Geoscience Reference
In-Depth Information
The implicit scheme (4.25) is unconditionally stable, whereas the explicit scheme
(4.24) is stable if
t
≤
x
/
|
u
|
(4.26)
which is called the CFL (Courant-Friedrichs-Lewy) condition.
Leapfrog scheme
Using the central difference scheme (4.13) for both the temporal and spatial terms in
Eq. (4.22) results in the so-called leapfrog scheme:
f
n
+
1
i
f
n
−
1
i
u
f
i
+
1
−
f
i
−
1
−
S
i
+
=
(4.27)
2
t
2
x
The leapfrog scheme is second-order accurate in time and space. If the CFL condi-
tion (4.26) is satisfied, the leapfrog scheme is neutrally stable. However, the leapfrog
scheme is a three-level scheme, which requires an alternative method for the first
time step.
Lax scheme and Lax-Wendroff scheme
Replacing
f
i
in scheme (4.23) by a weighted average value of
f
i
−
1
,
f
i
, and
f
i
+
1
yields
the Lax scheme:
ψ
f
n
+
1
i
f
i
1
2
f
i
−
1
+
f
i
+
1
)
−
+
(
1
−
ψ)(
∂
f
t
=
(4.28)
∂
t
where
is a spatial weighting coefficient.
Applying the Lax scheme (4.28) for the convection equation (4.22) leads to
ψ
f
n
+
1
i
1
2
f
i
f
i
−
1
+
f
i
+
1
)
−
ψ
+
(
1
−
ψ)(
u
f
i
+
1
−
f
i
−
1
+
=
S
(4.29)
t
2
x
u
2
t
2
x
2
and
S
0, the difference equation (4.29) becomes the
Lax-Wendroff scheme, which is second-order accurate in time and space.
For the homogeneous convection-diffusion equation
If
ψ
=
1
−
/
=
2
f
∂
f
u
∂
f
c
∂
+
x
=
ε
(4.30)
∂
t
∂
∂
x
2
the Lax-Wendroff scheme is
t
f
i
−
1
−
f
n
+
1
i
u
f
i
+
1
−
f
i
−
1
2
f
i
f
i
+
1
f
i
+
−
1
2
u
2
+
=
ε
+
(4.31)
c
t
2
x
x
2
