Chebyshev and Fourier Spectral Methods

The most common methods taught in introductory courses on numerical methods for partial differential equations are finite difference and finite element methods. These methods are widely taught and used, but there is another less well known class of methods for the numerical solution of PDEs that has important advantages in some applications. These spectral methods are the subject of this classic textbook and reference. The revised second edition is avaialabe as an inexpensive Dover reprint.

The basic idea in spectral methods is to write the solution \(u(x)\) as a linear combination of orthogonal basis functions \[u(x)=\sum_{i=0}^{n} a_{n} \phi_{n}(x).\] In practice, the basis functions are often Chebyshev polynomials, or sometimes sines and cosines forming a Fourier series. For these series, there are simple formulas for the coefficients in the expansion of \(u'(x)\) and higher order derivatives of \(u(x)\). If we have a linear differential operator \(Lu\), we can write the series coefficients of \(Lu\) as a matrix vector product \(La\). Sometimes, these sums are more efficiently evaluated by using Fast Fourier Transform (FFT) methods.

In the so-called pseusdospectral method, we discretize a PDE \(Lu=f\) by requiring \(Lu\) to exactly interpolate \(f(x)\) at \(n+1\) appropriately chosen collocation points. This produces a linear system of \(n+1\) equations for the unknown coefficients \(a_{0}, a_{1}, \ldots, a_{n}\). In the spectral method, we instead minimize \(\| Lu-f \|\) by orthogonal projection as in the Galerkin finite element method. The approach can also be extended in a straight forward way to deal with multiple space dimensions and nonlinear equations. In the spectral method, time dependent PDEs are converted into systems of ordinary differential equations. The spectral method can also be used to solve PDE eigenvalue problems.

The difference between spectral methods and conventional finite element methods is that in the spectral method we use a single high order polynomial to represent the solution rather than piecing together a solution from piecewise polynomial solutions. In comparison with finite element methods, the resulting linear systems of equations are dense rather than sparse. However, because of the high order of the interpolating polynomial it is often possible to obtain an accurate solution using a much smaller number of equations and variables than with a finite element method. For problems with smooth solutions, spectral methods can often produce far more accurate solutions than finite element or finite difference methods.

This book provides a very clear and accessible introduction to spectral methods. However, there is so much material here that it may be overwhelming to beginners. The book is mostly likely to serve as a useful reference to readers who are implementing spectral methods. For readers who are interested in a broader introduction to approximation by Fourier and Chebyshev series that briefly introduces spectral methods for PDEs, the recent Approximation theory and approximation practice by Lloyd Trefethen would be a good choice.

Brian Borchers is a professor of Mathematics at the New Mexico Institute of Mining and Technology. His interests are in optimization and applications of optimization in parameter estimation and inverse problems.