You, Xiong; Zhang, Yonghui; Zhao, Jinxi Trigonometrically-fitted Scheifele two-step methods for perturbed oscillators. (English) Zbl 1262.65077 Comput. Phys. Commun. 182, No. 7, 1481-1490 (2011). 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. Cited in 14 Documents 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 Keywords:trigonometrically-fitted methods; Scheifele two-step methods; perturbed oscillators; linear operators; G-function; Runge-Kutta method; stability; numerical experiments Citations:Zbl 0221.65135 PDFBibTeX XMLCite \textit{X. You} et al., Comput. Phys. Commun. 182, No. 7, 1481--1490 (2011; Zbl 1262.65077) Full Text: DOI References: [1] Scheifele, G., Z. Angew. Math. Phys., 22, 186 (1971) [2] González, A. B.; Martín, P.; Farto, J. M., Numer. Math., 82, 635 (1999) [3] Franco, J. M., Appl. Numer. Math., 56, 1040 (2006) [4] Franco, J. M., Comput. Phys. Comm., 147, 770 (2002) [5] Franco, J. M., J. Comput. Appl. Math., 173, 389 (2005) [6] Wu, X.; You, X.; Xia, J., Comput. Phys. Comm., 180, 2250 (2009) [7] Yang, H.; Wu, X.; You, X.; Fang, Y., Comput. Phys. Comm., 180, 1777 (2009) [8] Martín, P.; Ferrándiz, J. M., SIAM J. Numer. Anal., 34, 359 (1997) [9] Coleman, J. P., IMA J. Numer. Anal., 23, 197 (2003) [10] Franco, J. M., J. Comput. Appl. Math., 187, 41 (2006) [11] Van de Vyver, H., Internat. J. Modern Phys. C, 17, 663 (2006) [12] Van de Vyver, H., J. Comput. Appl. Math., 209, 33 (2007) [13] Van de Vyver, H., J. Comput. Appl. Math., 224, 415 (2009) [14] Vanden Berghe, G.; Van Daele, M.; Van de Vyver, H., J. Comput. Appl. Math., 159, 217 (2003) [15] Fang, Y.; Song, Y.; Wu, X., Comput. Phys. Comm., 179, 801 (2008) [16] Yang, H.; Wu, X., Appl. Numer. Math., 58, 1375 (2008) [17] Van Dooren, R., J. Comput. Phys., 16, 186 (1974) [18] Farto, J. M.; González, A. B.; Martín, P., Comput. Phys. Comm., 111, 110 (1998) [19] Wu, X.; You, X.; Shi, W.; Wang, B., Comput. Phys. Comm., 181, 1873 (2010) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.