In this work, we consider the numerical solution of the nonlinear Schrödinger equation with a highly oscillatory potential (NLSE-OP). The NLSE-OP is a model problem which frequently occurs in recent studies of some multiscale dynamical systems, where the potential introduces wide temporal oscillations to the solution and causes numerical difficulties. We aim to analyze rigorously the error bounds of the splitting schemes for solving the NLSE-OP to a fixed time. Our theoretical results show that the Lie–Trotter splitting scheme is uniformly and optimally accurate at the first order provided that the oscillatory potential is integrated exactly, while the Strang splitting scheme is not. Our results apply to general dispersive or wave equations with an oscillatory potential. The error estimates are confirmed by numerical results.
Mots-clés : Nonlinear Schrödinger equation, highly oscillatory potential, Lie–Trotter splitting, Strang splitting, error estimates, uniformly accurate
@article{M2AN_2020__54_5_1491_0,
author = {Su, Chunmei and Zhao, Xiaofei},
title = {On time-splitting methods for nonlinear {Schr\"odinger} equation with highly oscillatory potential},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1491--1508},
publisher = {EDP-Sciences},
volume = {54},
number = {5},
year = {2020},
doi = {10.1051/m2an/2020006},
mrnumber = {4116683},
zbl = {1451.65163},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2020006/}
}
TY - JOUR AU - Su, Chunmei AU - Zhao, Xiaofei TI - On time-splitting methods for nonlinear Schrödinger equation with highly oscillatory potential JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2020 SP - 1491 EP - 1508 VL - 54 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2020006/ DO - 10.1051/m2an/2020006 LA - en ID - M2AN_2020__54_5_1491_0 ER -
%0 Journal Article %A Su, Chunmei %A Zhao, Xiaofei %T On time-splitting methods for nonlinear Schrödinger equation with highly oscillatory potential %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2020 %P 1491-1508 %V 54 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2020006/ %R 10.1051/m2an/2020006 %G en %F M2AN_2020__54_5_1491_0
Su, Chunmei; Zhao, Xiaofei. On time-splitting methods for nonlinear Schrödinger equation with highly oscillatory potential. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 5, pp. 1491-1508. doi : 10.1051/m2an/2020006. http://www.numdam.org/articles/10.1051/m2an/2020006/
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