Talk:Spectral method
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[edit] Numeric Application
I moved the section added recently here. I'm afraid there are too much issues with it. Firstly, it concerns almost the same topic as the previous section, but it is not linked to it in any way. Indeed, it's written as if it were about something completely different. Secondly, it contains a mistake: the Fourier transform of uux is not − iku2; the multiplication becomes a convolution. This makes it actually an interesting example, which would be useful if treated correctly, but it needs to be rewritten. As it stands, it's not an improvement. -- Jitse Niesen (talk) 05:01, 16 May 2007 (UTC)
- Section copied from the article
For the purposes of solving various differential equations numerically, a spectral decomposition may prove to be quite useful and effective in favour of a finite difference method. A common method of approximating a solution to a differential equation, is by expressing a given ODE or PDE in terms of the solution function, U. Doing so within R-space, can be quite cumbersome. Consider the KdV equation
- Ut + Uxxx + 6UUx = 0.
However, within a Fourier space (F-space), using the property of the Fourier Transform:
- u'(k) = − iku(k),
where
- F{U(x)} = u(k),
and
- F − 1{u(k)} = U(x),
then the above differential equation can now be re-written as:
- ut + uxxx + 6uux = 0
- ut − ik3u − ik6u2 = 0.
Solving such an equation numerically (solving for ut) is relatively simple, and becomes much less computationally expensive. Furthermore, if the functional form of the solution changes for other initial conditions or parameter values, the expression within the Fourier space remains valid. If the equation were still within R-space, as the functional form of U changes, the derivatives would have to be recalculated.
===Error in the form of Aliasing===
Within the context of numerical methods, the Fourier Transform is implemented by use of the Fast Fourier Transform, where a finite series is used to approximate the true integral. The true integral of the Fourier Transform can be expressed as an infinite series, but for computational purposes must be truncated to a fixed value, N. The error of a spectral decomposition occurs in the form of aliasing, which is as a result of N being too small.
- That's a pretty bad section, actually. Good to have it removed. Loisel 09:08, 16 May 2007 (UTC)
Is this more of an objection to talking about the applications of spectral method (as per it not flowing with the article). If so, I should find a better place for it. Perhaps taking a step back from it being so concrete and remaining more general. Thoughts/suggestions?
[edit] Spectral decomposition
From the article:
However, whereas the spectral method is based on the eigendecomposition of the particular boundary value problem, the spectral element method does not use that information and works for arbitrary elliptic boundary value problems.
That's not exactly true: a spectral method is almost never based on the exact eigenfunctions of the BVP. Instead, the "spectrum" used is a simpler one, typically the Fourier basis or the Chebyshev or Legendre polynomials (for nonperiodic cases).
The math stays relatively simple with FFT-based transforms, and it avoids the expensive whammy of calculating the eigenfunctions in the first place. Indeed, spectral methods can be used themselves to approximate the eigenvalues/eigenfunctions of a differential operator, pretty straightforwardly.
Majromax 04:57, 19 June 2007 (UTC)
