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

Lubich, Christian

doi:10.1090/S0025-5718-08-02101-7

On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations
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by Christian Lubich;

Math. Comp. 77 (2008), 2141-2153

DOI: https://doi.org/10.1090/S0025-5718-08-02101-7

Published electronically: February 19, 2008

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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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