A brief introduction to pseudo-spectral methods: application to diffusion problems

TL;DR

This paper provides an accessible introduction to pseudo-spectral methods, emphasizing Fourier discretizations and their application to diffusion problems, with theoretical insights and practical considerations.

Contribution

It offers a concise overview of spectral and pseudo-spectral methods, including Fourier, Tchebyshev, and Legendre approaches, tailored for diffusion problems.

Findings

Fourier-type discretizations effectively solve diffusion problems.

Pseudo-spectral methods have broad applicability in numerical analysis.

Theoretical foundations support practical implementation of spectral methods.

Abstract

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.