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

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Christian Lubich
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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.## References

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## Additional Information

**Christian Lubich**- Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
- MR Author ID: 116445
- Email: lubich@na.uni-tuebingen.de
- 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.
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**77**(2008), 2141-2153 - MSC (2000): Primary 65M15
- DOI: https://doi.org/10.1090/S0025-5718-08-02101-7
- MathSciNet review: 2429878