skip to main content
research-article

Accurate Numerical Integration of Perturbed Oscillatory Systems in Two Frequencies

Published: 01 August 2009 Publication History

Abstract

Highly accurate long-term numerical integration of nearly oscillatory systems of ordinary differential equations (ODEs) is a common problem in astrodynamics. Scheifele’s algorithm is one of the excellent integrators developed in the past years to take advantage of special transformations of variables such as the K-S set. It is based on using expansions in series of the so-called G-functions, and generalizes the Taylor series integrators but with the remarkable property of integrating without truncation error oscillations in one basic known frequency. A generalization of Scheifele’s method capable of integrating exactly harmonic oscillations in two known frequencies is developed here, after introducing a two parametric family of analytical φ-functions. Moreover, the local error contains the perturbation parameter as a factor when the algorithm is applied to perturbed problems. The good behavior and the long-term accuracy of the new method are shown through several examples, including systems with low- and high-frequency constituents and a perturbed satellite orbit. The new methods provide significantly higher accuracy and efficiency than a selection of well-reputed general-purpose integrators and even recent symplectic or symmetric integrators, whose good behavior in the long-term integration of the Kepler problem and the other oscillatory systems is well stated in recent literature.

References

[1]
Bettis, D. G. 1970a. Numerical integration of products of Fourier and ordinary polynomials. Numer. Math. 14, 421--434
[2]
Bettis, D. G. 1970b. Stabilization of finite difference methods of numerical integration. Celest. Mech. 2, 282--295.
[3]
Brankin, R. W., Gladwell, I., Dormand, J. R., Prince, P. J., and Seward, W. L. 1989. Algorithm 670: A Runge-Kutta-Nyström code. ACM Trans. Math. Softw. 15, 1, 31--40.
[4]
Cash, J. R., Raptis, A. D., and Simos, T. E. 1990. A sixth-order exponentially fitted method for the numerical solution of the radial Schrödinger equation. J. Comput. Phys. 91, 413--423.
[5]
Coleman, J. P. and Ixaru, L. G. R. 1996. P-stability and exponential-fitting methods for y’’ = f(x, y). IMA J. Numer. Anal. 16, 179--199.
[6]
De Meyer, H., Vanden Berghe, G., and Vanthournout, J. 1991. Modified backward differentiation methods of the Adams-type based on exponential interpolation. Comput. Math. Appl. 2-3, 171--179.
[7]
Denk, G. 1993. A new numerical method for the integration of highly oscillatory second-order ordinary differential equations. Appl. Numer. Math. 13, 57--67.
[8]
Deuflhard, P. 1979. A study of extrapolation methods based on multistep schemes without parasitic solutions. Z. Angew. Math. Phys. 30, 177--189.
[9]
Dormand, J. R., El-Mikkawy, M. E. A., and Prince, P. J. 1987. High-order embedded Runge-Kutta-Nyström formulae. IMA J. Numer. Anal. 7, 4, 423--430.
[10]
Dormand, J. R., El-Mikkawy, M. E. A., and Prince, P. J. 1991. High-order embedded Runge-Kutta-Nyström formulae. IMA J. Numer. Anal. 11, 2, 297.
[11]
Fairén, V., Martín, P., and Ferrándiz, J. M. 1994. Numerical traking of small desviations from analytically known periodic orbits. Comput. Phys. 8, 4, 455--461.
[12]
Ferrándiz, J. M. 1988. A general canonical transformation increasing the number of variables with application to the two-body problem. Celest. Mech. 41, 343--357.
[13]
Ferrándiz, J. M. and Novo, S. 1991. Improved Bettis methods for long-term prediction. In Predictability, stability and chaos in N-body dynamical systems. Plenum Publishing Corporation, NATO ASI Series C. Plenum, New York, 515--521.
[14]
Ferrándiz, J. M., Vigo, J., and Martín, P. 1991. Reducing the error growth in the numerical propagation of satellite orbits. ESA SP-326. European Space Agency, Paris, France, 49--54.
[15]
Franco, J. M., Correas, J. M., and Petriz, F. 1991. Métodos adaptados de tipo Störmer-Cowell de orden elevado. Rev. Internac. Métod. Numér. Cálc. Diseñ. Ing. 7, 193--216.
[16]
Fukushima, T. 2003. Efficient orbit integration by scaling for Kepler energy consistency. Astronom. J. 126, 1097--1111.
[17]
Fukushima, T. 2004. Further simplification of the manifold correction method for orbit integration. Astronom. J. 128, 1446--1454.
[18]
García-Alonso, F. 2003. Algoritmos para la integración de problemas oscilatorios en varias frecuencias. Doctoral dissertation. University of Alicante, Alicante, Spain.
[19]
Hairer, E. and Hairer, M. 2002. GniCodes-Matlab programs for geometric numerical integration. Front. Numer. Anal. 199--240.
[20]
Hairer, E., Lubich, C., and Wanner, G. 2002. Geometric numerical integration structure-preserving algorithms for ordinary differential equations. Springer Series in Computional Mathematics 31. Springer, Berlin, Germany.
