[Aerterliusi]

I have some conceptual doubts about spectral methods, and the following is my current understanding of spectral methods. I am not sure if my understanding is correct for this type of method:

Strictly speaking, spectral methods = FT(Fourier Transform)、FS(Fourier Series)、DFT, then discretely solving the transformed equations, and then transforming back.

More generally, spectral methods involve transforming a function into another function, or expanding a function into a combination of a special set of functions, which can include not just Fourier bases, such as Legendre polynomials, Chebychev polynomials and etc. The subsequent process is the same as before, where the equation is transformed into another function's equation, discretely solved, and then transformed back.(the above is my guess because I have not yet seen an concrete example.)

Is my understanding of this type/family of method accurate? If not, could you explain what your understanding of spectral methods is?"

btw, I have heard the term 'pseudo-spectral methods', and I have briefly looked into the content of this method. However, I am curious as to why it is called 'pseudo-spectral'. What exactly are 'true spectral methods'?

[LuckyTran]

Yes you are correct in the sense that spectral methods are solving the same equations in frequency domain, which is like doing a Fourier transform. It is not necessary however that the solution is not obtained via inverse transforming because the goal is not necessarily to obtain the solution in time domain.

Legendre polynomials and Chebyshev polynomials don't inherently have anything to do with spectral methods and are just techniques for solving discrete equations whether they be time-domain or spectral domain. But legendre and chebyshev polynomials lend themselves to being great basis functions for spectral methods. For the sake of determining whether a method is pseudo-spectral or spectral spectral, you can forget what basis functions are actually used.

Actually the method of transforming into Fourier, solving the equations, and transforming back, is an example of a pseudo-spectral method. If your method of quadrature is done entirely in spectral space then it is a truly spectral method. If your method of quadrature involves going back to time-domain, then it is pseudo-spectral.

[Aerterliusi]

Originally Posted by LuckyTran

Yes you are correct in the sense that spectral methods are solving the same equations in frequency domain, which is like doing a Fourier transform. It is not necessary however that the solution is not obtained via inverse transforming because the goal is not necessarily to obtain the solution in time domain.


Legendre polynomials and Chebyshev polynomials have almost nothing to do with spectral methods and are just techniques for solving discrete equations whether they be time-domain or spectral domain.


Actually the method of transforming into Fourier, solving the equations, and transforming back, is an example of a pseudo-spectral method.

Okay, thank you. Now I know that my general understanding is not incorrect.

[gnwt4a]

any approximation method whose asymptotic error bound decreases faster than any algebraic power of the degrees-of-freedom, is called spectral. global polynomial expansions are best known, but spectral elements (local poly expansions) also have this property (for suitably smooth solutions).

for some types of bcs there are optimal (=lowest asymptotic error of any method) approximations: for periodic bcs are fourier; for double-Dirichlet chebs.

pseudo-spectral refers to an approximate method for calculating the convolutions arising from the nonlinear terms. it suffers from aliasing errors, but the originator S. Orszag, showed how to remove those in fourier expansions - at a cost. the pseudo-spectral technique is a key enabling technology of usable SMs.
--

[Aerterliusi]

Originally Posted by gnwt4a

any approximation method whose asymptotic error bound decreases faster than any algebraic power of the degrees-of-freedom, is called spectral. global polynomial expansions are best known, but spectral elements (local poly expansions) also have this property (for suitably smooth solutions).

for some types of bcs there are optimal (=lowest asymptotic error of any method) approximations: for periodic bcs are fourier; for double-Dirichlet chebs.

pseudo-spectral refers to an approximate method for calculating the convolutions arising from the nonlinear terms. it suffers from aliasing errors, but the originator S. Orszag, showed how to remove those in fourier expansions - at a cost. the pseudo-spectral technique is a key enabling technology of usable SMs.
--

Thank you. After some studying, I also learned that spectral methods have excellent convergence properties, with convergence speeds faster than any polynomial.

As for pseudo-spectral methods, I looked at the wiki and through the examples given, I understand that while a "true" spectral method processes nonlinear terms by performing a Fourier transform on each variable and then performing operations such as multiplication or squaring, pseudo-spectral methods perform these operations directly in physical space and then perform a Fourier transform on the "whole" term. As for the alignment issue you mentioned, I am not quite familiar with it, but it is said to have been well resolved.

[FMDenaro]

My experience is that SM are good but ... when you work on it you realize the issues.

Conservation is not ensured and aliasing requires proper treatment that reduces the formal spectral accuracy.


I have no particular experience in spectral element (many years ago I had the chance to see Patera illustrating it and Nekton in use), may if they are proper inserted in an integral form that could be a better way than SM.