Geoscience Reference
In-Depth Information
4.5 Conservative Entropy Fix
To avoid nonphysical results derived from the linearization, an entropy correction
to filter sonic rarefactions is needed. A version of the Harten-Hyman entropy fix is
applied (LeVeque
2002
).
In the case of a left transonic rarefaction, characterized by l
i
<
l
j
, with
0
<
l
U
j
, the initial jump associated to l
ð
1
Þ
l
i
¼
l
UðÞ
and l
j
¼
is decomposed in two
k
new jumps:
l
i
l
j
l
k
l
k
l
k
¼
l
k
¼
l
i
l
j
;
l
j
l
i
(21)
l
j
l
i
with l
k
þ
l
k
¼
l
k
, preserving the original value of the state
U
i
and, in conse-
quence, the stability region. For a right transonic rarefaction, l
i
<
l
j
, the
0
<
procedure is analogous, this time conserving the
U
i
state and its stability region.
l
k
l
j
l
k
l
i
l
k
¼
l
k
¼
l
i
l
i
;
(22)
l
j
l
i
l
j
l
i
4.6 Numerical Scheme
The final formulation of the scheme is written as:
(
)
e
ðnÞ
ik
A
i
X
L
k
X
3
3
Dt
m
þ
1
m
l
ðnÞ
ik
a
ðnÞ
b
ðnÞ
ik
Vhi
¼
Vhi
ik
(23)
k¼
1
n¼
1
5 Results
Numerical solutions for the Riemann problem are presented. Four dam-break
problem examples are analyzed, with the intention of demonstrating some proper-
ties of the scheme, namely the effects of being of first order in its diffusive behavior,
its ability to cope with dry bed and wetting fronts, and the effectiveness of the
entropy corrections.
The mesh was unstructured with an average cell side of 1.0 m. The value of the
CFL number was 0.9.
