I was reading up on Spectral Methods for PDEs. In all the descriptions I read, while the position component is approximated via a Fourier series or other methods, the time component is still discretized and solved via a time-step procedure (finite difference, etc.).
Is there any reason why the time component is also not approximated via a closed form solution?
Edit: I found one paper which does use a polynomial approximation even for the time dimension but my question remains as to why it's not done in general. Is it because chaotic dynamics means the number of terms required for the representation will be too large?