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

Cano, B.; Durán, A.

doi:10.1090/S0025-5718-03-01546-1

A technique to construct symmetric variable-stepsize linear multistep methods for second-order systems
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by B. Cano and A. Durán PDF

Math. Comp. 72 (2003), 1803-1816 Request permission

Abstract:

Some previous works show that symmetric fixed- and variable-stepsize linear multistep methods for second-order systems which do not have any parasitic root in their first characteristic polynomial give rise to a slow error growth with time when integrating reversible systems. In this paper, we give a technique to construct variable-stepsize symmetric methods from their fixed-stepsize counterparts, in such a way that the former have the same order as the latter. The order and symmetry of the integrators obtained is proved independently of the order of the underlying fixed-stepsize integrators. As this technique looks for efficiency, we concentrate on explicit linear multistep methods, which just make one function evaluation per step, and we offer some numerical comparisons with other one-step adaptive methods which also show a good long-term behaviour.

References

V. I. Arnol′d, Mathematical methods of classical mechanics, Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, [1989?]. Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein; Corrected reprint of the second (1989) edition. MR 1345386
M. P. Calvo and J. M. Sanz-Serna, The development of variable-step symplectic integrators, with application to the two-body problem, SIAM J. Sci. Comput. 14 (1993), no. 4, 936–952. MR 1223281, DOI 10.1137/0914057
M. P. Calvo, M. A. López-Marcos, and J. M. Sanz-Serna, Variable step implementation of geometric integrators, Appl. Numer. Math. 28 (1998), no. 1, 1–16. MR 1639818, DOI 10.1016/S0168-9274(98)00035-X
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.
Cano, B., Integración numérica de órbitas periódicas con métodos multipaso, PhD Thesis, Universidad de Valladolid, 1996.
Cano, B. and Durán, 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).
B. Cano and J. M. Sanz-Serna, Error growth in the numerical integration of periodic orbits, with application to Hamiltonian and reversible systems, SIAM J. Numer. Anal. 34 (1997), no. 4, 1391–1417. MR 1461789, DOI 10.1137/S0036142995281152
B. Cano and J. M. Sanz-Serna, Error growth in the numerical integration of periodic orbits by multistep methods, with application to reversible systems, IMA J. Numer. Anal. 18 (1998), no. 1, 57–75. MR 1492048, DOI 10.1093/imanum/18.1.57
Evans, N. W. and Tremaine, S., Linear multistep methods for integrating reversible differential equations, Astron. J 118 1888 (1999).
E. Hairer, Variable time step integration with symplectic methods, Appl. Numer. Math. 25 (1997), no. 2-3, 219–227. Special issue on time integration (Amsterdam, 1996). MR 1485817, DOI 10.1016/S0168-9274(97)00061-5
E. Hairer, S. P. Nørsett, and G. Wanner, Solving ordinary differential equations. I, 2nd ed., Springer Series in Computational Mathematics, vol. 8, Springer-Verlag, Berlin, 1993. Nonstiff problems. MR 1227985
Ernst Hairer and Daniel Stoffer, Reversible long-term integration with variable stepsizes, SIAM J. Sci. Comput. 18 (1997), no. 1, 257–269. Dedicated to C. William Gear on the occasion of his 60th birthday. MR 1433387, DOI 10.1137/S1064827595285494
Weizhang Huang and Benedict Leimkuhler, The adaptive Verlet method, SIAM J. Sci. Comput. 18 (1997), no. 1, 239–256. Dedicated to C. William Gear on the occasion of his 60th birthday. MR 1433386, DOI 10.1137/S1064827595284658
T. E. Hull, W. H. Enright, B. M. Fellen, and A. E. Sedgwick, Comparing numerical methods for ordinary differential equations, SIAM J. Numer. Anal. 9 (1972), 603–637; errata, ibid. 11 (1974), 681. MR 351086, DOI 10.1137/0709052
Hut, P., Makino, J. and McMillan, S., Building a better leapfrog, Astrophys. J., 443 (1995), pp. L93–L96.
Kahan, W., Unconventional numerical methods for trajectory calculations, Mathematics Dept., and Elect. Eng. & Computer Science Dept., University of California, 1993.
Benedict Leimkuhler, Reversible adaptive regularization: perturbed Kepler motion and classical atomic trajectories, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 357 (1999), no. 1754, 1101–1133. MR 1694704, DOI 10.1098/rsta.1999.0366
M. P. Laburta, Construction of starting algorithms for the RK-Gauss methods, J. Comput. Appl. Math. 90 (1998), no. 2, 239–261. MR 1624346, DOI 10.1016/S0377-0427(98)00003-X
Quinlan, G. D. and Tremaine, S., Symmetric multistep methods for the numerical integration of planetary orbits, Astron. J., 100 (1990), pp. 1694–1700.
Sebastian Reich, Backward error analysis for numerical integrators, SIAM J. Numer. Anal. 36 (1999), no. 5, 1549–1570. MR 1706731, DOI 10.1137/S0036142997329797
J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian problems, Applied Mathematics and Mathematical Computation, vol. 7, Chapman & Hall, London, 1994. MR 1270017