Composition methods for differential equations with processing

SIAM Journal on Scientific Computing

Volumn 27, Issue 6, 2006, Pages 1817-1843

  Blanes, S ^a   Casas, F ^b   Murua, A ^c  

^a UNIVERSITAT POLITÈCNICA DE VALÈNCIA   (Spain)

^b UNIVERSITAT JAUME I   (Spain)

^c UNIVERSITY OF THE BASQUE COUNTRY UPV EHU   (Spain)

Author keywords

Cheap postprocessor; Composition methods; Geometric integrators; Initial value problems; Processing technique

Indexed keywords

BOUNDARY VALUE PROBLEMS; INTEGRATION; PROBLEM SOLVING;

CHEAP POSTPROCESSOR; COMPOSITION METHODS; GEOMETRIC INTEGRATORS; PROCESSING TECHNIQUE;

DIFFERENTIAL EQUATIONS;

EID: 33751200668     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/030601223     Document Type: Article

References (26)

1. [1 S. BLANES, High order numerical integrators for differential equations using composition and processing of low order methods, Appl. Numer. Math., 37 (2001), pp. 289-306.
2. S. BLANKS, F. CASAS, AND A. MURUA, On the numerical integration of ordinary differential equations by processed methods, SIAM J. Numer, Anal., 42 (2004), pp. 531-552.
3. S. BLANKS, F. CASAS, AND J. ROS, Symplectic integration with processing: A general study, SIAM J. Sci. Comput., 21 (1999), pp. 711-727.
4. S. BLANKS, F. CASAS, AND J. Ros, High-order Runge-Kutta-Nyström geometric integrators with processing, Appl. Numer. Math., 39 (2001), pp. 245-259.
5. S. BLANKS AND P.C. MOAN, Practical symplectic partitioned Runge-Kutta and Runge-Kutta-Nyström methods, J. Comput. Appl. Math., 142 (2002), pp. 313-330.
6. J. BUTCHER, The effective order of Runge-Kutta methods, in Conference on the Numerical Solution of Differential Equations, Lecture Notes in Math. 109, Springer-Verlag, Berlin, 1969, pp. 133-139.
7. J. BUTCHER AND J. M. SANZ-SERNA, The number of conditions for a Runge-Kutta method to have effective order p, Appl. Numer. Math., 22 (1996), pp. 103-111.
8. E. HAIRER, C. LUBICH, AND G. WANNER, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Ser. Comput. Math. 31, Springer-Verlag, Berlin, 2002.
9. S.-J. KANG AND M.-H. KIM, Free Lie algebras, generalized Witt formula, and the denominator entity, J. Algebra, 183 (1996), pp. 560-594.
10. W. KAHAN AND R. C. Li, Composition constants for raising the order of unconventional schemes for ordinary differential equations, Math. Comp., 66 (1997), pp. 1089-1099.
11. P. V. KOSKLEFF, Formal Calculus for Lie Methods in Hamiltonian Mechanics, Ph.D. thesis, Lawrence Berkeley Laboratory LBID-2030, Tech. report UC-405, Berkeley, CA, 1994.
12. 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), pp. 223-238.
13. R. I. MCLACHLAN, On the numerical integration of ordinary differential equations by symmetric composition methods, SIAM J. Sci. Comput., 16 (1995), pp. 151-168.
14. R. I. MCLACHLAN, More on symplectic correctors, in Integration Algorithms and Classical Mechanics, Fields Inst. Commun. 10, J. E. Marsden, G. W. Patrick, and W. F. Shadwick, eds., AMS, Providence, RI, 1996, pp. 141-149.
15. R. I. MCLACHLAN, Families of high order composition methods, Numer. Algorithms, 31 (2002), pp. 233-246.
16. R. I. MCLACHLAN AND R. QUISPEL, Splitting methods, Acta Numer., 11 (2002), pp. 341-434.
17. H. MUNTHE-KAAS AND B. OWREN, Computations in a free Lie algebra, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), pp. 957-981.
18. A. MURUA AND J. M. SANZ-SERNA, Order conditions for numerical integrators obtained by composing simpler integrators, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), pp. 1079-1100.
19. P. J. OLVER, Applications of Lie Groups to Differential Equations, 2nd ed., Springer-Verlag, New York, 1993.
20. I. P. OMELYAN, I. M. MRYGLOD, AND R. FOLK, Construction of high-order force-gradient algorithms for integration of motion in classical and quantum systems, Phys. Rev. E, 66 (2002), 026701.
21. J. M. SANZ-SERNA AND M. P. CALVO, Numerical Hamiltonian Problems, Chapman and Hall, London, 1994.
22. M. SUZUKI, General theory of fractal path integrals with applications to many-body and statistical physics, J. Math. Phys., 32 (1991), pp. 400-407.
23. M. SUZUKI, Quantum Monte Carlo methods and general decomposition theory of exponential operators and symplectic integrators, Phys. A, 205 (1994), pp. 65-79.
24. V. S. VARADARAJAN, Lie Algebras and Their Representations, Springer-Verlag, New York, 1984.
25. J. WISDOM, M. HOLMAN, AND J. TOUMA, Symplectic correctors, in Integration Algorithms and Classical Mechanics, Fields lust. Commun. 10, J. E. Marsden, G. W. Patrick, and W. F. Shadwick, eds., AMS, Providence, RI, 1996, pp. 217-244.
26. H. YOSHIDA, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), pp. 262-268.