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Randomized exponential integrators for modulated nonlinear Schrödinger equations

Hofmanová, Martina; Knöller, Marvin; Schratz, Katharina

Abstract:

We consider the nonlinear Schrödinger equation with dispersion modulated by a (formal) derivative of a $\alpha$-Hölder continuous time-dependent function. Due to the highly oscillatory nature of the problem classical numerical methods face severe order reduction in non-smooth regimes $\alpha < 1$. In this work, we develop a new randomized exponential integrator based on a stratified Monte Carlo approximation which allows us to average the high oscillations in the problem and obtain improved error bounds of order $\alpha + 1/2$. In addition, the new approach allows us to treat a far more general class of modulations than the available literature. Numerical results underline our theoretical findings and show the favorable error behavior of our new scheme compared to classical methods.


Volltext §
DOI: 10.5445/IR/1000088600
Veröffentlicht am 14.12.2018
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht/Preprint
Publikationsjahr 2018
Sprache Englisch
Identifikator ISSN: 2365-662X
urn:nbn:de:swb:90-886009
KITopen-ID: 1000088600
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 17 S.
Serie CRC 1173 ; 2018/49
Projektinformation SFB 1173/1 (DFG, DFG KOORD, SFB 1173/1 2015)
Schlagwörter modulated Schrödinger equation, order reduction, highly oscillatory problems, exponential-type integrator
Nachgewiesen in arXiv
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