Overview

In the last chapter we discussed why grid points must cluster at boundaries for spectral methods based on polynomials. In particular, we introduced the Chebyshev points,

which cluster as required. In this chapter we shall use these points to construct Chebyshev differentiation matrices and apply these matrices to differentiate a few functions. The same set of points will continue to be the basis of many of our computations throughout the rest of the book.

Our scheme is as follows. Given a grid function v defined on the Chebyshev points, we obtain a discrete derivative w in two steps:

• Let p be the unique polynomial of degree ? N with p(x _j) = v _j, 0 ? j ? N.

• Set w _j = p ?(x _j).

This operation is linear, so it can be represented by multiplication by an ( N + 1) ( N + 1) matrix, which we shall denote by D _N:

Here N is an arbitrary positive integer, even or odd. The restriction to even N in this book (p. 18) applies to Fourier, not Chebyshev spectral methods.

To get a feel for the interpolation process, we take a look at N = 1 and N = 2 before proceeding to the general case.

Consider first N = 1. The interpolation points are x ₀ = 1 and x ₁ = ?1, and...