A Fourier Integrator for the Cubic Nonlinear Schrödinger Equation with Rough Initial Data

Marvin Knöller, Alexander Ostermann, Katharina Schratz

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Standard numerical integrators suffer from an order reduction when applied to nonlinear Schrödinger equations with low-regularity initial data. For example, standard Strang splitting requires the boundedness of the solution in $H^{r+4}$ in order to be second-order convergent in $H^r$, i.e., it requires the boundedness of four additional derivatives of the solution. We present a new type of integrator that is based on the variation-of-constants formula and makes use of certain resonance based approximations in Fourier space. The latter can be efficiently evaluated by fast Fourier methods. For second-order convergence, the new integrator requires two additional derivatives of the solution in one space dimension and three derivatives in higher space dimensions. Numerical examples illustrating our convergence results are included. These examples demonstrate the clear advantage of the Fourier integrator over standard Strang splitting for initial data with low regularity.
Original languageEnglish
Pages (from-to)1967-1986
Number of pages20
JournalSIAM Journal on Numerical Analysis
Volume57
Issue number4
Early online date13 Aug 2019
DOIs
Publication statusPublished - 2019

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