In the previous chapters, we have repeatedly referred to the exponential convergence rate of spectral methods for analytic functions. This is discussed in more detail in Section 4.1. When functions are not smooth, PS theory is much less clear. An approximation can appear very good in one norm and, at the same time, very bad in another. As illustrated in Section 4.2, PS performance can also be very impressive in many cases that are “theoretically questionable” – this is exploited in most major PS applications. Sections 4.3–4.5 describe differentiation matrices in more detail, their influence on time stepping procedures, and linear stability conditions. A fundamentally different kind of instability, specific to nonlinear equations, is discussed in Section 4.6. Very particular distributions of the nodes yield spectacular accuracies for Gaussian quadrature formulas. PS methods are often based on such formulas, presumably with the hope of obtaining a correspondingly enhanced accuracy when approximating differential equations. In the examples of Section 4.7, we see little evidence for this.

Smoothness of a function is a rather vague concept. Increasingly severe requirements include:

• a finite number of continuous derivatives;

• infinitely many derivatives; and

• analyticity – allowing continuation as a differentiate complex function away from the real axis.

In the limit of N (number of nodes or gridpoints) tending to infinity, these cases give different asymptotic convergence rates for PS methods. In the first case, the rate becomes polynomial with the power corresponding to the number of derivatives that are available.