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Exponential splitting for unbounded operators

Authors: Eskil Hansen and Alexander Ostermann
Journal: Math. Comp. 78 (2009), 1485-1496
MSC (2000): Primary 65M15, 65J10, 65L05, 35Q40
Published electronically: January 22, 2009
MathSciNet review: 2501059
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Abstract: We present a convergence analysis for exponential splitting methods applied to linear evolution equations. Our main result states that the classical order of the splitting method is retained in a setting of unbounded operators, without requiring any additional order condition. This is achieved by basing the analysis on the abstract framework of (semi)groups. The convergence analysis also includes generalizations to splittings consisting of more than two operators, and to variable time steps. We conclude by illustrating that the abstract results are applicable in the context of the Schrödinger equation with an external magnetic field or with an unbounded potential.

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  • 1. S. Blanes and F. Casas, On the necessity of negative coefficients for operator splitting schemes of order higher than two, Appl. Numer. Math. 54 (2005), 23-37. MR 2134093 (2006b:65085)
  • 2. M. Crandall and A. Majda, The method of fractional steps for conservation laws, Numer. Math. 34 (1980), 285-314. MR 571291 (81j:65101)
  • 3. R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 1/3/5, Springer, Berlin, 1990/90/92. MR 1036731 (90k:00004)
  • 4. S. Descombes and M. Schatzman, Strang's formula for holomorphic semi-groups, J. Math. Pures Appl. 81 (2002), 93-114. MR 1994884 (2005g:35008)
  • 5. B. O. Dia and M. Schatzman, Commutateurs de certains semi-groupes holomorphes et applications aux directions alternées, M2AN Math. Model. Numer. Anal. 30 (1996), 343-383. MR 1391710 (97e:47055)
  • 6. H. O. Fattorini, The Cauchy Problem, Addison-Wesley, Reading, 1983. MR 692768 (84g:34003)
  • 7. E. Hairer, Ch. Lubich, and G. Wanner, Geometric Numerical Integration. Structure-preserving Algorithms for Ordinary Differential Equations, second ed., Springer, Berlin, 2006. MR 2221614 (2006m:65006)
  • 8. E. Hansen and A. Ostermann, Dimension splitting for evolution equations, Numer. Math. 108 (2008), 557-570. MR 2369204
  • 9. W. Hundsdorfer and J. Verwer, Stability and convergence of the Peaceman-Rachford ADI method for initial-boundary value problems, Math. Comp. 53 (1989), 81-101. MR 969489 (90h:65195)
  • 10. T. Jahnke and Ch. Lubich, Error bounds for exponential operator splittings, BIT 40 (2000), 735-744. MR 1799313 (2001k:65143)
  • 11. K. H. Karlsen and N. H. Risebro, An operator splitting method for nonlinear convection-diffusion equations, Numer. Math. 77 (1997), 365-382. MR 1469677 (98i:65073)
  • 12. Ch. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comp. 77 (2008), 2141-2153. MR 2429878
  • 13. R. I. McLachlan and G. R. Quispel, Splitting methods, Acta Numer. 11 (2002), 341-434. MR 2009376 (2004f:34001)
  • 14. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. MR 710486 (85g:47061)
  • 15. M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, New York, 1972. MR 0493419 (58:12429a)
  • 16. Z. H. Teng, On the accuracy of fractional step methods for conservation laws in two dimensions, SIAM J. Numer. Anal. 31 (1994), 43-63. MR 1259965 (95c:65141)
  • 17. S. Teufel, Adiabatic Perturbation Theory in Quantum Dynamics, Lecture Notes in Mathematics, vol. 1821, Springer, Berlin, 2003. MR 2158392 (2006g:81061)
  • 18. M. Thalhammer, High-order exponential operator splitting methods for time-dependent Schrödinger equations, SIAM J. Numer. Anal. 46 (2008), 2022-2038. MR 2399406

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Additional Information

Eskil Hansen
Affiliation: Institut für Mathematik, Universität Innsbruck, Technikerstraße 13, A-6020 Innsbruck, Austria

Alexander Ostermann
Affiliation: Institut für Mathematik, Universität Innsbruck, Technikerstraße 13, A-6020 Innsbruck, Austria

Keywords: Exponential splitting, splitting schemes, convergence, nonstiff order, unbounded operators, Schr\"{o}dinger equation
Received by editor(s): February 29, 2008
Received by editor(s) in revised form: August 19, 2008
Published electronically: January 22, 2009
Additional Notes: This work was supported by the Austrian Science Fund under grant M961-N13.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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