A brief introduction to pseudo-spectral methods: application to diffusion problems
It serves as a tutorial for PhD students, but offers no novel contributions or quantitative results.
This paper provides an introductory overview of pseudo-spectral methods, focusing on Fourier-type discretizations for diffusion problems, with some discussion of Tchebyshev and Legendre approaches for non-periodic cases.
The topic of these notes could be easily expanded into a full one-semester course. Nevertheless, we shall try to give some flavour along with theoretical bases of spectral and pseudo-spectral methods. The main focus is made on Fourier-type discretizations, even if some indications on how to handle non-periodic problems via Tchebyshev and Legendre approaches are made as well. The applications presented here are diffusion-type problems in accordance with the topics of the PhD school.