Randomized exponential integrators for modulated nonlinear Schrödinger equations

Martina Hofmanová, Marvin Knöller, Katharina Schratz

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the nonlinear Schrödinger equation with dispersion modulated by a (formal) derivative of a time-dependent function with fractional Sobolev regularity of class Wα,2 for some α ∈ (0, 1). Due to the loss of smoothness in the problem, classical numerical methods face severe order reduction. In this work, we develop and analyze a new randomized exponential integrator based on a stratified Monte Carlo approximation. The new discretization technique averages the high oscillations in the solution allowing for improved convergence rates of order α + 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.

Original languageEnglish
Pages (from-to)2143-2162
Number of pages20
JournalIMA Journal of Numerical Analysis
Volume40
Issue number4
Early online date14 Jan 2020
DOIs
Publication statusPublished - Oct 2020

Keywords

  • Error analysis
  • Modulated Schrödinger
  • Randomized exponential integrator

ASJC Scopus subject areas

  • Mathematics(all)
  • Computational Mathematics
  • Applied Mathematics

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