Exponential Fourier collocation methods for solving first-order differential equations

TL;DR

This paper introduces exponential Fourier collocation methods (EFCMs) for efficiently solving first-order ODEs, extending existing methods and demonstrating high accuracy, energy preservation, and computational robustness through theoretical analysis and numerical experiments.

Contribution

The paper presents a new class of EFCMs that generalize and extend existing collocation methods, enabling high-order solutions for first-order differential equations.

Findings

EFCMs can achieve arbitrarily high order accuracy.

The proposed EFCM(2,2) is efficient and robust in numerical tests.

EFCMs preserve quadratic invariants and Hamiltonian energy effectively.

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).