Volume 54, Number 5, September-October 2020
|Page(s)||1491 - 1508|
|Published online||26 June 2020|
On time-splitting methods for nonlinear Schrödinger equation with highly oscillatory potential
Zentrum Mathematik, Technische Universität München, 85748 Garching bei München, Germany
2 School of Mathematics and Statistics & Hubei Key Laboratory of Computational Science, Wuhan University, 430072 Wuhan, P.R. China
* Corresponding author: firstname.lastname@example.org
Accepted: 20 January 2020
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.
Mathematics Subject Classification: 65L05 / 65L20 / 65L70
Key words: Nonlinear Schrödinger equation / highly oscillatory potential / Lie–Trotter splitting / Strang splitting / error estimates / uniformly accurate
© EDP Sciences, SMAI 2020
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