Geography Reference
In-Depth Information
300ms
−
1
and for a 1-km
vertical grid interval a time step of only a few seconds would be permitted.
wave described in that set of equations. In that case c
≈
13.3.4
Implicit Time Differencing
The spurious computational mode introduced by the leapfrog time differencing
scheme does not occur in a number of other explicit schemes, such as the Euler
backward scheme discussed in Problem 13.3. Such schemes do, however, retain
the time step limitation imposed by the CFL condition. This restriction and the
computational mode are eliminated by utilization of an alternative finite differ-
encing scheme called the
trapezoidal implicit scheme
. For the linear advection
equation (13.6) this scheme can be written in the form
ˆ
ˆ
q
m,s
q
m
−
1,s
+
1
2δx
ˆ
q
m
−
1,s
2δx
q
m,s
+
1
−ˆ
q
m
+
1,s
+
1
−ˆ
q
m
+
1,s
−ˆ
c
2
=−
+
δt
(13.19)
Substituting the trial solution (13.12) into (13.19) yields
1
B
s
−
i (σ/2) sin p
B
s
+
1
=
(13.20)
1
+
i (σ/2) sin p
where as before σ
=
cδt/δxandp
=
kδx. Defining
tan θ
p
≡
(
σ/2
)
sin p
(13.21)
and eliminating the common term B
s
in (13.20), it can be shown that
1
exp
−
2iθ
p
−
i tan θ
p
B
=
=
(13.22)
1
+
i tan θ
p
so that the solution may be expressed simply as
A exp
ik
mδx
2θ
p
s/k
q
m,s
=
ˆ
−
(13.23)
Equation (13.19) involves only two time levels. Hence, unlike (13.14) the solu-
tion yields only a single mode, which has phase speed c
=
2θ
p
/(kδt). According
to (13.21), θ
p
remains real for all values of δt. [This should be contrasted to the
situation for the explicit scheme given by (13.17).] Thus, the implicit scheme is
absolutely stable. The truncation errors, however, can become large if θ
p
is not
kept sufficiently small (see Problem 13.9). A disadvantage of the implicit scheme is
that the integration cannot proceed by marching through the grid, as in the explicit
