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### Is the pseudospectral approximation with variable coefficients stable?

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.

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