On the necessity of negative coefficients for operator splitting schemes of order higher than two

Applied Numerical Mathematics

Volumn 54, Issue 1, 2005, Pages 23-37

  Blanes, Sergio ^a   Casas, Fernando ^a  

^a UNIVERSITAT JAUME I   (Spain)

Author keywords

Composition methods; Effective order; Numerical integrators; Splitting methods

Indexed keywords

APPROXIMATION THEORY; COMPUTATIONAL COMPLEXITY; DIFFERENTIAL EQUATIONS; MATHEMATICAL MODELS; MATHEMATICAL OPERATORS; PROBLEM SOLVING; THEOREM PROVING;

COMPOSITION METHODS; EFFECTIVE ORDER; NUMERICAL INTEGRATORS; SPLITTING METHODS;

NUMERICAL ANALYSIS;

EID: 16644378169     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2004.10.005     Document Type: Article

References (23)

1. S. Blanes High order numerical integrators for differential equations using composition and processing of low order methods Appl. Numer. Math. 37 2001 289 306
2. S. Blanes, F. Casas, and J. Ros Symplectic integration with processing: A general study SIAM J. Sci. Comput. 21 1999 711 727
3. S. Blanes, F. Casas, and A. Murua On the numerical integration of ordinary differential equations by processed methods SIAM J. Numer. Anal. 42 2004 531 552
4. J. Butcher The effective order of Runge-Kutta methods J.L. Morris Conference on the Numerical Solution of Differential Equations Lecture Notes in Mathematics vol. 109 1969 Springer Berlin 133 139
5. S.A. Chin Quantum statistical calculations and symplectic corrector algorithms Phys. Rev. E 69 2004 046118
6. H. De Raedt Product formula algorithms for solving the time dependent Schrödinger equation Comput. Phys. Rep. 7 1987 1 72
7. A.J. Dragt, and D.T. Abell Symplectic maps and computation of orbits in particle accelerators J.E. Marsden G.W. Patrick W.F. Shadwick Integration Algorithms and Classical Mechanics 1996 American Mathematical Society Providence, RI 59 85
8. E. Forest, and R. Ruth Fourth-order symplectic integration Physica D 43 1990 105 117
9. D. Goldman, and T.J. Kaper Nth-order operator splitting schemes and nonreversible systems SIAM J. Numer. Anal. 33 1996 349 367
10. E. Hairer, C. Lubich, and G. Wanner Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations Springer Series in Computational Mathematics vol. 31 2002 Springer Berlin
11. P.-V. Koseleff, Formal Calculus for Lie Methods in Hamiltonian Mechanics, Ph.D. Thesis, Lawrence Berkeley Laboratory LBID-2030, Tech. Report UC-405, Berkeley, CA, 1994
12. B.J. Leimkuhler, S. Reich, and R.D. Skeel Integration methods for molecular dynamics J.P. Mesirov K. Schulten D.W. Sumners Mathematical Approaches to Biomolecular Structure and Dynamics IMA Volumes in Mathematics and Its Applications vol. 82 1996 Springer New York 161 185
13. M.A. López-Marcos, J.M. Sanz-Serna, and R.D. Skeel Explicit symplectic integrators using Hessian-vector products SIAM J. Sci. Comput. 18 1997 223 238
14. R.I. McLachlan Explicit Lie-Poisson integration and the Euler equations Phys. Rev. Lett. 71 1993 3043 3046
15. R.I. McLachlan On the numerical integration of ordinary differential equations by symmetric composition methods SIAM J. Sci. Comput. 16 1995 151 168
16. R.I. McLachlan, and P. Atela The accuracy of symplectic integrators Nonlinearity 5 1992 541 562
17. R.I. McLachlan, and R. Quispel Splitting methods Acta Numer. 11 2002 341 434
18. H. Munthe-Kaas, and B. Owren Computations in a free Lie algebra R. Soc. London Philos. Trans. Ser. A Math. Phys. Engrg. Sci. 357 1999 957 981
19. G. Rowlands A numerical algorithm for Hamiltonian systems J. Comput. Phys. 97 1991 235 239
20. Q. Sheng Solving linear partial differential equations by exponential splitting IMA J. Numer. Anal. 9 1989 199 212
21. G.J. Sussman, and J. Wisdom Chaotic evolution of the Solar system Science 257 1992 56 62
22. M. Suzuki General theory of fractal path integrals with applications to many-body theories and statistical physics J. Math. Phys. 32 1991 400 407
23. J. Wisdom, M. Holman, and J. Touma Symplectic correctors J.E. Marsden G.W. Patrick W.F. Shadwick Integration Algorithms and Classical Mechanics 1996 American Mathematical Society Providence, RI 217 244