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## The Pseudospectral Fourier Approximation

0

subject to periodic boundary conditions and prescribed initial data

To solve this problem by the pseudospectral Fourier method, we proceed as before, this time projecting (meth_ps.1) with the pseudospectral projection , to obtain for

Here, commutes with multiplication by a constant, but unlike the spectral case, it does not commute with differentiation, i.e., by the aliasing relation (app_ps.2) we have

where as

The difference between these two expressions is a pure aliasing error, i.e., we have for , see (app_ps.13)

which is spectrally small. Sacrificing such spectrally small errors, we are led to the pseudospectral approximation of (meth_ps.1)

subject to initial conditions

Here, is an N-degree trigonometric polynomial which satisfies a nearby equation satisfied by the interpolant of the exact solution . That is, satisfies (meth_ps.5) modulo spectrally small truncation error

where by (meth_ps.3), , and by (app_ps.17) it is indeed spectrally small

The stability proof of (meth_ps.5) follows along the lines of the spectral stability, and spectral convergence follows using Duhammel's principle for the stable numerical solution operator. That is, the error equation for is

whose solution is

Hence, by stability

this together with the estimate of the pseudospectral projection yields

To carry out the calculation of (meth_ps.5) we can compute the discrete Fourier coefficients which obey the ODE,

as was done with the spectral case; alternatively, we can realize our approximate interpolant at the 2N+1 equidistant points , and (meth_ps.5) amounts to a coupled (2N+1) - ODE system in the real space