A technique to construct symmetric variable-stepsize linear multistep methods for second-order systems

Mathematics of Computation

Volumn 72, Issue 244, 2003, Pages 1803-1816

  Cano, B ^a   Duran A ^a  

^a UNIVERSITY OF VALLADOLID   (Spain)

Author keywords

Efficiency; Error growth; Explicit linear multistep methods; High order methods; Reversible second order systems; Symmetric integrators; Variable stepsizes

Indexed keywords

EID: 0141872533     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-03-01546-1     Document Type: Article

References (21)

1. Arnold, V. I., Mathematical Methods of Classical Mechanics, 2nd. Ed., Springer, New York, 1989. MR 96c:70001
2. Calvo, M. P. and Sanz-Serna, J. M., The development of variable-step symplectic integrators, with application to the two-body problem, SIAM J. Sci. Comput., 14 (1993), pp. 936-952. MR 94g:65068
3. Calvo, M. P., López-Marcos, M. A. and Sanz-Serna, J. M., Variable step implementation of geometric integrators. Appl. Num. Math., 28 (1998), pp. 1-16. MR 99d:65236
4. Calvo, M. P., High order initial interants for implicit Runge-Kutta methods: an improvement for variable-step symplectic integrators, IMA J. Num. Anal. 22 (2002), pp. 153-166.
5. Cano, B., Integración numérica de órbitas periódicas con métodos multipaso, PhD Thesis, Universidad de Valladolid, 1996.
6. Cano, B. and Duran, A., Analysis of variable-stepsize linear multistep methods with special emphasis on symmetric ones, Math. Comp., posted on May 29, 2003, PII S 0025- 5718(03)01538-2 (to appear in print).
7. Cano, B. and Sanz-Serna, J.M., Error growth in the numerical integration of periodic orbits, with application to Hamiltonian and reversible systems, SIAM J. Num. Anal., 34 (1997), pp. 1391-1417. MR 98i:65052
8. Cano, B. and Sanz-Serna, J.M., Error growth in the numerical integration of periodic orbits by multistep methods, with application to reversible systems, IMA J. Num. Anal., 18 (1998), pp. 57-75. MR 99d:65237
9. Evans, N. W. and Tremaine, S., Linear multistep methods for integrating reversible differential equations, Astron. J 118 1888 (1999).
10. Hairer, E., Variable time step integration with symplectic methods, Appl. Numer. Math., 25 (1997), pp. 219-227. MR 99a:65083
11. Hairer, E., Nörsett, S. P. and Wanner, W. G., Solving Ordinary Differential Equations I. Nonstiff Problems, 2nd ed., Berlin, Springer, 1993. MR 94c:65005
12. Hairer, E. and Stoffer, D., Reversible long-term integration with variable stepsizes, SIAM J. Sci. Comput., 18 (1997), pp. 257-269. MR 97m:65118
13. Huaug, W. and Leimkuhler, B. The Adaptive Verlet method, SIAM J. Sci. Comput. 18 (1997), pp. 239-256. MR 98g:65063
14. Hull, T. E., Enright, W. H. Fellen, B. M. and Sedgwick, A. E., Comparing numerical methods for ordinary differential equations, SIAM J. Numer. Anal, 9 (1972), pp. 603-637. MR 50:3577
15. Hut, P., Makino, J. and McMillan, S., Building a better leapfrog, Astrophys. J., 443 (1995), pp. L93-L96.
16. Kahan, W., Unconventional numerical methods for trajectory calculations, Mathematics Dept., and Elect. Eng. & Computer Science Dept., University of California, 1993.
17. Leimkuhler, B., Reversible adaptive regularization I: Perturbed Kepler motion and classical atomic trajectories. R. Soc. Lond. Philos, Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), pp. 1101-1133. MR 2000b:70002
18. Laburta, M. P., Construction of starting algorithms for the RK-Gauss methods, J. Comp. Appl. Math., 90 (1998), pp. 239-261. MR 99d:65215
19. Quinlan, G. D. and Tremaine, S., Symmetric multistep methods for the numerical integration of planetary orbits, Astron. J., 100 (1990), pp. 1694-1700.
20. Reich, S., Backward error analysis of numerical integrators, SIAM J. Numer. Anal. 36 (1999), pp. 1549-1570. MR 2000f:65060
21. Sanz-Serna, J. M and Calvo, M. P., Numerical Hamiltonian problems, Chapman & Hall, London, 1994. MR 95f:65006