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Proficient trigonometrical-fitted two-derivative multistep collocation methods in predictor-corrector approach:

Khai Chien Lee1, Muhammad Naeim Mohd Aris1, Ishak Hashim1

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A new trigonometrically-fitted two-derivative multistep collocation (TF-TDMC) method effectively solves oscillatory second-order ordinary differential equations. This frequency-adapted method achieves zero-stability and outperforms existing techniques, demonstrating superior accuracy and efficiency.

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CollocationPredictor-correctorSecond-order ordinary differential equationsTrigonometrical-fittedTrigonometrical-fitted two-derivative multistep collocation method in predictor-corrector modeTwo-derivative multistep collocation

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Area of Science:

  • Numerical Analysis and Computational Mathematics
  • Applied Mathematics and Dynamical Systems

Background:

  • Second-order ordinary differential equations (ODEs) with oscillatory solutions pose significant challenges for traditional numerical methods.
  • Existing methods often struggle with accuracy and efficiency when dealing with the inherent frequency-dependent nature of oscillatory systems.

Purpose of the Study:

  • To develop an efficient and accurate numerical method for solving second-order ODEs with oscillatory solutions.
  • To introduce a trigonometrically-fitted technique to adapt the method to the specific frequencies of the solutions.
  • To analyze the stability properties and demonstrate the superiority of the proposed method.

Main Methods:

  • Development of a two-derivative multistep collocation (TDMC) method using Legendre polynomials.
  • Incorporation of a trigonometrically-fitting technique to create frequency-dependent coefficients.
  • Implementation in a predictor-corrector framework with rigorous stability analysis, including zero-stability verification.

Main Results:

  • The proposed trigonometrically-fitted two-derivative multistep collocation (TF-TDMC) method achieves zero-stability.
  • Numerical experiments show the TF-TDMC method significantly outperforms existing methods in terms of maximum global error.
  • The method demonstrates high efficiency and accuracy across various step sizes, including for the perturbed Kepler problem.

Conclusions:

  • The TF-TDMC method provides an effective and efficient direct solver for second-order ODEs with oscillatory solutions.
  • The frequency-fitting technique is crucial for enhancing accuracy and performance.
  • The TF-TDMC method is a superior alternative for solving challenging oscillatory problems, including those found in celestial mechanics.