Geoscience Reference
In-Depth Information
QUICK scheme
As shown in Fig. 4.18, QUICK (Quadratic Upwind Interpolation for Convective Kine-
matics) scheme, proposed by Leonard (1979), approximates the face value
φ
w
by fitting
a parabolic curve through the values of
φ
at points
WW
,
W
, and
P
when
F
w
≥
0, and
at
W
,
P
, and
E
when
F
w
<
0:
⎧
⎨
⎩
1
8
φ
3
4
φ
3
8
φ
−
+
+
P
F
w
≥
0
WW
W
φ
=
(4.119)
w
3
8
φ
3
4
φ
1
8
φ
+
−
F
w
<
0
W
P
E
A similar expression can be derived for the value of
φ
at face
e
. The interpolation
scheme (4.119) has a third-order truncation error.
The QUICK scheme is widely applied, but it may have numerical oscillations where
the function
φ
changes sharply.
Figure 4.18
Quadratic profie in QUICK scheme.
SOUCUP scheme
SOUCUP (Composite Second-Order Upwind/Central Difference/First-Order Upwind)
scheme was proposed by Zhu and Rodi (1991). When
F
w
≥
0, the SOUCUP scheme
approximates the face value
φ
w
as
⎧
⎨
⎩
≤
φ
1.5
φ
−
0.5
φ
WW
0
≤
0.5
W
W
< φ
φ
=
(4.120)
0.5
(φ
+
φ
)
0.5
≤
1
w
P
W
W
φ
W
otherwise
φ
where
.
Fig. 4.19 shows the relation between
φ
W
and
φ
w
for the SOUCUP scheme. When
=
(φ
−
φ
)/(φ
−
φ
)
W
W
WW
P
WW
≤
φ
0
≤
0.5,
φ
w
is approximated by the second-order upwind scheme. When 0.5
<
W
φ
φ
0or
φ
≤
1,
φ
w
is determined by the central scheme. When
<
>
1,
φ
w
is
W
W
W
approximated by the first-order upwind scheme.
