Geoscience Reference
In-Depth Information
by combining high-order and low-order fluxes with a limiter imposed to constrain the
gradients of flux functions and to prevent the formation of local extrema. One typical
approach is given below.
Integrating Eq. (9.30) over the
i
th control volume in Fig. 9.1 yields the following
general conservative scheme:
−
t
n
+
1
n
i
f
i
+
1
/
2
−
f
i
−
1
/
2
)
φ
=
φ
x
i
(
(9.32)
i
The intercell flux
f
i
+
1
/
2
is constructed as (see Wang
et al
., 2000)
f
i
2
1
2
t
f
i
+
1
/
2
=
f
i
+
1
−
2
+
x
i
(
a
i
+
1
/
2
)
ϕ(
r
)
+
(
1
−
ϕ(
r
))ψ(
a
i
+
1
/
2
)
φ
i
+
1
/
(9.33)
n
i
n
i
where
φ
=
φ
−
φ
;
a
i
+
1
/
2
is the characteristic speed, defined as
a
i
+
1
/
2
=
i
+
1
/
2
+
1
f
i
+
1
/
2
/φ
2
when
φ
=
0 and
a
i
+
1
/
2
=
a
(φ
)
when
φ
=
0; and
+
/
+
/
+
/
i
1
i
1
2
i
i
1
2
ψ(
a
i
+
1
/
2
)
is the dissipative function, defined by Harten (1983) as
|
a
i
+
1
/
2
|
,
|
a
i
+
1
/
2
|≥
ε
ψ(
a
i
+
1
/
2
)
=
(9.34)
2
2
[
(
a
i
+
1
/
2
)
+
ε
]
/
2
ε
or
ε
,
|
a
i
+
1
/
2
|
<ε
where
is a small positive number.
The function
ε
ϕ(
r
)
in Eq. (9.33) is a limiter, with
r
defined as
2
=
[|
a
i
+
1
/
2
−
σ
|−
t
(
a
i
+
1
/
2
−
σ
)
/
x
i
−
σ
]
ϕ
i
+
1
/
2
−
σ
r
,
σ
=
sign
(
a
i
+
1
/
2
)
(9.35)
2
[|
a
i
+
1
/
2
|−
t
(
a
i
+
1
/
2
)
/
x
i
]
ϕ
i
+
1
/
2
is essential to obtain monotonic solutions. Many limiters
have been published in the literature. For example,
The limiter function
ϕ(
r
)
0 yields the upwind scheme
that is a first-order TVD scheme. The commonly used limiters in second-order TVD
schemes include Roe's minmod limiter
ϕ(
r
)
=
ϕ(
)
=
(
)
r
min mod
1,
r
(9.36)
van Leer's monotonic limiter
r
+|
r
|
ϕ(
r
)
=
(9.37)
1
+|
r
|
van Leer's MUSCL limiter
ϕ(
r
)
=
max
[
0, min
(
2
r
,2
)
, 0.5
(
1
+
r
)
]
(9.38)
and Roe's superbee limiter
ϕ(
r
)
=
max
[
0, min
(
2
r
,1
)
, min
(
r
,2
)
]
(9.39)
