
i
n
i
n
i
φ
=
φ
−
C
i
−
1
/
2
(φ
−
φ
)
+
D
i
+
1
/
2
(φ
−
φ
)
(9.29)
i
−
1
+
1
are used to solve the following equation for the scalar hyperbolic conservation law:
∂φ
∂
+
∂
f
(φ)
=
0
(9.30)
t
∂
x
where
f
is the flux. According to Harten (1983), the sufficient conditions for any
scheme written as Eq. (9.29) to be TVD are
(φ)
C
i
−
1
/
2
≥
0,
D
i
+
1
/
2
≥
0,
and 0
≤
C
i
−
1
/
2
+
D
i
+
1
/
2
≤
1
(9.31)
Harten (1983), Sweby (1984), and Yee (1987) developed a variety of TVD schemes
that respect conditions (9.31). Most of the TVD schemes are essentially constructed
