148
3. Beyond the One-WayWaveEquation
ployed to solve (3.88) and (3.89), a frozen-coefficient stability analysis will yield
the same stability condition for each scheme. An analysis of truncation error, perfonned by substituting Taylor series expansions for c and 1/1 into the differentialdifference equations, shows that both schemes are accurate to 0 [(ö'x)2]. Is there
any practical difference between (3.88) and (3.89)? There is, but the difference
is not obvious unless one considers problems in which c and r/J are poorly resolved on the numerical mesh or situations where the true solution has additional
conservation properties (such as advection in a nondivergent fiow) that are not
automatically retained by finite-difference approximations.
First consider the problems that can arise when there are large-arnplitude poorly
resolved perturbations in r/J or c. Under such circumstances, both of the preceding numerical approximations can exhibit serious instabilities. The structure and
growth rates of the unstable perturbations generated by each scheme can , however, be very different. Perhaps the most useful way to explore these instabilities
is to examine the aliasing error produced by (3.88) and (3.89).
3.5.1 Aliasing Error
Aliasing error occurs when a short-wavelength fiuctuation is sampled at discrete intervals and misinterpreted as a longer-wavelength oscillation . The shortest
wavelength that can be represented on a numerical grid is twice the grid interval; all shorter wavelengths will be aliased. Figure 3.8 illustrates the aliasing of a
4ö'x13 wave into a 4ö'x wave. The apparent equivalence of the 4ö'x13 and 4ö'x
waves follows from the fact that for all integers n, the relation
(3.90)
is satisfied at all spatial locations j Ö,X on the discrete mesh. In the case shown
in Fig. 3.8, the wave number of the aliased wave is k = (2rr)/(4ö'xI3) =
3rr/(2ö'x), and the wave number of the resolved wave is -rr/(2ö'x), so that
(3.90) applies with n = -1 . The change in the sign of the wave number during aliasing is visible in Fig. 3.8 as the 180
0 phase shift between the original and
the aliased wave.
Aliasing error may occur when the initial data are represented on a discrete
grid or projected onto a truncated series of Fourier expansion functions . Aliasing error can also occur during the computation of the product ca1/1lax on a
finite-resolution numerical grid. In order to illustrate how the product of two spatially varying functions may introduce aliasing error, suppose that the product
r/J(x)X(x) is computed at a discrete set of grid points. If r/J = eik)x and X = e i k2X,
then r/J X = ei(kl +k2)X. Since r/J and X were representable on the numerical grid,
Ikll, Ik21 ::: rr1S». It is possible, however, that the wave number of their product
lies in the range tt1Sx < Ikl + k21 ::: 2rr1ö'x, in which case the product cannot be resolved on the numerical mesh and will misrepresented as a longer wave.
The wave number k into which a binary product is aliased is deterrnined by the
3. Beyond the One-WayWaveEquation
ployed to solve (3.88) and (3.89), a frozen-coefficient stability analysis will yield
the same stability condition for each scheme. An analysis of truncation error, perfonned by substituting Taylor series expansions for c and 1/1 into the differentialdifference equations, shows that both schemes are accurate to 0 [(ö'x)2]. Is there
any practical difference between (3.88) and (3.89)? There is, but the difference
is not obvious unless one considers problems in which c and r/J are poorly resolved on the numerical mesh or situations where the true solution has additional
conservation properties (such as advection in a nondivergent fiow) that are not
automatically retained by finite-difference approximations.
First consider the problems that can arise when there are large-arnplitude poorly
resolved perturbations in r/J or c. Under such circumstances, both of the preceding numerical approximations can exhibit serious instabilities. The structure and
growth rates of the unstable perturbations generated by each scheme can , however, be very different. Perhaps the most useful way to explore these instabilities
is to examine the aliasing error produced by (3.88) and (3.89).
3.5.1 Aliasing Error
Aliasing error occurs when a short-wavelength fiuctuation is sampled at discrete intervals and misinterpreted as a longer-wavelength oscillation . The shortest
wavelength that can be represented on a numerical grid is twice the grid interval; all shorter wavelengths will be aliased. Figure 3.8 illustrates the aliasing of a
4ö'x13 wave into a 4ö'x wave. The apparent equivalence of the 4ö'x13 and 4ö'x
waves follows from the fact that for all integers n, the relation
(3.90)
is satisfied at all spatial locations j Ö,X on the discrete mesh. In the case shown
in Fig. 3.8, the wave number of the aliased wave is k = (2rr)/(4ö'xI3) =
3rr/(2ö'x), and the wave number of the resolved wave is -rr/(2ö'x), so that
(3.90) applies with n = -1 . The change in the sign of the wave number during aliasing is visible in Fig. 3.8 as the 180
0 phase shift between the original and
the aliased wave.
Aliasing error may occur when the initial data are represented on a discrete
grid or projected onto a truncated series of Fourier expansion functions . Aliasing error can also occur during the computation of the product ca1/1lax on a
finite-resolution numerical grid. In order to illustrate how the product of two spatially varying functions may introduce aliasing error, suppose that the product
r/J(x)X(x) is computed at a discrete set of grid points. If r/J = eik)x and X = e i k2X,
then r/J X = ei(kl +k2)X. Since r/J and X were representable on the numerical grid,
Ikll, Ik21 ::: rr1S». It is possible, however, that the wave number of their product
lies in the range tt1Sx < Ikl + k21 ::: 2rr1ö'x, in which case the product cannot be resolved on the numerical mesh and will misrepresented as a longer wave.
The wave number k into which a binary product is aliased is deterrnined by the