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SUMMARY:Low-regularity Fourier integrators for the nonlinear Schrödinger 
 equation - Katharina Schratz (KIT)
DTSTART:20181129T150000Z
DTEND:20181129T160000Z
UID:TALK114022@talks.cam.ac.uk
CONTACT:6743
DESCRIPTION:A large toolbox of numerical schemes for the nonlinear Schröd
 inger equation has been established\, based on different discretization te
 chniques such as discretizing the variation-of-constants formula (e.g.\, e
 xponential integrators) or splitting the full equation into a series of si
 mpler subproblems (e.g.\, splitting methods).  In many situations these cl
 assical schemes  allow a precise and efficient approximation. This\, howev
 er\, drastically changes whenever "non-smooth'' phenomena enter the scene 
 such as for problems at low-regularity and high oscillations. Classical sc
 hemes fail to capture the oscillatory parts within the solution which lead
 s to severe instabilities and loss of convergence. In this talk I present 
 a new class of Fourier integrators for the nonlinear Schrödinger equation
  at low-regularity. The key idea in the construction of the new schemes is
  to tackle and hardwire the underlying structure of resonances into the nu
 merical discretization.​ These terms are the cornerstones of theoretical
  analysis of the long time behaviour of differential equations and their n
 umerical discretizations (cf. modulated Fourier Expansion\; Hairer\, Lubic
 h & Wanner) and offer the new schemes strong geometric structure at low re
 gularity.​
LOCATION:MR14\, Centre for Mathematical Sciences
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