Spectral methods solve elliptic PDEs numerically with errors bounded by an exponentially decaying function of the number of modes when the solution is analytic. For time dependent PDEs, almost all focus has been on low-order finite difference schemes for the time derivative and spectral schemes for spatial derivatives. This is not ideal since the time discretization error

destroys the spectral convergence in space. Space-time spectral methods are new methods which apply spectral discretization in both space and time. This is a series talk and consists of two parts.

Part-II: in the second talk, we show an analysis for the Schrodinger, wave, airy, beam and Stokes equations. These analyses require sharp estimates of the spectrum of spectral derivative matrices. We also show numerical experiments for many common nonlinear PDEs: Allen--Cahn, Cahn--Hilliard, Burgers, reaction-diffusion, KdV, Sine Gordon, Kuramoto--Shivashinsky, Navier--Stokes, MHD.