Talk:Pseudo-spectral method

Proposed changes
I have a few issues with this article. I'm certainly prepared to do some editting, but I figured I'd post here first.

1.) The diagram: you appear to be getting significant aliasing error in your solution. Does the simulation continue to be stable?  In my experience, with that kind of high-frequency oscillation, you're pretty darn close to getting NaN's in your solution.

2.) From this page, the mathematics behind the PS method is unclear.  Something more than just taking about solving the Schroedinger equation in momentum space is necessary.

3.) Quantum scattering off an arbitrary potential is not the most transparent example I can think of to demonstrate this method, insofar as 80% of the article deals with quantum scattering rather than the PS method.  I think it would be better to focus on the mechanics of the PS method, delete the current example in its entirety, and then use, say, an advective type equation like

$$\frac{du}{dt} = v \frac{du}{dx}$$

which doesn't really get into much in the way of nitty-gritty physics.

Any comments and such? If I don't hear back from anyone in the next couple days, I'll go ahead and make these changes.

J. Langton 21:24, 9 July 2006 (UTC)


 * I especially agree with points 2 and 3. An improved diagram would be nice too, of course. I encourage you to work on it! JJL 22:57, 9 July 2006 (UTC)

This article has almost nothing to do with pseudospectral methods except indirectly mentioning the applicability of the FFT when time-stepping the Schrödinger equation with the split-operator method.I think this needs a total rewrite which explains the method starting with Lagrange polynomials and their derivatives. The emphasis should be on point-space solutions as opposed to coefficient-space solutions with classical spectral methods.

Gregvw 09:26, 4 May 2007 (UTC)


 * Scattering from a square well is absolutely the wrong problem to use here. The square well is discontinuous, introducing Gibbs oscillations in the Fourier components.  This reduces the "spectral accuracy" to essentially zeroth-order, since Gibbs oscillations don't uniformly converge.  For a rewrite, I'd recommend Poisson's equation $$((f(x) u')' = g(x))$$ for some smoothly varying (periodic?) f.  That would show the benefits of the pseudospectral (collocation) method directly -- the convolution would be done by multiplication in physical space. Majromax 05:01, 19 June 2007 (UTC)


 * Agreed, this article is pretty useless and hasn't helped me learn anything about the method. I'd like to see a simple example that shows how an O(N^2) operation can be reduced to O(N log N) one, rather than have it buried in quantum mechanics, and discussion on aliasing etc. Leperous (talk) 14:04, 23 June 2008 (UTC)


 * Going from O(N^2) to O(N ln N) is more likely to be relevant to FFT. Apparently I've been meaning to clean up this article for the last two years, but hopefully I'll get around to it this week. J. Langton (talk) 15:27, 18 August 2008 (UTC)

Hot damn, this article is worthless; I got the impression that the article things that "pseudo-spectral methods" are actually split-operator propagation. Makes no sense at all, and should be deleted. 130.126.230.234 (talk) 17:36, 3 March 2009 (UTC)
 * If you think it should be deleted and someone disagrees and removes the "prod" tag, the proper procedure is to take the matter to WP:AfD. But if the article could be rewritten to become a worthwhile article on the topic identified in its title, then that, rather than deletion, is the appropriate remedy. Michael Hardy (talk) 17:37, 6 March 2009 (UTC)

The previous posters are right that the article in its current form is useless, even wrong. I would give it a try at completely rewriting it, but this will take a few days. Avg08 (talk) 08:59, 22 November 2013 (UTC)

PROD
Teh subject itself is quite notable, in my opinion, regardless of the quality of the article. JJL (talk) 22:03, 5 March 2009 (UTC)

Rewrite
I have now completely rewritten the article, and it should be hugely improved over the original version. The first time I encountered pseudo-spectral methods, I read a thorough discussion about the topic, which largely failed to give me the big picture (grid-based method and what the error is). I tried to avoid this throughout the article, to mainly present the central idea in a simple way. However, there are some issues that could be done differently. Since I have written this article, however, I might not be the best person to judge on changes. Avg08 (talk) 09:27, 2 December 2013 (UTC)
 * I have completely avoided the notion of the pseudo-spectral basis. I.e., when you give the function at the grid points, you could also rewrite this as an expansion in a basis. I think it would detract too much from the simple line of argument that I presented in the article, but maybe it could be incorporated without a loss of focus. And the article would certainly be more thorough if the pseudo-spectral basis was included.
 * I have not updated or done anything with the references at all. They need to be scrutinized.
 * There are some minor issues with respect to other articles. For example, I started all sums at zero because this seems naturally when you define then the polynomial or plane-wave basis. However, for example the Gaussian quadrature article uses sums starting at one, which introduces some minor inconsistencies in the notation.
 * The error discussion is plain lousy. I hope the article conveys the important idea (how many grid points you need), but at this point I was lacking both the formal apparatus and the motivation.
 * There is probably a strong physicist's touch to the article. I tried not to do so, and e.g. sticked to the notation of the spectral method, but I would bet that there are enough issues buried to make a mathematician cry.