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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.

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