[21]
Ixaru, L. Gr., Vanden Berghe, G., and De Meyer, H. 2001. Exponentially fitted variable two-step BDF algorithm for first order ODEs. Comput. Phys. Commun. 140, 346--357.
[22]
Jain, M. K. 1984. Numerical Solution of Differential Equations. Wiley Eastern, New York, NY.
[23]
Lambert, J. D. 1991. Numerical Methods for Ordinary Differential Systems. John Wiley and Sons, New York, NY.
[24]
Martín, P. and Ferrándiz, J. M. 1995. Behavior of the SMF method for the numerical integration of satellite orbits. Celest. Mech. 63, 29--40.
[25]
Martín, P. and Ferrándiz, J. M. 1997. Multistep numerical methods based on Scheifele G-functions with application to satellite dynamics. SIAM J. Numer. Anal. 34, 359--375.
[26]
Natesan, S., Vigo-Aguiar, J., and Ramanujam, N. 2003. A numerical algorithm for singular perturbation problems exhibiting weak boundary layers. Comput. Math. Appl. 45, 469--479.
[27]
Neta, B. and Ford, C. H. 1984. Families of methods for the ordinary differential equations based on trigonometric polynomials. J. Comput. Appl. Math. 10, 33--38.
[28]
Norsett, S. P. 1969. An A-stable modification of the Adams-Bashforth methods. In Proceedings of the Conference on the Numerical Solution of Differential Equations. A. Dold and B. Eckmann, Eds. Lecture Notes in Mathematics, vol. 109. Springer-Verlag, Berlín, Germany. 214--219.
[29]
Richardson, L. and Panovskyj, A. 1993. A family of implicit Chebyshev methods for the numerical integration of first-order differential equations. In Proceedings of the AAS/AIAA Conference on Spaceflight Mechanics, (Victoria, BC, Canada).
[30]
Richardson, L. and Vigo, J. 1995. Adapted Chebyshev methods for the numerical integration of perturbed oscillators. In Proceedings of the AAS/AIAA Conference on Spaceflight Mechanics, (Albuquerque, NM).
[31]
Scuro, S. R. and Chin, S. A. 2005. Forward symplectic integrators and the long-time phase error in periodic motions. Phys. Rev. E 71, 056703, 1--12.
[32]
Simos, T. E. 1990. A four-step method for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 30, 251--255.
[33]
Simos, T. E. and Raptis, A. D. 1992. A four-order Bessel fitting method for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 43, 313--322.
[34]
Simos, T. E. and Vigo-Aguiar, J. 2001a. A symmetric high order method with minimal phase-lag for the numerical solution of the Schodinger equation. Int. J. Mod. Phys. C 12, 7, 1035--1042.
[35]
Simos, T. E. and Vigo-Aguiar, J. 2001b. On the construction of efficient methods for second order IVPS with oscillating solution. Int. J. Mod. Phys. C 12, 1453--1476.
[36]
Simos, T. E. and Vigo-Aguiar, J. 2003. Exponentially fitted symplectic integrator. Phys. Rev. E 67, 1, 016701 Part 2, 1--7.
[37]
Scheifele, G. 1971. On numerical integration of perturbed linear oscillating systems. Zeit. Angew. Math. Phys. 22, 186--210.
[38]
Stiefel, E. L. and Bettis, D. G. 1969. Stabilization of Cowell’s method. Numer. Math. 13, 154--175.
[39]
Stiefel, E. L. and Scheifele, G. 1971. Linear and Regular Celestial Mechanics. Springer-Verlag, Berlin, Germany.
[40]
Taff, L. 1985. Celestial Mechanics: A Computational Guide for the Practitioners. John Wiley and Sons, New York, NY.
[41]
Van de Vyver, H. 2007. An adapted explicit hybrid method of Numerov type for the numerical integration of perturbed oscillators. Appl. Math. Comput. 186, 2, 1385--1394.
[42]
Vanden Berghe, G. L., Ixaru, Gr., and Van Daele, M. 2001. Optimal implicit exponentially-fitted Runge-Kutta methods. Comput. Phys. Commun. 140, 346--357
[43]
Vigo-Aguiar J. and Andres-Pérez, F. 2001a. Backward differentiation formulae adapted to scalar linear equations. Appl. Math. Lett. 14, 639--643.
[44]
Vigo-Aguiar, J. and Ferrándiz, J. M. 1998a. A general procedure for the adaptation of multistep algorithms to the integration of oscillatory problems. SIAM J. Numer. Anal. 35, 1684--1708.
[45]
Vigo-Aguiar, J. and Ferrándiz, J. M. 1998b. VSVO adapted multistep methods for the numerical integration of second order differential equations. Appl. Math. Lett. 11, 83--89.
[46]
Vigo-Aguiar, J., Ferrándiz, J. M., and Simos, T. E. 2000. Encke methods adapted to regularizing variables. Int. J. Mod. Phys. A 15, 25, 3993--4010.
[47]
Vigo-Aguiar, J. and Simos, T. E. 2001a. An exponentially fitted and trigonometrically fitted method for the numerical solution of orbital problems. Astron. J. 122, 3, 1656--1660.
[48]
Vigo-Aguiar, J., Simos, T. E., and Ferrándiz, J. M. 2004. Controlling the error growth in long-term numerical integration of perturbed oscillations in one or several frequencies. Proc. R. Soc. Lond. A, 460, 561--567.
[49]
Vigo-Aguiar, J., Simos, T. E., and Tocino, A. 2001b. An adapted symplectic integrator for Hamiltonian problems. Int. J. Mod. Phys. C 12, 2, 225--234.
[50]
Yoshida, H. 1990. Construction of higher order symplectic integrators. Phys. Lett. A 150, 262--268.

