Geoscience Reference
In-Depth Information
where the a
ðnÞ
ik
coefficients, the wave strengths, are:
n
ik
D
ik
h
h
2
1
2
c
ik
u
ik
D
ik
hhi
a
ð
1
Þ
ik
D
ik
u
i
hhi
¼
t
ik
1
c
ik
u
ik
D
ik
hhi
a
ð
2
Þ
ik
¼
D
ik
u
i
hhi
(12)
n
ik
D
ik
h
h
2
þ
1
2
c
ik
u
ik
D
ik
hhi
a
ð
1
Þ
ik
¼
D
ik
u
i
hhi
T
.
where
t
¼ð
j
;
i
Þ
4.2 Flux Vector Splitting
FVS aims at generalizing upwinding schemes for nonlinear systems and uses the
fact that the flux differences can be split explicitly has incoming and outgoing
contributions for each cell, where l
¼
1
2
ð
l
jj
l
Þ
. If the fluxes are expressed in the
local system eigenvector base, we have:
þ
X
X
X
3
3
3
l
ðnÞ
ik
a
ðnÞ
ik
e
ðnÞ
l
ðnÞ
ik
a
ðnÞ
ik
e
ðnÞ
l
ðnÞ
ik
a
ðnÞ
ik
e
ðnÞ
D
ik
E
h
n
i¼
¼
þ
(13)
ik
ik
ik
n¼
1
n¼
1
n¼
1
In order to estimate a time step,
Dt
, that ensures stability, a CFL condition is
used. In 1D, the imposition is that
Dt
is chosen small enough so that there is no
interaction of waves from neighboring Riemann problems. In 2D, the framework
is the same and a geometrical parameter that is the equivalent to distance in the 1D
case must be computed. According to the work of Murillo et al. (
2006
),
min
ð
w
i
;
w
j
Þ
A
i
max
Dt
l
Dt
l
w
i
¼
L
ki
Þ
;
Dt
CFL
;
¼
(14)
l
m
ð
max
It is however possible to find values of
h
<
0 for some region of the domain,
which corresponds to a nonphysical solution.
Murillo (
2006
) defines two parameters that represent the intermediate state of
the solution of the local Riemann problem,
h
i
and
h
j
:
a
ð
1
Þ
b
ð
1
Þ
ik
=
l
ð
1
Þ
a
ð
3
Þ
b
ð
3
Þ
ik
=
l
ð
3
Þ
h
i
¼
h
j
h
i
þ
ik
;
¼
h
j
ik
(15)
ik
ik
0, the
Dt
can be locally defined, in order
to contain the influence of the RP. Only in subcritical flow cases does this need to be
computed.
In order to control the occurrence of
h
<
h
j
h
j
w
j
2l
k
h
i
h
i
w
i
2l
k
ðDt
; Dt
; Dt
l
Dt
¼
Dt
¼
;
h
i
;
Dt
¼
min
Þ
(16)
h
j
