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Trigonometrically-fitted Scheifele two-step methods for perturbed oscillators. (English) Zbl 1262.65077

Summary: A new family of trigonometrically-fitted Scheifele two-step (TFSTS) methods for the numerical integration of perturbed oscillators is proposed and investigated (cf. [G. Scheifele, Z. Angew. Math. Phys. 22, 186–210 (1971; Zbl 0221.65135)]). An essential feature of TFSTS methods is that they are exact in both the internal stages and the updates when solving the unperturbed harmonic oscillator \(y^{\prime \prime}= - \omega ^{2}y\) for known frequency \(\omega \). Based on the linear operator theory, the necessary and sufficient conditions for TFSTS methods of up to order five are derived. Two specific TFSTS methods of orders four and five, respectively, are constructed and their stability and phase properties are examined. In the five numerical experiments carried out, the new integrators are shown to be more efficient and competent than some well-known methods in the literature.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations

Citations:

Zbl 0221.65135
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References:

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