High-order exponential operator splitting methods for time-dependent schrödinger equations

SIAM Journal on Numerical Analysis

Volumn 46, Issue 4, 2008, Pages 2022-2038

  Thalhammer, Mechthild ^a  

^a UNIVERSITY OF INNSBRUCK   (Austria)

Author keywords

Convergence; Exponential operator splitting; High order methods; Schr dinger equations; Stability

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; LINEAR SYSTEMS; MATHEMATICAL OPERATORS;

CONVERGENCE; DINGER EQUATIONS; ERROR BOUNDS; EXPONENTIAL OPERATOR SPLITTING; EXPONENTIAL OPERATORS; HIGH-ORDER METHODS; INITIAL VALUES; LINEAR DIFFERENTIAL EQUATIONS; SPLITTING METHODS; UNBOUNDED OPERATORS;

LINEAR EQUATIONS;

EID: 55349108570     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/060674636     Document Type: Article

References (21)

1. W. BAO, D. JAKSCH, AND P. MARKOWICH, Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comput. Phys., 187 (2003), pp. 318-342.
2. C. BESSE, B. BIDÉGARAY, AND S. DESCOMBES, Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 40 (2002), pp. 26-40.
3. S. BLANES AND P.C. MOAN, Practical symplectic partitioned Runge-Kutta and Runge-Kutta-Nyström methods, J. Comput. Appl. Math., 142 (2002), pp. 313-330.
4. S. BLANES, F. CASAS, AND A. MURUA, On the Linear Stability of Splitting Methods, Numerical Analysis Reports 8/2006, University of Cambridge, Cambridge, UK, 2006.
5. E. CELLEDONI AND N. SAEFSTROEM, A symmetric splitting method for rigid body dynamics, Model. Identif. Control, 27 (2006), pp. 95-108.
6. P. CHARTIER AND A. MURUA, Preserving First Integrals and Volume Forms of Additively Split Systems, INRIA Report RR-6016, Rennes, France, 2006.
7. K.J. ENGEL AND R. NAGEL, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000.
8. E. HAIRER, CH. LUBICH, AND G. WANNER, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer, Berlin, 2002.
9. M. HOCHBRUCK AND CH. LUBICH, On Magnus integrators for time-dependent Schrödinger equations, SIAM J. Numer. Anal., 41 (2003), pp. 945-963.
10. E. HILLE AND R.S. PHILLIPS, Functional Analysis and Semi-Groups, AMS, Providence, RI, 1957.
11. T. JAHNKE AND CH. LUBICH, Error bounds for exponential operator splittings, BIT, 40 (2000), pp. 735-744.
12. R. KOZLOV, A. KVÆRNØ, AND B. OWREN, The Behaviour of the Local Error in Splitting Methods Applied to Stiff Problems, Preprint Numerics 1/2003, University of Trondheim, Trondheim, Norway, 2003; also available online from http://www.math.ntnu.no/ ~bryn/trliste.imf (6th report).
13. R. KOZLOV AND B. OWREN, Order Reduction in Operator Splitting Methods, Preprint Numerics 6/1999, University of Trondheim, Trondheim, Norway, 1999.
14. CH. LUBICH, On Splitting Methods for Schrödinger-Poisson and Cubic Nonlinear Schrödinger Equations, preprint, Universität Tübingen, Tübingen, Germany, 2007.
15. R. MCLACHLAN AND R. QUISPEL, Geometric integration of DDEs, J. Phys., 39 (2006), pp. 5251-5286.
16. R. MCLACHLAN AND R. QUISPEL, Splitting methods, Acta Numer., 11 (2002), pp. 341-434.
17. A. PAZY, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
18. Q. SHENG, Global error estimates for exponential splitting, IMA J. Numer. Anal., 14 (1994), pp. 27-56.
19. B. SPORTISSE AND J.G. VERWER., A Note on Operator Splitting in the Stiff Linear Case, Technical report MAS-R9830, CWI Amsterdam, Amsterdam, The Netherlands, 1998.
20. G. STRANG, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), pp. 506-517.
21. H.F. TROTTER, On the product of semi-groups of operators, Proc. Amer. Math. Soc., 10 (1959), pp. 545-551.