Let us turn to the variable coefficient case,
The pseudospectral approximation takes the form
subject to initial conditions
It can be solved as a coupled ODE system in the Fourier space, and at the same
time it
can be realized at the 2N+1 so-called collocation points
with initial conditions
The truncation error of this model is spectrally small in the sense that
satisfies
where
is spectrally small
Hence, if the approximation (meth_ps.12) is stable then
spectral convergence follows. Is
the approximation (meth_ps.12) stable?
The presence of aliasing errors
makes this stability question an intricate one -
here is a brief explanation.
Trying to follow the differential and spectral
setup, we should multiply by , integrate by parts and
hope for the best. However,
here is not orthogonal to
(-- otherwise this would enable us to estimate
in terms of
and we are done); more precisely, for
we only have that ; yet leaves us with an additional contribution which
is not necessarily bounded in terms of ,
and this argument fails short of a straightforward
stability proof by Gronwall's inequality. To shed a different light on
this difficulty, we can turn to the Fourier space;
we write (meth_ps.17) in the
form
and Fourier transform to get for the kth Fourier coefficient
i.e.,
This time, is unbounded.
This difficulty appears when we confine ourselves to the
discrete framework: multiplying (meth_ps.18) by
and trying to sum
by parts we arrive at
but the first term on the right does not vanish in this case - it equals,
by the aliasing relation, to
and a loss of one derivative is reflected by the factor 2N+1
inside the right summation. This does not prove an instability
as much as it shows the failure of disproving it along these lines.