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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?

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up vote 10 down vote accepted

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.

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I'm not sure I follow -- time would be just another dimension. Why is nonlinearity in case of time so much harder to deal with than in case of the spatial dimensions? – user7306 Feb 10 '14 at 21:49
in particular, in normal analysis, wouldn't we have a lot of problems coupled over the entire space? Why is that less of an issue compared to being coupled over time? – user7306 Feb 10 '14 at 21:56
You have the same problem if you do a modal analysis in space with nonlinear problems where the individual modes all couple. Of course, this is no different than when you do a local approach such as the finite element method where you also have to solve a global problem if you have a boundary value problem. But, and here's the difference, time dependent problems are not boundary value problems -- they are typically initial value problems and, consequently, you can use a time stepping approach. This corresponds to being able to "sweep" problems that have a designated transport direction. – Wolfgang Bangerth Feb 10 '14 at 22:57
In both cases, if you have a designated direction of information flow (i.e., a transport direction in space, or the time arrow in time), sweeping methods and time stepping methods are far better than globally coupled methods. – Wolfgang Bangerth Feb 10 '14 at 22:58
Ok. Thanks! Do you know of a few good references for this stuff? – user7306 Feb 11 '14 at 0:28

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".

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