Trigonometrically-Fitted Methods: A Review

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 fr...

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Published inMathematics (Basel) Vol. 7; no. 12; p. 1197
Main Authors Chun, Changbum, Neta, Beny
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.12.2019
Subjects
Algebra
Boundary value problems
Differential equations
Error reduction
Floodgates
Interpolation
Mathematics
Methods
Numerical methods
Ordinary differential equations
Polynomials
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ISSN2227-7390
2227-7390
DOI10.3390/math7121197

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Summary: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.
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ISSN:2227-7390
2227-7390
DOI:10.3390/math7121197