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Open AccessReview

Trigonometrically-Fitted Methods: A Review

by 1 and 2,*
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea
Naval Postgraduate School, Department of Applied Mathematics, Monterey, CA 93943, USA
Author to whom correspondence should be addressed.
Mathematics 2019, 7(12), 1197;
Received: 28 October 2019 / Revised: 27 November 2019 / Accepted: 2 December 2019 / Published: 6 December 2019
(This article belongs to the Special Issue Computational Mathematics, Algorithms, and Data Processing)
Numerical methods for the solution of ordinary differential equations are based on polynomial interpolation. In 1952, Brock and Murray have suggested exponentials for the case that the solution is known to be of exponential type. In 1961, Gautschi came up with the idea of using information on the frequency of a solution to modify linear multistep methods by allowing the coefficients to depend on the frequency. Thus the methods integrate exactly appropriate trigonometric polynomials. This was done for both first order systems and second order initial value problems. Gautschi concluded that “the error reduction is not very substantial unless” the frequency estimate is close enough. As a result, no other work was done in this direction until 1984 when Neta and Ford showed that “Nyström’s and Milne-Simpson’s type methods for systems of first order initial value problems are not sensitive to changes in frequency”. This opened the flood gates and since then there have been many papers on the subject. View Full-Text
Keywords: second order initial value problems; linear multistep methods; Obrechkoff schemes; trigonometrically fitted second order initial value problems; linear multistep methods; Obrechkoff schemes; trigonometrically fitted
MDPI and ACS Style

Chun, C.; Neta, B. Trigonometrically-Fitted Methods: A Review. Mathematics 2019, 7, 1197.

AMA Style

Chun C, Neta B. Trigonometrically-Fitted Methods: A Review. Mathematics. 2019; 7(12):1197.

Chicago/Turabian Style

Chun, Changbum; Neta, Beny. 2019. "Trigonometrically-Fitted Methods: A Review" Mathematics 7, no. 12: 1197.

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