An Introduction to Numerical Methods and Analysis, 2nd Edition

CHAPTER 10

AN INTRODUCTION TO SPECTRAL METHODS

In Chapters 6 and 9 we discussed using finite difference and finite element methods to get approximate solutions to ordinary and partial differential equations. Both approaches can produce very accurate approximations, and a lot of important computations are being done these days using both of these techniques.

But in §4.9 we did something different—we defined our approximation as a linear combination of smooth functions, and imposed the differential equation at a discrete set of points, which we called collocation points. Also in Chapter 4, we observed that the Chebyshev polynomials can produce very accurate approximations to functions. So we have an obvious question to ask and explore: Can we use the Chebyshev polynomials to produce approximate solutions to differential equations that are as accurate as the approximations in §4.11.2 might suggest? The answer is “yes” (we wouldn’t have posed the question if it couldn’t be done), and the resulting body of work is generally known, collectively, as spectral methods, although a variety of terms are used.¹ The presentation here relies heavily on the work of Boyd [2] and Trefethen [9].²

It should be said that the methods as presented here are in some ways suboptimal. “True” spectral methods rely on some techniques not covered in this text, in order to achieve their full efficiency, but it was thought worthwhile to expose students (and perhaps some instructors) to these techniques in a way ...