Mathematical foundations of spectral methods for time-dependent PDEs

TL;DR

The paper reframes spectral methods for time‑dependent PDEs as a choice of an orthonormal basis $\Phi$ in a Hilbert space $\mathcal{H}$, so the solution is $u(x,t)=\sum_{n=0}^\infty \hat{u}_n(t)\varphi_n(x)$ and the Galerkin projection yields a truncated ODE system for the coefficients. It analyzes two principal basis families—T‑systems (Cauchy/real line) and W‑systems (Dirichlet/finite interval)—and their differentiation‑matrix structures, fast expansion maps, and linear‑algebra strategies, linking stability, convergence, and computational efficiency to basis choice. It shows that T‑systems offer tridiagonal differentiation matrices enabling $O(N)$ operations and fast $f(\mathcal{D}_N)$ evaluations, while W‑systems deliver dense but semiseparable matrices that still permit efficient products and solves, with exponential convergence for analytic data in favorable settings. The results provide a principled, geometry‑aware framework for designing stable, fast, and structure‑preserving spectral methods for time‑dependent PDEs across boundary conditions, clarifying when Fourier bases are optimal and how to tailor bases for nonperiodic problems.

Abstract

The contention of this paper is that a spectral method for time-dependent PDEs is basically no more than a choice of an orthonormal basis of the underlying Hilbert space. This choice is governed by a long list of considerations: stability, speed of convergence, geometric numerical integration, fast approximation and efficient linear algebra. We subject different choices of orthonormal bases, focussing on the real line, to these considerations. While nothing is likely to improve upon a Fourier basis in the presence of periodic boundary conditions, the situation is considerably more interesting in other settings. We introduce two kinds of orthonormal bases, T-systems and W-systems, and investigate in detail their features. T-systems are designed to work with Cauchy boundary conditions, while W-systems are suited to zero Dirichlet boundary conditions.

Figures (2)

• Figure 4.1: Pointwise errors in $[-1,1]$ for ultraspherical W-systems and the functions $f(x)=\sin\pi x$ (on the left) and $f(x)=\cos^2(\pi x/2)$. In each case $=\alpha=1,2,3,4$ corresponds to green, blue, yellow and red lines.
• Figure 4.2: Pointwise errors in $[-0,\infty)$ for Laguerre W-systems and the functions $f(x)=x{\mathrm e}^{-x}/(1+x)$ (on the left) and $f(x)=x{\mathrm e}^{-2x}\sin x$. In each case $=\alpha=1,2,3,4$ corresponds to green, blue, yellow and red lines.