Geoscience Reference
In-Depth Information
4.3 Bottom Source Terms
The bottom source terms are dealt with in the same conservative manner: they are
projected in the local eigenvectors base and split into positive and negative
contributions:
!
þ
X
L
k
X
X
3
3
3
b
ðnÞ
e
ðnÞ
ik
b
ðnÞ
e
ðnÞ
ik
A
i
Hhi
ik
~
þ
ik
~
(17)
k¼
1
n¼
1
n¼
1
where
k
;
1
2
c
ik
p
b
r
w
b
ð
1
Þ
ik
b
ð
2
Þ
ik
b
ð
3
Þ
ik
b
ð
1
Þ
ik
¼
¼
0
;
¼
(18)
=ð Þ
k
, in order to
optimize the pressure balance in various cases, where a difference of performance
in the alternative formulations of pressure force terms is noted (Murillo and Garc
´
a-
Navarro
2010
). This discretization should follow a steady-state equilibrium criteria,
i.e., it must allow the scheme to preserve steady states such as still-water equilib-
rium (well-balanced scheme, LeVeque
2002
).
These requirements on the source terms of the numerical solution can lead to
extremely small time steps, as
Dt
or
Dt
can be various orders of magnitude
smaller than
Dt
l
Special attention must be given in the actual definition of
p
b
r
w
. Such a situation can be avoided by means of a reconstruction
of the approximate solution, forcing positive values on
h
i
and
h
j
by reducing the
numerical source term instead of the time step size (Murillo and Garc´a-Navarro
2010
).
4.4 Wetting and Drying Algorithm
Regions with updated negative values of water depth near wet/dry interfaces, with
or without discontinuous bed levels can occur. According to the
Dt
criteria in
(
16
), the time step in those cases becomes nil. To ensure positivity and conservation
in the solution in all cases, a redistribution of the fluxes is proposed. The flux in a
general intercell edge
k
is computed as:
Þ
i;k
¼ D E
ð
D E
h
n
i
h
H
n
i
ð
h
n
i
h
H
n
i
Þ
i;k
if
h
j
¼
0 and
h
j
<
0
;
set
(19)
Þ
j;k
¼
ð
D E
h
n
i
h
H
n
i
0
Þ
j;k
¼ D E
ð
D E
h
n
i
h
H
n
i
ð
h
n
i
h
H
n
i
Þ
j;k
if
h
i
¼
0 and
h
i
<
0
;
set
(20)
Þ
i;k
¼
ð
D E
h
n
i
h
H
n
i
0
Otherwise, normal updating is used.
