0
We return to the scalar constant coefficient case
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