Abstract
We propose a multistep method for solving special second-order ordinary differential equations with damped oscillatory solutions. The proposed methods integrate exactly (with only round-off error) ordinary polynomials and the product of trigonometric functions at a frequency ω by exponentials of a parameter g. When ω=g=0 they reduce to the classical Nyströn and Cowell methods. Although there exist several methods with these properties, the proposed method allows independent computation of predictor and corrector which motivates parallel implementation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. R. Cash, A. D. Raptis, and T. E. Simos. A sixth-order exponentially fitted method for the numerical solution of the radial Schrödinger Equation. J. Comp. Physics, 91:413-423, 1990.
J. P. Coleman and L. Gr. Ixaru. P-stability and exponential-fitting methods for y″ = f( x, y). IMA J. Num. Anal., 16:179-199, 1996.
G. Denk. A new numerical method for the integration of highly oscillatory second-order ordinary differential equations. App. Num. Math., 13:57-67, 1993.
J. J. Dongarra, S. W. Otto, M. Snir, and D. Walker. A message passing standardfor MPP and workstations. Commun. ACM, 39(3):84-90, 1996.
J. M. Ferrándiz and J. Vigo. A new numerical methodimproving the integration of time in KS regularization. J. Guidance, Control Dynamics, 19:742-745, 1996.
Henrici P. Discrete Variable Methods in Ordinary Differential Equations. Wiley, New York, 1962.
D. L. Richardson and J. Vigo. Adapted Chebyshev methods for the numerical integration of perturbed oscillators. In Proceedings AAS/AIAA Spaceflight Mechanics meeting, Albuquerque, New Mexico. Univelt Incorporate, San Diego. 1995.
T. E. Simos. A four-step method for the numerical solution of the Schrödinger equation. J. Comp. and Appl. Math., 30:251-255, 1990.
E. L. Stiefel and G. Scheifele. Linear and Regular Celestial Mechanics. Springer, Berlin, 1971.
J. Vigo-Aguiar and J. M. Ferráandiz. A general procedure for the adaptation of multistep algorithm to the integration of oscillatory problems. SIAM J. Numer. Anal., 35(4):1684-1708, August 1998.
J. Vigo-Aguiar and J. M. Ferráandiz. Higher-order variable-step algorithms adapted to the accurate numerical integration of perturbed problems. Computers in Physics, 12(5):467-470, October 1998.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Vigo-Aguiar, J., Quintales, L.M. A Parallel ODE Solver Adapted to Oscillatory Problems. The Journal of Supercomputing 19, 163–171 (2001). https://doi.org/10.1023/A:1011175722328
Issue Date:
DOI: https://doi.org/10.1023/A:1011175722328