Exponential Fourier Collocation Methods for Solving First-Order Differential Equations

Exponential Fourier Collocation Methods for Solving First-Order Differential Equations

Year:    2017

Author:    Bin Wang, Xinyuan Wu, Fanwei Meng, Yonglei Fang

Journal of Computational Mathematics, Vol. 35 (2017), Iss. 6 : pp. 711–736

Abstract

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 first-order 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).

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jcm.1611-m2016-0596

Journal of Computational Mathematics, Vol. 35 (2017), Iss. 6 : pp. 711–736

Published online:    2017-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    26

Keywords:    First-order differential equations Exponential Fourier collocation methods Variation-of-constants formula Structure-preserving exponential integrators Collocation methods.

Author Details

Bin Wang

Xinyuan Wu

Fanwei Meng

Yonglei Fang