In this paper we discuss the efficient implementation of RKN-type Fourier collocation methods, which are used when solving second-order differential equations. The proposed implementation relies on an alternative formulation of the methods and the blended formulation. The features and effectiveness of the implementation are confirmed by the performance of the methods on two numerical tests.

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a b s t r a c tA new kind of trigonometrically fitted embedded pair of explicit ARKN methods for the numerical integration of perturbed oscillators is presented in this paper. This new pair is based on the trigonometrically fitted ARKN method of order five derived by Yang and Wu in [H.L. Yang, X.Y. Wu, Trigonometrically-fitted ARKN methods for perturbed oscillators, Appl. Numer. Math. 9 (2008) 1375-1395]. We analyze the stability properties, phase-lag (dispersion) and dissipation of the higher-order method of the new pair. Numerical experiments carried out show that our new embedded pair is very competitive in comparison with the embedded pairs proposed in the scientific literature.

In this paper, a novel class of exponential Fourier collocation methods (EFCMs) is presented for solving systems of first-order ordinary differential equations. These so-called exponential Fourier collocation methods are based on the variation-of-constants formula, incorporating a local Fourier expansion of the underlying problem with collocation methods. We discuss in detail the connections of EFCMs with trigonometric Fourier collocation methods (TFCMs), the well-known Hamiltonian Boundary Value Methods (HBVMs), Gauss methods and Radau IIA methods. It turns out that the novel EFCMs are an essential extension of these existing methods. We also analyse the accuracy in preserving the quadratic invariants and the Hamiltonian energy when the underlying system is a Hamiltonian system. Other properties of EFCMs including the order of approximations and the convergence of fixed-point iterations are investigated as well. The analysis given in this paper proves further that EFCMs can achieve arbitrarily high order in a routine manner which allows us to construct higher-order methods for solving systems of firstorder ordinary differential equations conveniently. We also derive a practical fourth-order EFCM denoted by EFCM(2,2) as an illustrative example. The numerical experiments using EFCM(2,2) are implemented in comparison with an existing fourth-order HBVM, an energy-preserving collocation method and a fourth-order exponential integrator in the literature. The numerical results demonstrate the remarkable efficiency and robustness of the novel EFCM(2,2).Mathematics subject classification: 65L05, 65L20, 65M20, 65M70.

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