Geography Reference
In-Depth Information
to obtain more precise information on accuracy from examination of the solution
(13.14). Note that for θ
p
0. The part of the solution
proportional to C is the
physical mode
. The part proportional to D is called the
computational mode
because it has no counterpart in the analytic solution to the
original differential equation. It arises because centered time differencing has
turned a differential equation that is first order in time into a second-order finite
difference equation. The accuracy of the finite difference solution depends not only
on the smallness of D and the closeness of C to unity, but on the match between
the phase speed of the physical mode and the phase speed in the analytic solution.
The phase of the physical mode is given in (13.14) by
→
0,C
→
1, and D
→
(p/δx)
mδx
θ
p
sδx/p
=
k
x
c
t
−
=
−
−
pm
θ
p
s
where c
θ
p
δx/(pδt) is the phase speed of the physical mode. Its ratio to the
true phase speed is
=
c
/c
sin
−
1
(σ sin p) / (σp)
=
θ
p
δx/ (pcδt)
=
so that c
/c
0. The dependence of c
/c and
→
1asσp
→
|
D
|
/
|
C
|
on wavelength
is shown in Table 13.1 for the particular case where σ
0.75.
It is clear from Table 13.1 that phase speed and amplitude errors both increase
as the wavelength decreases. Short waves move slower than long waves in the
finite difference solution, even though in the original equation all waves move at
speed c. This dependence of phase speed on wavelength in the difference solution
is called
numerical dispersion
. It is a serious problem for numerical modeling of
any advected field that has sharp gradients (and hence large-amplitude short-wave
components).
Short waves also suffer from having significant amplitude in the computational
mode. This mode, which has no counterpart in the solution to the original differen-
tial equation, propagates opposite to the direction of the physical mode and changes
sign from one time step to the next. This behavior makes it easy to recognize when
the computational mode has significant amplitude.
=
Table 13.1
Phase and Amplitude Accuracy of Centered Difference Solution of the
Advection Equation as a Function of Resolution for
σ
=
0.75
c
/c
L/δx
p
θ
p
|
D
|
/
|
C
|
2
π
π
—
∞
4
π/2
0.848
0.720
0.204
8
π/4
0.559
0.949
0.082
16
π/8
0.291
0.988
0.021
32
π/16
0.147
0.997
0.005
