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Numerical Algorithms

, Volume 70, Issue 1, pp 61–91 | Cite as

Defect-based local error estimators for high-order splitting methods involving three linear operators

  • Winfried Auzinger
  • Othmar Koch
  • Mechthild Thalhammer
Original Paper

Abstract

Prior work on high-order exponential operator splitting methods is extended to evolution equations defined by three linear operators. A posteriori local error estimators are constructed via a suitable integral representation of the local error involving the defect associated with the splitting solution and quadrature approximation via Hermite interpolation. In order to prove asymptotical correctness, a multiple integral representation involving iterated defects is deduced by repeated application of the variation-of-constant formula. The error analysis within the framework of abstract evolution equations provides the basis for concrete applications. Numerical examples for initial-boundary value problems of Schrödinger and of parabolic type confirm the asymptotical correctness of the proposed a posteriori error estimators.

Keywords

Linear evolution equations Time integration methods High-order exponential operator splitting methods Local error A posteriori local error estimators 

Mathematics Subject Classification (2010)

65J10 65L05 65M12 65M15 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Winfried Auzinger
    • 1
  • Othmar Koch
    • 1
  • Mechthild Thalhammer
    • 2
  1. 1.Institute for Analysis and Scientific ComputingVienna University of TechnologyWienAustria
  2. 2.Institut für MathematikLeopold–Franzens Universität InnsbruckInnsbruckAustria

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