Spectral methods are a popular choice for the numerical approximation of nonlinear evolution equations. One of the advantages of spectral methods is the rapid convergence, making it possible to achieve high accuracy in computations with relatively few grid points. Indeed, it can often be proved that spectral projections feature convergence rates that are higher than any algebraic power.
In this lecture, we will focus on the KdV equation. It will be shown that when the initial data is analytic, then the convergence rate is actually exponential. This result agrees well with numerical experiments which also exhibit exponential convergence.