Trigonometrically-Fitted Methods: A Review

Changbum Chun and Beny Neta
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Changbum Chun: Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea
Beny Neta: Naval Postgraduate School, Department of Applied Mathematics, Monterey, CA 93943, USA

Mathematics, 2019, vol. 7, issue 12, 1-20

Abstract: 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.

Keywords: second order initial value problems; linear multistep methods; Obrechkoff schemes; trigonometrically fitted (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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