TY - JOUR
T1 - Randomized exponential integrators for modulated nonlinear Schrödinger equations
AU - Hofmanová, Martina
AU - Knöller, Marvin
AU - Schratz, Katharina
N1 - Funding Information:
Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 258734477 – SFB 1173.
Publisher Copyright:
© 2020 Oxford University Press. All rights reserved.
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.
PY - 2020/10
Y1 - 2020/10
N2 - 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.
AB - 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.
KW - Error analysis
KW - Modulated Schrödinger
KW - Randomized exponential integrator
UR - http://www.scopus.com/inward/record.url?scp=85100997557&partnerID=8YFLogxK
U2 - 10.1093/IMANUM/DRZ050
DO - 10.1093/IMANUM/DRZ050
M3 - Article
AN - SCOPUS:85100997557
SN - 0272-4979
VL - 40
SP - 2143
EP - 2162
JO - IMA Journal of Numerical Analysis
JF - IMA Journal of Numerical Analysis
IS - 4
ER -