Geoscience Reference
In-Depth Information
n
i
=
36
f
i
−
1
−
2
f
i
f
i
+
1
n
i
n
i
4
f
+
6
(∂
f
/∂
x
)
−
(∂
f
/∂
x
)
∂
+
1
−
1
−
∂
x
4
x
4
x
3
n
i
+
2
f
24
∂
x
4
−
O
(
)
(4.49)
x
2
∂
x
2
Eqs. (4.46)-(4.49) can be derived using the Taylor series expansion. Note that
Eq. (4.46) needs to be solved using a direct or iterative method to compute the first
derivative at each point.
Higher-order (up to eighth-order) difference schemes can be derived by adding the
first and second derivatives at points
i
1 in Eq. (4.45). The approach
described above can also be applied in the derivation of high-order schemes for the
diffusion and convection-diffusion equations by substituting relevant relations for
∂
−
1 and
i
+
t
k
into Eq. (4.44) (Wu, 1993).
However, the above high-order difference schemes must be treated specially at
boundary points because external points or boundary values for the first and/or
second derivatives are involved. Furthermore, they usually need a uniform mesh that
is difficult to conform to the irregular and movable boundaries of river flow. There-
fore, the numerical schemes of higher than fourth-order accuracy are rarely used in
computational river dynamics.
k
f
/∂
4.2.2 Finite difference method for multidimensional
problems on regular grids
4.2.2.1 Discretization of multidimensional steady problems
It is straightforward to extend the aforementioned 1-D finite difference schemes to the
discretization of 2-D and 3-D differential equations on regular grids. For example, on
the rectangular grid shown in Fig. 4.6, applying the upwind difference scheme (4.17)
for the convection terms and the central difference scheme (4.14) for the diffusion
terms in the 2-D steady convection-diffusion equation
2
f
2
f
u
x
∂
f
u
y
∂
f
∂
x
2
+
∂
x
+
y
=
ε
+
S
(4.50)
c
∂
∂
∂
∂
y
2
yields
G
yi
,
j
=
ε
c
f
i
−
1,
j
−
2
f
i
,
j
+
f
i
+
1,
j
f
i
,
j
−
1
−
2
f
i
,
j
+
f
i
,
j
+
1
G
xi
,
j
+
+
+
S
i
,
j
(4.51)
x
2
y
2
where
x
and
y
are the grid spacings in the
x
- and
y
-directions, respectively;
G
xi
,
j
is
set as
u
x
(
f
i
,
j
−
f
i
−
1,
j
)/
x
when
u
x
≥
0 and
u
x
(
f
i
+
1,
j
−
f
i
,
j
)/
x
when
u
x
<
0; and
G
yi
,
j
is
u
y
(
f
i
,
j
−
f
i
,
j
−
1
)/
y
when
u
y
≥
0 and
u
y
(
f
i
,
j
+
1
−
f
i
,
j
)/
y
when
u
y
<
0.
