Geoscience Reference
In-Depth Information
t
2
,
x
2
The truncation error of the Lax-Wendroff scheme (4.31) is
O
(
)
as well. Its
stability condition is
u
2
t
2
x
2
(see Fletcher, 1991).
+
2
ε
t
≤
c
Crank-Nicholson scheme
Applying Eq. (4.23) to the time derivative and a weighted average of the central
difference scheme (4.14) between time levels
n
and
n
+
1 to the diffusion term in
the 1-D diffusion equation
2
f
∂
f
c
∂
=
ε
x
2
+
S
(4.32)
∂
t
∂
yields
f
n
+
1
i
2
f
n
+
1
i
f
n
+
1
i
f
n
+
1
i
f
i
−
+
−
−
1
+
1
S
n
+
1
i
=
θ
ε
+
c
t
x
2
f
i
−
1
−
2
f
i
f
i
+
1
+
S
i
+
(
1
−
θ)
ε
+
(4.33)
c
x
2
where
θ
is a temporal weighting factor. When
θ
=
0, Eq. (4.33) is an explicit scheme,
θ>
θ
=
/
and when
2, Eq. (4.33) is called the
Crank-Nicholson scheme, which is second-order accurate in time and space.
0, Eq. (4.33) is an implicit scheme. For
1
Preissmann scheme
Preissmann (1961) proposed an implicit scheme based on two levels in time and two
points in space, as shown in Fig. 4.5. This scheme replaces the continuous function
f
and its time and space derivatives by
f
n
+
1
i
f
n
+
1
i
f
i
+
1
+
(
f
i
f
=
θ
[
ψ
+
(
1
−
ψ)
]+
(
1
−
θ)
[
ψ
1
−
ψ)
]
(4.34)
+
1
Figure 4.5
Computational element in the Preissmann scheme.