Title: Solving Partial Differential Equations Exactly Over Polynomials

Speaker: Keaton Burns, Instructor of Applied Mathematics, MIT

Abstract: Numerical simulations of partial differential equations (PDEs) have become an indispensable tool across science and engineering. For problems in simple geometries, spectral methods are a powerful class of techniques that produce extremely accurate solutions for broad ranges of equations. But many flavors of spectral methods exist with different properties and performance, and choosing the best method for a complex nonlinear problem is often challenging.

Here we will present a framework that unifies different polynomial spectral methods, from classical "collocation" to modern "ultraspherical" schemes, by examining the exact equations satisfied by the discrete solutions they produce. By understanding how these equations are perturbed from the original PDE (called the "tau correction"), we can precisely categorize existing methods and derive new solvers that robustly implement new boundary conditions, eliminate spurious numerical modes, and satisfy exact conservation laws.

This approach conceptually separates *what* discrete model a spectral scheme solves from *how* it solves it, allowing for much more freedom when building and optimizing numerical solvers. Along the way, we will illustrate these advantages with examples from the Dedalus Project, an open-source package for solving PDEs with modern spectral methods.

**Please note, this event will be held in-person, as well as online via Zoom at the following link: https://northwestern.zoom.us/j/92082212753

Special Note: This colloquium is on Tuesday at 11:15am, the new regular time for the applied math colloquium.

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