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Trigonometrically-Fitted Methods: A Review

Author

Listed:
  • Changbum Chun

    (Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea)

  • Beny Neta

    (Naval Postgraduate School, Department of Applied Mathematics, Monterey, CA 93943, USA)

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.

Suggested Citation

  • Changbum Chun & Beny Neta, 2019. "Trigonometrically-Fitted Methods: A Review," Mathematics, MDPI, vol. 7(12), pages 1-20, December.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:12:p:1197-:d:294883
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    References listed on IDEAS

    as
    1. T. E. Simos & Jesus Vigo Aguiar, 2001. "A Symmetric High Order Method With Minimal Phase-Lag For The Numerical Solution Of The Schrödinger Equation," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 12(07), pages 1035-1042.
    2. Franco, J.M. & Rández, L., 2016. "Explicit exponentially fitted two-step hybrid methods of high order for second-order oscillatory IVPs," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 493-505.
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