140
3. Beyond the One-Way Wave Equation
(a)
-
.
LFB
(b)
...........
---o
I
o
,
\
\
LFT
,
LFF
' "
"...
'"
-1
-1.5
-1.0
-0.5
AAt
o
-1.5
-1.0
-0.5
AAt
o
FIGURE 3.7. (a) Regions of useful stability in the X-iV plane if the oscillatory forcing is integrated using leapfrog differencing and the damping is integrated using the forward (LFF),
trapezoidal (LFf), or backward (LFB) methods . (b) Region of absolute stability when both
the oscillatory forcing and the damping are both integrated using the forward (F), the second-order (AB2), or the third-order (AB3) Adams-Bashforth scheme . Also shown in (b) is
the region of absolute stability when the integration of the oscillatory forcing is third-order
Adams-Bashforth and the damping is integrated using the trapezoidal method (AB3T).
Note the difference in the horizontal and vertical scales.
The portion of the i-w plane satisfying this inequality is the circle of unit radius
centered at (-1,0) plotted in Fig. 3.7b. As discussed in Section 2.3, forward
differencing generates unstable solutions to the pure advection problem. This can
also be deduced from Fig. 3.7b by noting that the region of absolute stability does
not inc1ude a finite segment of the waxis. Forward differencing will therefore fail
to generate a stable numerical solution to the advection-diffusion problem unless
the diffusion is relatively large.
One might attempt to improve the stability of the numerical solution to the
low-viscosity advection-diffusion problem by using leapfrog time-differencing,
which as demonstrated in Section 2.3.4 is stable in the zero-viscosity limit. The
two amplification factors associated with the leapfrog approximation to (3.75) are
Alf = iw + A
-
±
[ (iw + A)
- 2
+ 1
] 1/2 .
The conditions under which both the computational and physical modes are stable
can be most easily established by defining a complex number such that
i cos = iw + i.
Then
Alf = i cos ± sin =
and thus the two amplification factors associated with the leapfrog scheme have
magnitudes
I and
I. One of these will exceed unity unless is real, and
3. Beyond the One-Way Wave Equation
(a)
-
.
LFB
(b)
...........
---o
I
o
,
\
\
LFT
,
LFF
' "
"...
'"
-1
-1.5
-1.0
-0.5
AAt
o
-1.5
-1.0
-0.5
AAt
o
FIGURE 3.7. (a) Regions of useful stability in the X-iV plane if the oscillatory forcing is integrated using leapfrog differencing and the damping is integrated using the forward (LFF),
trapezoidal (LFf), or backward (LFB) methods . (b) Region of absolute stability when both
the oscillatory forcing and the damping are both integrated using the forward (F), the second-order (AB2), or the third-order (AB3) Adams-Bashforth scheme . Also shown in (b) is
the region of absolute stability when the integration of the oscillatory forcing is third-order
Adams-Bashforth and the damping is integrated using the trapezoidal method (AB3T).
Note the difference in the horizontal and vertical scales.
The portion of the i-w plane satisfying this inequality is the circle of unit radius
centered at (-1,0) plotted in Fig. 3.7b. As discussed in Section 2.3, forward
differencing generates unstable solutions to the pure advection problem. This can
also be deduced from Fig. 3.7b by noting that the region of absolute stability does
not inc1ude a finite segment of the waxis. Forward differencing will therefore fail
to generate a stable numerical solution to the advection-diffusion problem unless
the diffusion is relatively large.
One might attempt to improve the stability of the numerical solution to the
low-viscosity advection-diffusion problem by using leapfrog time-differencing,
which as demonstrated in Section 2.3.4 is stable in the zero-viscosity limit. The
two amplification factors associated with the leapfrog approximation to (3.75) are
Alf = iw + A
-
±
[ (iw + A)
- 2
+ 1
] 1/2 .
The conditions under which both the computational and physical modes are stable
can be most easily established by defining a complex number such that
i cos = iw + i.
Then
Alf = i cos ± sin =
and thus the two amplification factors associated with the leapfrog scheme have
magnitudes
I and
I. One of these will exceed unity unless is real, and