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Properties and implementation of r-Adams methods based on mixed-type interpolation

Marnix Van Daele (UGent) , Guido Vanden Berghe (UGent) and Hans De Meyer (UGent)
(1995) COMPUTERS & MATHEMATICS WITH APPLICATIONS. 30(10). p.37-54
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Abstract
We investigate the properties of the coefficients of modified r-Adams methods for the integration of ODEs. The derivation of these methods is, in contrast with the classical Adams methods, not based on a polynomial interpolation theory, but rather starts from a mixed interpolation theory in which a parameter kappa is involved. It will be shown that the coefficients of the modified methods possess properties which make these methods very attractive. Further, we will discuss the role of so-called over-implicit modified r-Adams schemes in the construction of more general linear multistep methods. Our second goal is to show that the modified Adams-Bashforth/Adams-Moulton methods are very well suited to be implemented as a predictor-corrector pair. In particular, are will discuss the choice of the interpolation parameter when such a method is applied to general systems of equations. Numerical tests are performed to support the theory.
Keywords
INTERPOLATION, NUMERICAL INTEGRATION, MULTISTEP METHODS, ORDINARY DIFFERENTIAL-EQUATIONS, SCHRODINGER-EQUATION, NUMERICAL-SOLUTION, TRIGONOMETRIC POLYNOMIALS, MULTISTEP METHODS, FITTING METHODS, FAMILIES, ERROR

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MLA
Van Daele, Marnix, et al. “Properties and Implementation of R-Adams Methods Based on Mixed-Type Interpolation.” COMPUTERS & MATHEMATICS WITH APPLICATIONS, vol. 30, no. 10, 1995, pp. 37–54, doi:10.1016/0898-1221(95)00155-R.
APA
Van Daele, M., Vanden Berghe, G., & De Meyer, H. (1995). Properties and implementation of r-Adams methods based on mixed-type interpolation. COMPUTERS & MATHEMATICS WITH APPLICATIONS, 30(10), 37–54. https://doi.org/10.1016/0898-1221(95)00155-R
Chicago author-date
Van Daele, Marnix, Guido Vanden Berghe, and Hans De Meyer. 1995. “Properties and Implementation of R-Adams Methods Based on Mixed-Type Interpolation.” COMPUTERS & MATHEMATICS WITH APPLICATIONS 30 (10): 37–54. https://doi.org/10.1016/0898-1221(95)00155-R.
Chicago author-date (all authors)
Van Daele, Marnix, Guido Vanden Berghe, and Hans De Meyer. 1995. “Properties and Implementation of R-Adams Methods Based on Mixed-Type Interpolation.” COMPUTERS & MATHEMATICS WITH APPLICATIONS 30 (10): 37–54. doi:10.1016/0898-1221(95)00155-R.
Vancouver
1.
Van Daele M, Vanden Berghe G, De Meyer H. Properties and implementation of r-Adams methods based on mixed-type interpolation. COMPUTERS & MATHEMATICS WITH APPLICATIONS. 1995;30(10):37–54.
IEEE
[1]
M. Van Daele, G. Vanden Berghe, and H. De Meyer, “Properties and implementation of r-Adams methods based on mixed-type interpolation,” COMPUTERS & MATHEMATICS WITH APPLICATIONS, vol. 30, no. 10, pp. 37–54, 1995.
@article{195665,
  abstract     = {{We investigate the properties of the coefficients of modified r-Adams methods for the integration of ODEs. The derivation of these methods is, in contrast with the classical Adams methods, not based on a polynomial interpolation theory, but rather starts from a mixed interpolation theory in which a parameter kappa is involved. It will be shown that the coefficients of the modified methods possess properties which make these methods very attractive. Further, we will discuss the role of so-called over-implicit modified r-Adams schemes in the construction of more general linear multistep methods. Our second goal is to show that the modified Adams-Bashforth/Adams-Moulton methods are very well suited to be implemented as a predictor-corrector pair. In particular, are will discuss the choice of the interpolation parameter when such a method is applied to general systems of equations. Numerical tests are performed to support the theory.}},
  author       = {{Van Daele, Marnix and Vanden Berghe, Guido and De Meyer, Hans}},
  issn         = {{0898-1221}},
  journal      = {{COMPUTERS & MATHEMATICS WITH APPLICATIONS}},
  keywords     = {{INTERPOLATION,NUMERICAL INTEGRATION,MULTISTEP METHODS,ORDINARY DIFFERENTIAL-EQUATIONS,SCHRODINGER-EQUATION,NUMERICAL-SOLUTION,TRIGONOMETRIC POLYNOMIALS,MULTISTEP METHODS,FITTING METHODS,FAMILIES,ERROR}},
  language     = {{eng}},
  number       = {{10}},
  pages        = {{37--54}},
  title        = {{Properties and implementation of r-Adams methods based on mixed-type interpolation}},
  url          = {{http://doi.org/10.1016/0898-1221(95)00155-R}},
  volume       = {{30}},
  year         = {{1995}},
}
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