Cited By

View all
  • (2019)Symmetric collocation ERKN methods for general second-order oscillatorsCalcolo10.1007/s10092-019-0344-156:4Online publication date: 20-Nov-2019
  • (2017)A class of linear multi-step method adapted to general oscillatory second-order initial value problemsJournal of Applied Mathematics and Computing10.1007/s12190-017-1087-256:1-2(561-591)Online publication date: 14-Feb-2017
  • (2016)Multidimensional ARKN Methods for General Multi-frequency Oscillatory Second-Order IVPsStructure-Preserving Algorithms for Oscillatory Differential Equations II10.1007/978-3-662-48156-1_10(211-227)Online publication date: 27-Feb-2016
  • Show More Cited By

Recommendations

Reviews

Zahari Zlatev

The paper considers initial value problems for second-order ordinary differential equations of the type x " + α x = ε · f ( x , x ', t ). The parameter α is a given constant frequency, while ε is a perturbation parameter. Normally, high accuracy is required when such problems are treated numerically, because their solutions have oscillatory behavior. The paper discusses a new algorithm, an improvement on a well-known method proposed originally by Scheifele [1]. In Section 1, the authors briefly review Scheifele's method. Section 2 introduces φ-functions, and Section 3 studies their different properties. A one-step numerical integrator based on the φ-functions, which generalizes Scheifele's method, is derived in the fourth section. Section 5 gives many numerical examples. The results clearly show that the new method is both very accurate and fast. In Section 6, the authors introduce a problem from orbital dynamics to demonstrate the new method. The paper ends with Section 7, "Conclusions and Outlook."> Online Computing Reviews Service

Access critical reviews of Computing literature here

Become a reviewer for Computing Reviews.

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Mathematical Software
ACM Transactions on Mathematical Software  Volume 36, Issue 4
August 2009
140 pages
ISSN:0098-3500
EISSN:1557-7295
DOI:10.1145/1555386
Issue’s Table of Contents
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 August 2009
Accepted: 01 February 2009
Revised: 01 July 2008
Received: 01 July 2006
Published in TOMS Volume 36, Issue 4

Permissions

Request permissions for this article.

Check for updates

Author Tags

  1. Numerical solutions of ODEs
  2. highly oscillatory solutions
  3. long-term integration
  4. perturbed oscillators

Qualifiers

  • Research-article
  • Research
  • Refereed

Funding Sources

Contributors

Other Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

  • Downloads (Last 12 months)3
  • Downloads (Last 6 weeks)0
Reflects downloads up to 20 Jan 2025

Other Metrics

Citations

Cited By

View all
  • (2019)Symmetric collocation ERKN methods for general second-order oscillatorsCalcolo10.1007/s10092-019-0344-156:4Online publication date: 20-Nov-2019
  • (2017)A class of linear multi-step method adapted to general oscillatory second-order initial value problemsJournal of Applied Mathematics and Computing10.1007/s12190-017-1087-256:1-2(561-591)Online publication date: 14-Feb-2017
  • (2016)Multidimensional ARKN Methods for General Multi-frequency Oscillatory Second-Order IVPsStructure-Preserving Algorithms for Oscillatory Differential Equations II10.1007/978-3-662-48156-1_10(211-227)Online publication date: 27-Feb-2016
  • (2014)Order conditions for RKN methods solving general second-order oscillatory systemsNumerical Algorithms10.1007/s11075-013-9728-566:1(147-176)Online publication date: 1-May-2014
  • (2012)A new approach for multistep numerical methods in several frequencies for perturbed oscillatorsAdvances in Engineering Software10.1016/j.advengsoft.2011.10.00245:1(252-260)Online publication date: 1-Mar-2012

View Options

Login options

Full Access

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Media

Figures

Other

Tables

Share

Share

Share this Publication link

Share on social media