Geoscience Reference
In-Depth Information
The Euler scheme for the time-derivative terms results in simple, explicit time-
marching algorithms in Eqs. (9.10) and (9.11), even though it is only first-order
accurate in time. The accuracy can be enhanced by applying the MacCormack scheme
(Fennema and Chaudhry, 1990), Range-Kutta method, etc.
To complete the discretization in Eqs. (9.10) and (9.11), numerical schemes are
needed to evaluate the intercell fluxes. The often used schemes are the central differ-
ence scheme and MacCormack scheme with artificial diffusion fluxes (e.g., Fennema
and Chaudhry, 1990; Molls and Chaudhry, 1995), approximate Riemann solvers
(Godunov, 1959; Roe, 1981; Osher and Solomon, 1982; Harten
et al
., 1983; Toro
et al
., 1994), TVD (Total Variation Diminishing) schemes (Harten, 1983; Yee, 1987;
Garcia-Navarro
et al
., 1992; Wang
et al
., 2000), and upwind flux schemes (Ying
et al
.,
2004). Some of them are introduced briefly below.
9.1.1 Central difference scheme with artificial
diffusion flux
If the intercell fluxes are evaluated using the central difference scheme, spurious oscilla-
tions may appear in the solution near regions with sharp gradients and discontinuities
(see Molls and Chaudhry, 1995). To eliminate these oscillations, an artificial diffusion
(and/or dissipation) flux is usually added, thus yielding
1
2
(
F
i
+
1
/
2
=
F
i
F
i
+
1
)
+
D
i
+
1
/
2
+
(9.12)
where
D
i
+
1
/
2
is the artificial diffusion flux. Various formulations for
D
i
+
1
/
2
can be
found in the literature. The one suggested by Martinelli and Jameson (1988) consists
of second-order and fourth-order terms:
D
i
+
1
/
2
=
ρ(
[−
ε
(
2
)
i
n
i
n
i
)
+
ε
(
4
)
i
n
i
n
i
n
i
n
i
M
)
(
−
(
−
3
+
3
−
)
]
i
+
1
/
2
+
1
+
2
+
1
−
1
+
1
/
2
+
1
/
2
(9.13)
where
ρ(
M
)
is the spectral radius of the Jacobian matrix
M
, defined as
ρ(
M
)
=
gh
; and
ε
(
2
)
i
ε
(
4
)
i
|
q
|
/
h
+
2
and
2
are non-linear functions:
+
1
/
+
1
/
ε
(
2
)
i
ε
(
4
)
i
(κ
(
4
)
−
ε
(
2
)
i
=
κ
(
2
)
max
(χ
i
,
χ
i
+
1
)
,
=
max
[
0,
)
]
(9.14)
+
1
/
2
+
1
/
2
+
1
/
2
κ
(
2
)
and
κ
(
4
)
being coefficients, and
with
χ
=|
h
i
−
1
−
2
h
i
+
h
i
+
1
|
/(
h
i
−
1
+
2
h
i
+
h
i
+
1
)
.
i
In the 2-D case, in analogy to Eq. (9.12), the intercell fluxes are evaluated as
1
2
(
)
i
+
1
/
2,
j
[−
ε
(
2
)
i
F
i
+
1
/
2,
j
=
F
i
,
j
+
F
i
+
1,
j
)
+
ρ(
n
i
n
i
,
j
A
(
−
)
+
1
/
2,
j
+
1,
j
+
ε
(
4
)
i
n
i
n
i
n
i
,
j
n
i
(
−
3
+
3
−
)
]
(9.15a)
+
2,
j
+
1,
j
−
1,
j
+
1
/
2,
j
