Exponential Fourier Collocation Methods for Solving First-Order Differential Equations

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 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.

Volume-preserving exponential integrators and their applications

“…Proof In the light of Definition 2.2 and the result (30), we adapt the SSEI methods to the system (29) and then get the scheme (31). Based on Theorem 4.4, the volume preserving result of (31) is immediately obtained.…”

“…. , s, all one-stage and two-stage (with c 1 = c 2 ) ERKN integrators (31), and compositions thereof, are of volume preservation for solving the separable partitioned system (29).…”

Arbitrary-order functionally fitted energy-diminishing methods for gradient systems

“…[2,5]). Many effective methods have been derived for this stiff gradient system with a constant matrix G and we refer to [7,8,10,12,21,22,23,24,25] for example. The FFED method (2) for solving this stiff gradient system is defined as follows.…”

Exponential collocation methods for conservative or dissipative systems

“…One important example of them is the multi-frequency highly oscillatory Hamiltonian systems with the following Hamiltonian H(q, p) = 1 2 p ⊺M −1 p + 1 2 q ⊺K q + U (q), (5) whereK is a symmetric positive semi-definite stiffness matrix,M is a symmetric positive definite mass matrix, and U (q) is a smooth potential with moderately bounded derivatives. In recent decades, exponential integrators have been widely investigated and developed as an efficient approach to integrating (3), and we refer the reader to [3,8,9,11,12,13,19,22,31,34,47,56,60] for example. Exponential integrators make well use of the variation-of-constants formula (4), and their performance has been evaluated by a range of test problems.…”