Introduction to spectral methods

TL;DR

This paper provides an introductory overview of spectral methods, demonstrating their power through explicit solutions to simple problems, and covers mathematical foundations, basis functions, and various solver techniques.

Contribution

It offers a comprehensive introduction to spectral methods, including mathematical foundations, basis functions, and solver techniques, with detailed examples and discussions.

Findings

Spectral methods effectively solve simple problems with high accuracy.

Legendre and Chebyshev polynomials are key basis functions.

Multiple domain techniques enhance spectral method applications.

Abstract

This proceeding is intended to be a first introduction to spectral methods. It is written around some simple problems that are solved explicitly and in details and that aim at demonstrating the power of those methods. The mathematical foundation of the spectral approximation is first introduced, based on the Gauss quadratures. The two usual basis of Legendre and Chebyshev polynomials are then presented. The next section is devoted to one dimensional equation solvers using only one domain. Three different methods are described. Techniques using several domains are shown in the last section of this paper and their various merits discussed.