Geoscience Reference
In-Depth Information
1
2
(
)
i
,
j
+
1
/
2
[−
ε
(
2
)
i
,
j
G
i
+
1
/
2,
j
=
G
i
,
j
+
G
i
,
j
+
1
)
+
ρ(
n
i
,
j
n
i
,
j
(
−
)
B
+
1
/
2
+
1
+
ε
(
4
)
i
,
j
n
i
,
j
n
i
,
j
n
i
,
j
n
i
,
j
(
−
3
+
3
−
)
]
(9.15b)
+
2
+
1
−
1
+
1
/
2
where
ρ(
A
)
and
ρ(
B
)
are the spectral radii of matrices
A
and
B
, respectively.
9.1.2 Approximate Riemann solvers
For the 1-D problem, Godunov (1959) suggested determining the intercell fluxes in
Eq. (9.10) as
F
i
+
1
/
2
=
F
(
(
0
))
(9.16)
i
+
1
/
2
where
(
0
)
is the exact similarity solution
(
x
/
t
)
of the Riemann problem
i
+
1
/
2
i
+
1
/
2
⎧
⎨
∂
∂
+
∂
F
(
)
∂
=
0
t
x
(9.17)
⎩
L
,
if
x
<
0
(
x
,0
)
=
>
R
,
if
x
0
evaluated at
x
0.
Solving the exact Riemann problem (9.17) is rather complicated. The frequently
used methods are approximate Riemann solvers. Many of them were developed in
computational aerodynamics (Godunov, 1959; Roe, 1981; van Leer, 1982; Harten
et al
., 1983; Osher and Solomon, 1982; etc.) and later adopted in free-surface flow
simulation (Glaister, 1988; Alcrudo and Garcia-Navarro, 1993; Jha
et al
., 2000; etc.).
Here, the HLL and HLLC approximate Riemann solvers are introduced as examples.
More Riemann solvers were summarized by Toro (2001).
Harten, Lax, and van Leer (HLL, 1983) suggested an approximate Riemann solver
by directly finding an approximation to the intercell flux (9.16). The HLL approach
assumes estimates
S
L
and
S
R
for the smallest and largest signal velocities in the solution
of the Riemann problem (9.17) with data
/
t
=
n
n
i
=
i
,
=
1
and corresponding
L
R
+
fluxes
F
L
, as shown in Fig. 9.3. The following HLL numerical
flux is derived by applying the integral form of the conservation laws in appropriate
control volumes:
=
F
(
)
,
F
R
=
F
(
)
L
R
⎧
⎨
⎩
F
L
if
S
L
≥
0
S
R
F
L
−
S
L
F
R
+
S
R
S
L
(
−
)
R
L
F
hll
F
i
+
1
/
2
=
≡
if
S
L
≤
0
≤
S
R
(9.18)
S
R
−
S
L
F
R
if
S
R
≤
0
