Q:

Spectral Methods in time

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? 

A:

The reason is that with the exception of linear problems, if you do a Fourier (or other) decomposition in time, you end up with a significant number of problems that are coupled globally in time.  In other words, you have to solve lots of problems on the entire time interval concurrently.  That will typically bust your computational or memory budget.  The beauty of time stepping schemes is that you only need to consider one (or a very small number) if time steps at the same time and can forget everything about the more recent past. 

A:

For something with a spectral flavor in time, look at deferred correction methods, starting with this paper.   I would argue that they're not spectral in the usual sense of the word, but they give you a family of arbitrary-order Runge-Kutta methods, so if you think of "refining" by increasing the order (by adding more nodes), then the convergence can be spectral.   Of course, you could do the same thing with extrapolation methods, and nobody calls them "spectral".