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On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations


Author: Christian Lubich
Journal: Math. Comp. 77 (2008), 2141-2153
MSC (2000): Primary 65M15
DOI: https://doi.org/10.1090/S0025-5718-08-02101-7
Published electronically: February 19, 2008
MathSciNet review: 2429878
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Abstract: We give an error analysis of Strang-type splitting integrators for nonlinear Schrödinger equations. For Schrödinger-Poisson equations with an $ H^4$-regular solution, a first-order error bound in the $ H^1$ norm is shown and used to derive a second-order error bound in the $ L_2$ norm. For the cubic Schrödinger equation with an $ H^4$-regular solution, first-order convergence in the $ H^2$ norm is used to obtain second-order convergence in the $ L_2$ norm. Basic tools in the error analysis are Lie-commutator bounds for estimating the local error and $ H^m$-conditional stability for error propagation, where $ m=1$ for the Schrödinger-Poisson system and $ m=2$ for the cubic Schrödinger equation.


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Additional Information

Christian Lubich
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
Email: lubich@na.uni-tuebingen.de

DOI: https://doi.org/10.1090/S0025-5718-08-02101-7
Keywords: Split-step method, split-operator scheme, semilinear Schr\"odinger equations, error analysis, stability, regularity
Received by editor(s): January 9, 2007
Received by editor(s) in revised form: September 12, 2007
Published electronically: February 19, 2008
Additional Notes: This work was supported by DFG, SFB 382.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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