Geoscience Reference
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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.
 
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