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DTSTAMP:20260606T101658Z
SUMMARY:Imperial-UCL Numerics Seminar – NA Seminar: Keaton Burns (MIT)
DESCRIPTION:Solving PDEs exactly over polynomials\nAbstract: Numerical simu
 lations of partial differential equations (PDEs) are indispensable across 
 science and engineering. For simple geometries\, spectral methods are a po
 werful class of techniques that produce exceptionally accurate solutions f
 or wide ranges of equations. But many variations of these methods exist\, 
 each with distinct properties and performance\, and developing the best me
 thod for a complex nonlinear problem is often quite challenging.\nIn this 
 context\, we present a framework that unifies all polynomial and trigonome
 tric spectral methods\, from classical “collocation” to modern “ultr
 aspherical” schemes. In particular\, we examine the exact discrete equat
 ions solved by each method and characterize their deviation from the origi
 nal PDE in terms of perturbations called “tau corrections”. By analyzi
 ng these corrections\, we can precisely categorize existing methods and de
 sign new solvers that robustly accommodate new boundary conditions\, elimi
 nate spurious numerical modes\, and satisfy exact conservation laws.\nThis
  approach conceptually separates *what* discrete model a spectral scheme s
 olves from *how* it solves it. This separation provides much more freedom 
 when building and optimizing new numerical models. We will illustrate thes
 e advantages with various examples from fluid dynamics using Dedalus\, an 
 open-source package for solving PDEs with modern spectral methods.
URL:https://www.imperial.ac.uk/events/184152/imperial-ucl-numerics-seminar-
 na-seminar-keaton-burns-mit/
DTSTART;TZID=Europe/London:20241106T150000
DTEND;TZID=Europe/London:20241106T160000
LOCATION:145\, Huxley Building\, South Kensington Campus\, Imperial College
  London\, London\, SW7 2AZ\, United Kingdom
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