2.3 Time-Differencing
49
If the spatial structure of the solution to the advection equation
a1/r + c a1/r = 0
at
ax
is represented by a Fourier series (or a Fourier Integral), the amplitude of an individual Fourier mode bk(t) must satisfy
dbi.
-
= -ikcbk.
(2.29)
dt
The preceding is an ordinary differential equation that can be used to examine the
numerical error introduced by time-differencing in wave-propagation problems.
Equation (2.29) is a specific example of the oscillation equation
d1/r
dt
-
.
= IK1/r,
(2.30)
where K is areal constant representing a frequency. Integrating the osciIIation
equation over a time Ilt yields
(2.31)
Here the last relation defines an "exact amplification factor" A e , which is a complex number of modulus one. According to (2.31), 1/r moves Kilt radians around
a circle ofradius 11/r(to)I in the complex plane over the time interval Ilt.
Suppose that a finite-difference scheme is used to compute an approximate
solution 4J to the osciIIation equation. A numerical amplification factor may be
defined such that 4Jn+ 1 = A4Jn, where 4Jn is the numerical approximation to
1/r(nllt). The standard Von Neumann stability analysis seeks a yes-no answer to
the question, Is IAI IA e I = I? Additional information about the amplitude and
phase -speed error in the numerical solution can, however, be obtained by writing
the numerical amplification factor in the form IAle
i O
, where
lAI = (9t{A}2 +
and () = arctan
Phase -speed errors arise from the difference between the argument of the finitedifference amplification factor () and the correct value of Kilt. A useful way to
characterize phase-speed error is through the telative phase change R = ()/(K Ilt),
which is the ratio of the phase advance produced by one time step of the finitedifference scheme divided by the change in phase experienced by the true solution
over the same time intervaI. If R > 1, the finite-difference scheme is accelerating;
if R < 1, the scheme is decelerating .
Amplitude errors arise from the difference between the modulus of the finitedifference amplification factor IAland the correct value of unity. When IA I = I,
the scheme is neutral. If IAI< 1, the scheme is damping; and if IAI> I, it
is amplifying. Amplifying schemes are certainly unstable in the sense that they
generate approximate solutions to the osciIIation equation that blow up, whereas
49
If the spatial structure of the solution to the advection equation
a1/r + c a1/r = 0
at
ax
is represented by a Fourier series (or a Fourier Integral), the amplitude of an individual Fourier mode bk(t) must satisfy
dbi.
-
= -ikcbk.
(2.29)
dt
The preceding is an ordinary differential equation that can be used to examine the
numerical error introduced by time-differencing in wave-propagation problems.
Equation (2.29) is a specific example of the oscillation equation
d1/r
dt
-
.
= IK1/r,
(2.30)
where K is areal constant representing a frequency. Integrating the osciIIation
equation over a time Ilt yields
(2.31)
Here the last relation defines an "exact amplification factor" A e , which is a complex number of modulus one. According to (2.31), 1/r moves Kilt radians around
a circle ofradius 11/r(to)I in the complex plane over the time interval Ilt.
Suppose that a finite-difference scheme is used to compute an approximate
solution 4J to the osciIIation equation. A numerical amplification factor may be
defined such that 4Jn+ 1 = A4Jn, where 4Jn is the numerical approximation to
1/r(nllt). The standard Von Neumann stability analysis seeks a yes-no answer to
the question, Is IAI IA e I = I? Additional information about the amplitude and
phase -speed error in the numerical solution can, however, be obtained by writing
the numerical amplification factor in the form IAle
i O
, where
lAI = (9t{A}2 +
and () = arctan
Phase -speed errors arise from the difference between the argument of the finitedifference amplification factor () and the correct value of Kilt. A useful way to
characterize phase-speed error is through the telative phase change R = ()/(K Ilt),
which is the ratio of the phase advance produced by one time step of the finitedifference scheme divided by the change in phase experienced by the true solution
over the same time intervaI. If R > 1, the finite-difference scheme is accelerating;
if R < 1, the scheme is decelerating .
Amplitude errors arise from the difference between the modulus of the finitedifference amplification factor IAland the correct value of unity. When IA I = I,
the scheme is neutral. If IAI< 1, the scheme is damping; and if IAI> I, it
is amplifying. Amplifying schemes are certainly unstable in the sense that they
generate approximate solutions to the osciIIation equation that blow up, whereas