CMX Lunch Seminar

Global spectral methods are a classical technique for solving partial differential equations (PDEs) in simple geometries, including Fourier series in periodic domains and orthogonal polynomials in bounded domains. While traditional "collocation" approaches for polynomials are slow at scale, modern spectral methods reformulate these systems in sparse operator form, enabling fast and accurate solvers with near-FFT-like performance. However, such methods may suffer from poor conditioning, difficulties at coordinate singularities, and conservation issues that depend sensitively on their formulation.

In this context, we will present a "generalized tau" framework that unifies all polynomial and trigonometric spectral methods, from classical collocation to modern "ultraspherical" schemes. In particular, we examine the exact discrete equations solved by each method and characterize their deviation from the original PDE in terms of perturbations called "tau corrections." By analyzing these corrections, we can precisely categorize existing methods and design new solvers that robustly accommodate new boundary conditions, eliminate spurious numerical modes, and satisfy exact conservation laws.

We will demonstrate the capabilities of this system as implemented in Dedalus, an open-source Python framework for solving PDEs using sparse spectral methods. Dedalus provides a symbolic equation specification system that allows users to define their own PDEs and automatically constructs optimally sparse, parallelized, and differentiable solvers tailored to the chosen equations and geometry. We will present examples combining the generalized tau method with new spectral bases for curvilinear domains, providing fast and well-conditioned solvers for general tensor-valued PDEs in cylinders, disks, spheres, and balls.