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CATEGORIES:Lectures & Presentations
DESCRIPTION:Presenter: Keaton Burns\, Instructor in Applied Mathematics at
MIT\n\nDescription: Numerical simulations of partial differential equations
(PDEs) are indispensable within science and engineering. For simple geomet
ries\, spectral methods are a powerful class of techniques that produce ext
remely accurate solutions for broad ranges of equations. But many flavors o
f these methods exist\, each with different properties and performance\, an
d developing the best method for a complex nonlinear problem is often quite
challenging. \n\nHere we present a framework that unifies all polynomial a
nd trigonometric spectral methods\, from classical "collocation" to the mor
e recent "ultraspherical" schemes. In particular\, we examine the exact dis
crete equations solved by each method and characterize their deviation from
the original PDE in terms of perturbations called "tau corrections". By an
alyzing these corrections\, we can precisely categorize existing methods an
d design new solvers that robustly implement new boundary conditions\, elim
inate spurious numerical modes\, and satisfy exact conservation laws. \n\nT
his approach conceptually separates what discrete model a spectral scheme s
olves from how it solves it\, allowing for much more freedom when building
and optimizing numerical models. We will also illustrate these advantages w
ith examples from fluid dynamics using Dedalus Project\, an open-source pac
kage for solving PDEs with modern spectral methods. \n\nBio: Keaton is an I
nstructor 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. Keat
on’s work focuses on the development of high-order numerical methods\, thei
r implementation in open-source software\, and their application to problem
s in astrophysical\, geophysical\, and biological fluid dynamics. He is the
lead developer of Dedalus\, an open-source framework for solving PDEs usin
g global spectral methods.\n\nHosted by: Professor Marcella Gomez\n\nJoin u
s in person or on Zoom: https://ucsc.zoom.us/j/97598034090?pwd=c2N6K1hPRGp3
YTkyNVhjMVVVc0grZz09
DTEND:20230509T000000Z
DTSTAMP:20240810T104138Z
DTSTART:20230508T230000Z
GEO:37.000369;-122.063237
LOCATION:Jack Baskin Engineering\, BE 165
SEQUENCE:0
SUMMARY:AM Seminar: Solving partial differential equations exactly over pol
ynomials
UID:tag:localist.com\,2008:EventInstance_43076928025207
URL:https://calendar.ucsc.edu/event/solving_partial_differential_equations_
exactly_over_polynomials
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