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SUMMARY:Solving partial differential equations exactly over polynomials - 
 Keaton Burns (MIT)
DTSTART:20231109T150000Z
DTEND:20231109T160000Z
UID:TALK207691@talks.cam.ac.uk
CONTACT:Nicolas Boulle
DESCRIPTION:Numerical simulations of partial differential equations (PDEs)
  are indispensable across science and engineering. For simple geometries\,
  spectral methods are a powerful class of techniques that produce exceptio
 nally accurate solutions for wide ranges of equations. But many variations
  of these methods exist\, each with distinct properties and performance\, 
 and developing the best method for a complex nonlinear problem is often qu
 ite challenging.\n\nIn this context\, we present a framework that unifies 
 all polynomial and trigonometric spectral methods\, from classical "colloc
 ation" to the more recent "ultraspherical" schemes. In particular\, we exa
 mine the exact discrete equations solved by each method and characterize t
 heir deviation from the original PDE in terms of perturbations called "tau
  corrections". By analyzing these corrections\, we can precisely categoriz
 e existing methods and design new solvers that robustly accommodate new bo
 undary conditions\, eliminate spurious numerical modes\, and satisfy exact
  conservation laws.\n\nThis approach conceptually separates *what* discret
 e model a spectral scheme solves from *how* it solves it. This separation 
 provides much more freedom when building and optimizing new numerical mode
 ls. We will illustrate these advantages with some examples from fluid dyna
 mics using Dedalus\, an open-source package for solving PDEs with modern s
 pectral methods.\n
LOCATION:Centre for Mathematical Sciences\, MR14
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