The Azimuth Project
Spectral methods for PDEs (Rev #1)

Contents

Idea

Spectral methods are methods for the numerical approximation of partial differential equations. They are important for the solution of the Navier-Stokes equations in meteorology and in climate models.

The basic idea of spectral methods is to choose a finite set of functions and calculate the optimal approximation of the exact solution by these functions. These basis functions are often part of an orthonormal basis of a Hilbert space, and more specifically trigonometric functions, hence the name “spectral” methods.

Details

The following paragraph is meant as an introduction to the method for pure mathematicians with a background in functional analysis.

For illustrative purposes we will make some simplifying assumptions. Let’s assume that we have an infinite topological vector space T, its topological dual T *T^* and a (differential) operator

A:TT A: T \to T

with a unique solution of the equation

A(f)=0 A(f) = 0

We omit initial and boundary conditions for the moment. In order to calculate an approximation to the exact solution ff, we need to turn the infinite dimensional problem to a finite dimensional one.

The basic idea of spectral methods is to choose a finite dimensional subspace of T spanned by a given set of functions {g 1,...,g n}\{g_1, ..., g_n \}, which are called in this context trial, expansion or approximation functions.

References

  • Spectral methods, Wikipedia

  • Claudio Canuto, M. Yousuff Hussaini, Alfio Quarteroni, Thomas A. Zang: Spectral methods. Fundamentals in single domains. (Springer 2006, ZMATH)

  • Claudio Canuto, M. Yousuff Hussaini, Alfio Quarteroni, Thomas A. Zang: Spectral methods. Evolution to complex geometries and applications to fluid dynamics. (Springer 2007, ZMATH)

  • David A. Kopriva: Implementing spectral methods for partial differential equations. Algorithms for scientists and engineers. (Springer 2009, ZMATH)