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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";
Journal of Computational and Applied Mathematics, 255 (2013), ISBN: 978-3-902627-05-6; 384 - 403.

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://dx.doi.org/10.1016/j.cam.2013.04.043

http://www.sciencedirect.com/science/article/pii/S0377042713002495

Project Head Othmar Koch:
Adaptives Splitting für nichtlineare Schrödingergleichungen