Elsevier

Applied Numerical Mathematics

Volume 58, Issue 9, September 2008, Pages 1375-1395
Applied Numerical Mathematics

Trigonometrically-fitted ARKN methods for perturbed oscillators

https://doi.org/10.1016/j.apnum.2007.08.001Get rights and content

Abstract

In this paper, new and robust trigonometrically-fitted adapted Runge–Kutta–Nyström methods for the numerical integration of perturbed oscillators are presented, which combine the features of trigonometrically-fitted methods with ARKN methods. Based on the linear-operator theory, the necessary and sufficient order conditions for these methods are derived. The numerical experiments are accompanied to show the efficiency and competence of our methods in comparison with some well-known methods.

Cited by (43)

  • The tri-coloured free-tree theory for symplectic multi-frequency ERKN methods

    2023, Journal of Computational and Applied Mathematics
    Citation Excerpt :

    Multi-frequency ERKN methods have been widely used in many fields of science and engineering. They have been investigated for exponentially or trigonometrically fitted methods [2,4,10–12] and for two-step hybrid methods [13] solving oscillatory second-order differential equations, for energy-preserving methods [14] solving multi-frequency oscillatory Hamiltonian systems, for asymptotic methods solving highly oscillatory problems [15,16], for symplectic methods solving Hamiltonian ODEs [5,17–20], and for multisymplectic methods solving Hamiltonian PDEs [21]. The purpose of this section is to simplify the symplectic conditions for ERKN methods.

  • An approach to solving Maxwell's equations in time domain

    2023, Journal of Mathematical Analysis and Applications
    Citation Excerpt :

    Then, effective solvers for ODEs can be used. Typically, the solvers mean some effective integrators appeared in the literature, such as the Gautschi-type method [7], trigonometric Fourier collocation methods [34], extended Runge-Kutta(-Nystrom) methods [38,46,48–51,53,54], and the discrete energy-preserving methods (the AVF methods) [4,20–23,28,30,45]. All of them can be carried by replacing the differential operator by a suitable differentiation matrix in the operator-variation-of-constants formula for Maxwell's equations.

  • Symplectic and symmetric trigonometrically-fitted ARKN methods

    2019, Applied Numerical Mathematics
    Citation Excerpt :

    For more work we see [16,17,26]. Recently, in order to let ARKN methods behave better in integrating oscillatory problems (1), Yang et al. [28] modified the ARKN methods by introducing frequency depending coefficients into the terms in the internal stages and proposed trigonometrically-fitted ARKN (TFARKN) methods. Numerical tests of [28] have shown the superiority of these methods.

View all citing articles on Scopus

This Project Supported by the Natural Science Foundation of China 10771099.

View full text