A General Procedure For the Adaptation of Multistep Algorithms to the Integration of Oscillatory Problems

  • Authors:

  • Jesus Vigo-Aguiar;José M. Ferrándiz

  • Affiliations:

  • -;-

  • Venue:
  • SIAM Journal on Numerical Analysis
  • Year:
  • 1998

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Abstract

This paper introduces a general technique for the construction of multistep methods capable of integrating, without local truncation error, homogeneous linear ODEs with constant coefficients, including those, in particular, that result in oscillatory solutions. Moreover, these methods can be further adapted through coefficient modification for the exact integration of forced oscillations in one or more frequencies, even confluent ones that occur from nonhomogeneous terms in the differential equation. Our procedure allows the derivation of many of the existing codes with similar properties, as well as the improvement of others that in their original design were only able to integrate oscillations in a single frequency. The properties of the methods are studied within a general framework, and numerical examples are presented. These demonstrate the way in which the new algorithms perform distinctly better than the general purpose codes, particularly when integrating the class of equations with perturbed oscillatory solutions. The methods developed are mainly applicable to the accurate and efficient integration of problems for which the oscillation frequencies are known, as occurs in satellite orbit propagation. The underlying ideas have already been applied to the improvement of some Chebyshev methods that are not multistep.