Geoscience Reference
In-Depth Information
Figure 4.16
Stepwise profie in upwind scheme.
equation (4.108) become
a
W
=
D
w
+
(
)
max
F
w
,0
a
E
=
D
e
+
(
−
)
max
F
e
,0
(4.112)
a
P
=
a
W
+
a
E
+
(
F
e
−
F
w
)
It is evident that no negative coefficients would arise in Eq. (4.112). Thus, the
solution will always be physically realistic. However, the upwind scheme is only first-
order accurate and has strong numerical diffusion.
Hybrid scheme
As mentioned above, the central scheme is second-order accurate, but it may encounter
difficulties when
2
D
; while the upwind scheme can solve these difficulties
although it is only first-order accurate. Combining these two schemes leads to a hybrid
scheme, which has the advantages of both schemes. The concept is that when
|
F
|
>
|
F
|≤
2
D
,
the central scheme is used, and when
|
F
|
>
2
D
, the upwind scheme is used. Thus, the
coefficient
a
W
for the hybrid scheme is
⎧
⎨
⎩
F
w
,
if
P
w
>
2
a
W
=
D
w
+
F
w
/
2,
if
−
2
≤
P
w
≤
2
(4.113)
0
if
P
w
<
−
2
where
P
w
is the Peclet number at face
w
. The resulting discretized equation can then
be written as Eq. (4.108) with coefficients:
a
W
=
max
(
F
w
,
D
w
+
F
w
/
2, 0
)
a
E
=
max
(
−
F
e
,
D
e
−
F
e
/
2, 0
)
(4.114)
a
P
=
a
W
+
a
E
+
(
F
e
−
F
w
)
