Talk:Pseudo-spectral method

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[edit] 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)