Defect-based local error estimators for splitting methods, with application to Schrödinger equations, Part I: The linear case

• Published in: Journal of computational and applied mathematics Vol. 236; no. 10; pp. 2643 - 2659
• Main Authors: Auzinger, Winfried, Koch, Othmar, Thalhammer, Mechthild
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-04-2012
• Subjects: A posteriori local error estimates; A priori local error estimates; Defect correction; Exponential operator splitting methods; Linear evolution equations; Time-dependent linear Schrödinger equations; 65L05; 65M15; Exponential operator splitting methods; A priori local error estimates; Linear evolution equations; Time-dependent linear Schrödinger equations; Defect correction; A posteriori local error estimates; 65J10; 65M12
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2012.01.001

We introduce a defect correction principle for exponential operator splitting methods applied to time-dependent linear Schrödinger equations and construct a posteriori local error estimators for the Lie–Trotter and Strang splitting methods. Under natural commutator bounds on the involved operators we prove asymptotical correctness of the local error estimators, and along the way recover the known a priori convergence bounds. Numerical examples illustrate the theoretical local and global error estimates.