Baskin Engineering 1156 High Street, Santa Cruz, California 95064

Presenter: Keaton Burns, Instructor in Applied Mathematics at MIT

Description: Numerical simulations of partial differential equations (PDEs) are indispensable within science and engineering. For simple geometries, spectral methods are a powerful class of techniques that produce extremely accurate solutions for broad ranges of equations. But many flavors of these methods exist, each with different properties and performance, and developing the best method for a complex nonlinear problem is often quite challenging. 

Here we present a framework that unifies all polynomial and trigonometric spectral methods, from classical "collocation" to the more recent "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 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 models. We will also illustrate these advantages with examples from fluid dynamics using Dedalus Project, an open-source package for solving PDEs with modern spectral methods. 

Bio: Keaton is an Instructor in Applied Mathematics at MIT. He completed his Ph.D. in Physics at MIT and was a postdoc at the Simon Foundation’s Flatiron Institute. Keaton’s work focuses on the development of high-order numerical methods, their implementation in open-source software, and their application to problems in astrophysical, geophysical, and biological fluid dynamics. He is the lead developer of Dedalus, an open-source framework for solving PDEs using global spectral methods.

Hosted by: Professor Marcella Gomez

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