Geoscience Reference
In-Depth Information
f
n
+
1
i
f
i
+
1
f
n
+
1
i
f
i
−
−
∂
f
+
1
t
=
ψ
+
(
1
−
ψ)
(4.35)
∂
t
t
f
n
+
1
i
f
n
+
1
i
f
i
+
1
−
f
i
−
∂
f
+
1
x
=
θ
+
(
1
−
θ)
(4.36)
∂
x
x
where
are the weighting factors in time and space, respectively. The original
Preissmann scheme adopts
θ
and
ψ
ψ
=
1
/
2; thus, Eqs. (4.34)-(4.36) are the generalized
version.
Application of the Preissmann scheme in the 1-D simulation of unsteady open-
channel flows is discussed in detail in Section 5.2.2.
4.2.1.4 High-order difference schemes
The backward and forward difference schemes (4.11) and (4.12) based on two grid
points are the simplest asymmetric difference schemes, and the central difference
schemes (4.13) and (4.14) based on three grid points are the simplest symmetric
difference schemes. To enhance the accuracy of numerical discretization, one may
use more grid points in the difference formulation. For example, the following three-
point and four-point asymmetric difference schemes for the first derivative are derived
using the Taylor series expansion:
∂
i
=
f
f
i
−
2
−
4
f
i
−
1
+
3
f
i
x
2
+
O
(
)
(4.37)
∂
x
2
x
∂
f
f
i
−
2
−
6
f
i
−
1
+
3
f
i
+
2
f
i
+
1
x
3
i
=
+
O
(
)
(4.38)
∂
x
6
x
and the five-point symmetric difference schemes for the first and second derivatives are
∂
f
f
i
−
2
−
8
f
i
−
1
+
8
f
i
+
1
−
f
i
+
2
x
4
i
=
+
O
(
)
(4.39)
∂
x
12
x
2
f
∂
i
=
−
f
i
−
2
+
16
f
i
−
1
−
30
f
i
+
16
f
i
+
1
−
f
i
+
2
x
4
+
O
(
)
(4.40)
x
2
x
2
∂
12
By using schemes (4.37) and (4.38), the second-order and third-order upwind
schemes for the convection terms in Eqs.
(4.15) and (4.22) can be established
as follows:
⎨
u
f
i
−
2
−
4
f
i
−
1
+
3
f
i
u
∂
i
=
u
≥
0
f
2
x
(4.41)
∂
x
⎩
u
−
3
f
i
+
4
f
i
+
1
−
f
i
+
2
u
<
0
2
x
⎧
⎨
u
f
i
−
2
−
6
f
i
−
1
+
3
f
i
+
2
f
i
+
1
u
∂
u
≥
0
f
6
x
i
=
(4.42)
∂
x
⎩
u
−
2
f
i
−
1
−
3
f
i
+
6
f
i
+
1
−
f
i
+
2
<
u
0
6
x
