A unified framework for the study of high-order energy-preserving integrators for solving Poisson systems

As is known, a system of differential equations possessing a first integral can be written in a non-canonical Hamiltonian system that often leads to a Poisson system under moderate conditions. Therefore, energy-preserving numerical integrators for such systems have become a subject of intensive investigation. In this paper, we design and analyse two classes of energy-preserving functionally-fitted integrators of arbitrary order for Poisson systems with highly oscillatory solutions using a new framework. We also show that both the order and the stage order of the proposed integrators may be affected by the used quadrature formula in practice. Furthermore, the existence and uniqueness of energy-preserving functionally-fitted integrators, their implementation issues, and the conservation of Casimir functions are investigated in detail. Finally, numerical experiments are carried out with the fourth-order and sixth-order energy-preserving integrators proposed in this paper, including the Lotka-Volterra system and the charged-particle dynamics in a strong magnetic field, and the numerical results demonstrate the remarkable accuracy and efficiency of our high-order energy-preserving integrators compared with the other energy-preserving methods in the literature. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

In this paper, two classes of energy-preserving functionally-fitted integrators for Poisson systems with highly oscillatory solutions are designed and analyzed using a new framework. The study shows that the order and stage order of the integrators may be affected by the used quadrature formula. Furthermore, the existence and uniqueness of the integrators, their implementation issues, and the conservation of Casimir functions are investigated. Numerical experiments demonstrate the remarkable accuracy and efficiency of the proposed high-order energy-preserving integrators.