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W. Auzinger, O. Koch, M. Thalhammer:
"Defect-based local error estimators for splitting methods, with application to Schrödinger equations. Part II. Higher-order methods for linear problems";
in: "ASC Report 50/2012", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2012, ISBN: 978-3-902627-05-6, 1 - 37.

In this work, defect-based local error estimators for higher-order exponential operator splitting methods are constructed and analyzed in the context of time-dependent linear Schrödinger equations. The technically involved procedure is carried out in detail for a general three-stage third-order splitting method and then extended to the higher-order case. Asymptotical correctness of the a posteriori local error estimator is proven under natural commutator bounds for the involved operators, and along the way the known (non)stiff order conditions and a priori convergence bounds are recovered. The theoretical error estimates for higher-order splitting methods are confirmed by numerical examples for a test problem of Schrödinger type. Further numerical experiments for a test problem of parabolic type complement the investigations.

Linear evolution equations, Time-dependent linear Schrödinger equations, Time integration, Higher-order exponential operator splitting methods, Defect correction, A priori local error estimates, A posteriori local error estimates

http://www.asc.tuwien.ac.at/preprint/2012/asc50x2012.pdf

Project Head Othmar Koch:
Adaptives Splitting für nichtlineare Schrödingergleichungen