Abstract: In this talk we describe a new perspective on spectral methods for time-dependent PDEs. Basically, given an equation evolving in a separable Hilbert space, a spectral method is no more than a choice of an orthonormal basis. The choice of such a basis is governed by a raft of considerations: stability, speed of convergence, structure preservation and the ease of numerical algebra.
Focusing on a single space dimension, we distinguish between two cases: T-systems and W-systems. T-systems are defined on L2(R), can be characterized completely and possess a tridiagonal, skew-Hermitian differentiation matrix: this renders linear algebra very easy. W-systems act on L2(a, b), where (a, b) ⊂ R, and are defined directly from orthogonal polynomials. In their case the differentiation matrix is semi-separable of rank 1, again yielding itself to rapid linear algebra.
We describe the state of the art with both types of systems, with an emphasis on their approximation-theoretic features. Time allowing, we will mention a generalization to Galerkin–Petrov-type methods and to multivariate setting.