Geoscience Reference
In-Depth Information
Upwind schemes (4.41) and (4.42) have better accuracy and less numerical diffusion
than the first-order upwind scheme (4.17). However, they may produce numerical
oscillations where the function
f
varies rapidly.
Using the asymmetric difference scheme (4.37) for the time derivative yields the
three-level implicit scheme:
n
+
1
3
f
n
+
1
i
f
n
−
1
i
4
f
i
−
+
∂
f
=
(4.43)
∂
t
2
t
i
which is second-order accurate in time.
In addition, one may establish high-order schemes based on only two or three grid
points by including the first and second derivatives of the function in difference formu-
lations (Yang and Cunge, 1989; Wu, 1993). One approach is based on the expansion
of
f
n
+
1
as a Taylor series about
t
n
:
n
i
+
m
t
k
k
k
f
∂
f
n
+
1
i
f
i
t
m
+
1
=
+
O
(
)
(4.44)
!
t
k
∂
k
=
1
For the homogeneous convection equation (4.22) with
S
=
0 and a constant velocity
k
f
t
k
k
k
f
x
k
. Substituting this relation into Eq. (4.44)
u
, one can derive
∂
/∂
=
(
−
u
)
∂
/∂
yields
k
n
i
+
m
t
k
k
k
f
∂
f
n
+
1
i
f
i
t
m
+
1
=
+
!
(
−
u
)
O
(
)
(4.45)
x
k
∂
k
=
1
If the first and second derivatives in Eq. (4.45) are evaluated using the central dif-
ference schemes (4.13) and (4.14), Eq. (4.45) is exactly the Lax-Wendroff scheme.
A fourth-order accurate scheme can be obtained by using the five-point schemes (4.39)
and (4.40) for the first and second derivatives and constructing two fourth-order seven-
point schemes for the third and fourth derivatives in Eq. (4.45). However, to limit
the number of grid points involved, Wu (1993) suggested the following three-point
schemes for these derivatives:
∂
n
4
∂
n
i
+
∂
n
3
f
i
+
1
−
f
i
−
1
f
f
f
x
4
1
+
1
=
+
O
(
)
(4.46)
∂
x
∂
x
∂
x
x
i
−
i
+
n
i
=
∂
x
n
i
−
∂
x
n
i
2
f
i
−
1
−
2
f
i
f
i
+
1
2
f
+
f
/∂
f
/∂
∂
+
1
−
1
x
4
−
+
O
(
)
(4.47)
∂
x
2
x
2
2
x
n
3
∂
x
n
i
8
∂
x
n
i
+
∂
x
n
i
15
f
i
+
1
−
f
i
−
1
3
f
f
/∂
1
+
f
/∂
f
/∂
∂
−
+
1
x
4
=
−
+
(
)
O
x
3
x
3
x
2
∂
2
2
i
(4.48)
