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CATEGORIES:Applied and Computational Analysis
SUMMARY:Low-regularity Fourier integrators for the nonline
ar Schrödinger equation - Katharina Schratz (KIT)
DTSTART;TZID=Europe/London:20181129T150000
DTEND;TZID=Europe/London:20181129T160000
UID:TALK114022AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/114022
DESCRIPTION:A large toolbox of numerical schemes for the nonli
near Schrödinger equation has been established\, b
ased on different discretization techniques such a
s discretizing the variation-of-constants formula
(e.g.\, exponential integrators) or splitting the
full equation into a series of simpler subproblems
(e.g.\, splitting methods). In many situations t
hese classical schemes allow a precise and effici
ent approximation. This\, however\, drastically ch
anges whenever "non-smooth'' phenomena enter the s
cene such as for problems at low-regularity and hi
gh oscillations. Classical schemes fail to capture
the oscillatory parts within the solution which l
eads to severe instabilities and loss of convergen
ce. In this talk I present a new class of Fourier
integrators for the nonlinear Schrödinger equation
at low-regularity. The key idea in the constructi
on of the new schemes is to tackle and hardwire th
e underlying structure of resonances into the nume
rical discretization. These terms are the corners
tones of theoretical analysis of the long time beh
aviour of differential equations and their numeric
al discretizations (cf. modulated Fourier Expansio
n\; Hairer\, Lubich & Wanner) and offer the new sc
hemes strong geometric structure at low regularity
.
LOCATION:MR14\, Centre for Mathematical Sciences
CONTACT:
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