Spectral differentiation discussions

[yewalenikhil65]

Spectral differentiation is one of the traditional ways of approximating derivatives. We would like to have a short demonstration as docs/markdown files, and some efficient and generalized functions to obtain spectral differentiation operators/matrices using following basic/common ways:

• Fourier spectral methods
• Chebyshev spectral methods
• FFT Based differentiation

Its worth mentioning some good reads of Prof Lloyd N. Trefethen that could be helpful later:

1. Book, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations
2. Book, Spectral methods in MATLAB

[yewalenikhil65]

Before going after spectral differentiation operators, i thought its good to first introduce the implementations of DFT computing algorithms, from naive Ο(N²) , followed by (Ο(N²) but less than N², and then plain recursive Cooley-tuckey(decimation-in-time radix 2) FFT algorithm of Ο(N. log(N)). Currently this is 1D DFT implementation, and has focus on

1. Verification with standard functions of fft in packages like FFTW
2. Benchmarking the performance with packages like FFTW

The implementation can found here and for the further work, we will rely on the julia bindings of FFTW package

Note:

1. The naive Ο(N²) implementations are simple loop based, and we have not adopted its equivalent DFT matrix based Ο(N²).
2. @pravinjagtap You are welcome to include multi-threaded DFTs , or can take a look when I add them myself later
3. Cosmetics will be done, when we define spectral-operators along with the docs later !