72
2. Basic Finite-Difference Methods
If the oscillation equation (2.30) is integrated using a linear finitedifference scheme and if the truncation error of the resulting finitedifference approximation to the oscillation equation is of order r,
then as K tJ.t -+ 0 the amplitude change in each step of the numerical
solution is no worse than
where
n = r + l ,
{ n r + 2,
if r is odd;
if r is even;
and the relative phase change is no worse than
m =r,
if r is odd;
if r is even.
where
Switching from an even- to an odd-order scheme increases the order of accuracy
of the relative phase change without improving the order of accuracy of the amplitude error. Switching from odd to even order reduces the asymptotic amplitude
error without altering the order of the error in the relative phase change.
2.4 Space-Differencing
Having examined the errors associated with time-differencing in Section 2.3, let
us now consider the errors introduced when spatial derivatives are replaced with
finite differences . In order to isolate the influence of the spatial differencing, the
time dependence will not be discretized. Once again, our investigation will focus
on the constant-wind-speed advection equation
al/J
al/J
-+c-=O.
at
ax
(2.59)
If the x-dornain is periodic or unbounded, the spatial structure of the solution
may be represented by a Fourier series or a Fourier integral, and a solution for
each individual mode may be sought in the form of a traveling wave
l/J(x, t) = e i(kx-wt) .
Here k is the wave number, and co is the frequency. Substitution of this assumed
solution into (2.59) shows that the traveling wave will satisfy the goveming equation only if its frequency satisfies the dispersion relation
w = ck.
The wave travels with constant amplitude at aphase speed co] k = c. These waves
are nondispersive, meaning that their phase speed is independent of the wave
number. The energy associated with an isolated "packet" of waves propagates
at the group velocity aw/ak = c, which is also independent of wave number.
Readers unfamiliar with the concept of group velocity may wish to consult Gill
(1982) or Whitham (1974).
2. Basic Finite-Difference Methods
If the oscillation equation (2.30) is integrated using a linear finitedifference scheme and if the truncation error of the resulting finitedifference approximation to the oscillation equation is of order r,
then as K tJ.t -+ 0 the amplitude change in each step of the numerical
solution is no worse than
where
n = r + l ,
{ n r + 2,
if r is odd;
if r is even;
and the relative phase change is no worse than
m =r,
if r is odd;
if r is even.
where
Switching from an even- to an odd-order scheme increases the order of accuracy
of the relative phase change without improving the order of accuracy of the amplitude error. Switching from odd to even order reduces the asymptotic amplitude
error without altering the order of the error in the relative phase change.
2.4 Space-Differencing
Having examined the errors associated with time-differencing in Section 2.3, let
us now consider the errors introduced when spatial derivatives are replaced with
finite differences . In order to isolate the influence of the spatial differencing, the
time dependence will not be discretized. Once again, our investigation will focus
on the constant-wind-speed advection equation
al/J
al/J
-+c-=O.
at
ax
(2.59)
If the x-dornain is periodic or unbounded, the spatial structure of the solution
may be represented by a Fourier series or a Fourier integral, and a solution for
each individual mode may be sought in the form of a traveling wave
l/J(x, t) = e i(kx-wt) .
Here k is the wave number, and co is the frequency. Substitution of this assumed
solution into (2.59) shows that the traveling wave will satisfy the goveming equation only if its frequency satisfies the dispersion relation
w = ck.
The wave travels with constant amplitude at aphase speed co] k = c. These waves
are nondispersive, meaning that their phase speed is independent of the wave
number. The energy associated with an isolated "packet" of waves propagates
at the group velocity aw/ak = c, which is also independent of wave number.
Readers unfamiliar with the concept of group velocity may wish to consult Gill
(1982) or Whitham (1974).