Functionally-fitted energy-preserving methods for solving oscillatory nonlinear Hamiltonian systems

In the last few decades, numerical simulation for nonlinear oscillators has received a great deal of attention, and many researchers have been concerned with the design and analysis of numerical methods for solving oscillatory problems. In this paper, from the perspective of the continuous finite element method, we propose and analyze new energy-preserving functionally fitted methods, in particular trigonometrically fitted methods of an arbitrarily high order for solving oscillatory nonlinear Hamiltonian systems with a fixed frequency. To implement these new methods in a widespread way, they are transformed into a class of continuous-stage Runge--Kutta methods. This paper is accompanied by numerical experiments on oscillatory Hamiltonian systems such as the FPU problem and nonlinear Schr\"odinger equation. The numerical results demonstrate the remarkable accuracy and efficiency of our new methods compared with the existing high-order energy-preserving methods in the literature.Comment: 24 page

Functionally-fitted energy-preserving integrators for Poisson systems

In this paper, a new class of energy-preserving integrators is proposed and analysed for Poisson systems by using functionally-fitted technology. The integrators exactly preserve energy and have arbitrarily high order. It is shown that the proposed approach allows us to obtain the energy-preserving methods derived in BIT 51 (2011) by Cohen and Hairer and in J. Comput. Appl. Math. 236 (2012) by Brugnano et al. for Poisson systems. Furthermore, we study the sufficient conditions that ensure the existence of a unique solution and discuss the order of the new energy-preserving integrators.Comment: 19 page

Long-time oscillatory energy conservation of total energy-preserving methods for highly oscillatory Hamiltonian systems

For an integrator when applied to a highly oscillatory system, the near conservation of the oscillatory energy over long times is an important aspect. In this paper, we study the long-time near conservation of oscillatory energy for the adopted average vector field (AAVF) method when applied to highly oscillatory Hamiltonian systems. This AAVF method is an extension of the average vector field method and preserves the total energy of highly oscillatory Hamiltonian systems exactly. This paper is devoted to analysinganother important property of AAVF method, i.e., the near conservation of its oscillatory energy in a long term. The long-time oscillatory energy conservation is obtained via constructing a modulated Fourier expansion of the AAVF method and deriving an almost invariant of the expansion. A similar result of the method in the multi-frequency case is also presented in this paper

Exponential collocation methods for conservative or dissipative systems

In this paper, we propose and analyse a novel class of exponential collocation methods for solving conservative or dissipative systems based on exponential integrators and collocation methods. It is shown that these novel methods can be of arbitrarily high order and exactly or nearly preserve first integrals or Lyapunov functions. We also consider order estimates of the new methods. Furthermore, we explore and discuss the application of our methods in important stiff gradient systems, and it turns out that our methods are unconditionally energy-diminishing and strongly damped even for very stiff gradient systems. Practical examples of the new methods are derived and the efficiency and superiority are confirmed and demonstrated by three numerical experiments including a nonlinear Schr\"{o}dinger equation. As a byproduct of this paper, arbitrary-order trigonometric/RKN collocation methods are also presented and analysed for second-order highly oscillatory/general systems. The paper is accompanied by numerical results that demonstrate the great potential of this work

On the effectiveness of spectral methods for the numerical solution of multi-frequency highly-oscillatory Hamiltonian problems

Multi-frequency, highly-oscillatory Hamiltonian problems derive from the mathematical modelling of many real life applications. We here propose a variant of Hamiltonian Boundary Value Methods (HBVMs), which is able to efficiently deal with the numerical solution of such problems.Comment: 28 pages, 4 figures (a few typos fixed

A long-term numerical energy-preserving analysis of symmetric and/or symplectic extended RKN integrators for efficiently solving highly oscillatory Hamiltonian systems

The primary objective of this paper is to present a long-term numerical energy-preserving analysis of one-stage explicit symmetric and/or symplectic extended Runge--Kutta--Nystr\"{o}m (ERKN) integrators for highly oscillatory Hamiltonian systems. We study the long-time numerical energy conservation not only for symmetric integrators but also for symplectic integrators. In the analysis, we neither assume symplecticity for symmetric methods, nor assume symmetry for symplectic methods. It turns out that these both kinds of ERKN integrators have a near conservation of the total and oscillatory energy over a long term. To prove the result for symmetric integrators, a relationship between symmetric ERKN integrators and trigonometric integrators is established by using Strang splitting and based on this connection, the long-time conservation is derived. For the long-term analysis of symplectic ERKN integrators, the above approach does not work anymore and we use the technology of modulated Fourier expansion developed in SIAM J. Numer. Anal. 38 (2000) by Hairer and Lubich. By taking some novel adaptations of this essential technology for non-symmetric methods, we derive the modulated Fourier expansion for symplectic ERKN integrators. Moreover, it is shown that the symplectic ERKN integrators have two almost-invariants and then the near energy conservation over a long term is obtained

Energy-preserving integration of non-canonical Hamiltonian systems by continuous-stage methods

As is well known, energy is generally deemed as one of the most important physical invariants in many conservative problems and hence it is of remarkable interest to consider numerical methods which are able to preserve it. In this paper, we are concerned with the energy-preserving integration of non-canonical Hamiltonian systems by continuous-stage methods. Algebraic conditions in terms of the Butcher coefficients for ensuring the energy preservation, symmetry and quadratic-Casimir preservation respectively are presented. With the presented condition and in use of orthogonal expansion techniques, the construction of energy-preserving integrators is examined. A new class of energy-preserving integrators which is symmetric and of order $2m$ is constructed. Some numerical results are reported to verify our theoretical analysis and show the effectiveness of our new methods

Efficient energy-preserving methods for charged-particle dynamics

In this paper, energy-preserving methods are formulated and studied for solving charged-particle dynamics. We first formulate the scheme of energy-preserving methods and analyze its basic properties including algebraic order and symmetry. Then it is shown that these novel methods can exactly preserve the energy of charged-particle dynamics. Moreover, the long time momentum conservation is studied along such energy-preserving methods. A numerical experiment is carried out to illustrate the notable superiority of the new methods in comparison with the popular Boris method in the literature

Explicit symplectic adapted exponential integrators for charged-particle dynamics in a strong and constant magnetic field

This paper studies explicit symplectic adapted exponential integrators for solving charged-particle dynamics in a strong and constant magnetic field. We first formulate the scheme of adapted exponential integrators and then derive its symplecticity conditions. Based on the symplecticity conditions, we propose five practical explicit symplectic adapted exponential integrators. Two numerical experiments are carried out and the numerical results demonstrate the remarkable numerical behavior of the new methods.Comment: The content of this manuscript is included in a new paper of ours as one section. We will submit the new paper to arXi

Symmetric integrators based on continuous-stage Runge-Kutta-Nystrom methods for reversible systems

In this paper, we study symmetric integrators for solving second-order ordinary differential equations on the basis of the notion of continuous-stage Runge-Kutta-Nystrom methods. The construction of such methods heavily relies on the Legendre expansion technique in conjunction with the symmetric conditions and simplifying assumptions for order conditions. New families of symmetric integrators as illustrative examples are presented. For comparing the numerical behaviors of the presented methods, some numerical experiments are also reported