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Implicit symmetric and symplectic exponentially fitted modified Runge–Kutta–Nyström methods for solving oscillatory problems
 Bing Zhen Chen^{1}Email authorView ORCID ID profile and
 Wen Juan Zhai^{2}
https://doi.org/10.1186/s1366001819154
© The Author(s) 2018
 Received: 2 April 2018
 Accepted: 14 November 2018
 Published: 20 November 2018
Abstract
Symplectic exponentially fitted RK and RKN methods have been extensively studied by many researchers. Due to their good property, they have been applied to many problems such as pendulum, Morse oscillator, harmonic oscillator, Lennard–Jones oscillator, Kepler’s orbit problem, and so on. In this paper, we construct an implicit symmetric and symplectic exponentially fitted modified Runge–Kutta–Nyström (ISSEFMRKN) method. The new integrator integrates exactly differential systems whose solutions can be expressed as linear combinations of functions from the set \(\{\exp(\lambda t),\exp(\lambda t)\}\), \(\lambda\in\mathbb{C}\), or equivalently \(\{\sin(\omega t),\cos(\omega t)\}\) when \(\lambda=i\omega\), \(\omega \in\mathbb{R}\). When \(z=\lambda h\) approaches zero, the ISSEFMRKN method reduces to the classical symplectic, symmetric RKN integrator. Numerical experiments are accompanied to show the efficiency and competence of the new method compared with some efficient codes in the literature.
Keywords
 Implicit
 Symmetric
 Symplectic
 Exponentially fitted
 Modified Runge–Kutta–Nyström method
 Oscillatory problem
1 Introduction
If the solutions of ODEs satisfy a conservation law, such as dynamical systems for which total energy is conserved, the symplectic methods [8, 9, 30] should be considered. The term symplectic essentially means area preserving in a phase space. Approximate solutions generated by symplectic methods are conservative even at finite resolution, in contrast with numerical methods that generate approximate solutions that are conservative only in the limit as the time step size approaches zero. Symplectic methods have been applied to many problems such as pendulum, Morse oscillator, harmonic oscillator, Lennard–Jones oscillator, Kepler’s orbit problem, and so on. As pointed out in Chap. V and Chap. XI of [13], symmetric methods show a better long time behavior than nonsymmetric ones when solving the reversible differential system. So, some symmetric and symplectic RKN methods are proposed such as [24, 34].
During the last thirty years, many researchers have been working on exponentially fitted RK or RKN methods. This technique was first analyzed in theory by Gautschi [12] and Lyche [21]. Exponentially fitted methods which intend to integrate exactly differential systems whose solutions can be expressed as linear combinations of functions from \(\{\exp(\lambda t),\exp(\lambda t), \lambda\in\mathbb{C}\}\), or equivalently, from \(\{\sin(\omega t ),\cos(\omega t), \omega\in\mathbb{R}\}\) with \(\lambda=i\omega\), \(i^{2}=1\), share better behaviors when applied to oscillatory problems than nonsymplectic methods. Therefore, it has become an indispensable tool for solving oscillatory problems. The construction of exponentially fitted RK(N) methods is originally due to Paternoster [23], and a detailed exposition of exponentially fitted methods with an extensive bibliography on this subject can be found in Ixaru and Vanden Berghe [15].
Recently, the authors [35] have given a twostage implicit symmetric and symplectic exponentially fitted Runge–Kutta–Nyström method (ISSEFRKN). It shows a good behavior compared with some existing methods. Exactly, this method is not a complete exponential fitting method. It can be seen from the process of derivation that there are two different expressions of \(b_{1}\). So the authors make them as close as possible by choosing a parameter \(\theta=\pm\frac{\sqrt{3}}{6}\). To avoid this happening, we investigate the construction of twostage implicit symmetric and symplectic exponentially fitted modified Runge–Kutta–Nyström (ISSEFMRKN). Compared with the ISSEFRKN method, we add modified parameters for the term h in the internal stages. Consequently, we can obtain unique expression of every coefficient which is not true for ISSEFRKN. The new method ISSEFMRKN also reduces to the classical symplectic, symmetric RKN integrator when the parameter z approaches zero.
This paper is organized as follows: In Sect. 2 we present the notations and definitions to be used in the rest of the paper as well as some previous results on symmetric and symplectic methods. In Sect. 3 we make a study of the local truncation error, obtaining the order conditions (up to fifth order) for this class of methods. In Sect. 4, we derive the new twostage implicit symplectic and symmetric EFMRKN integrator based on the former conditions. In Sect. 5, we devote to some numerical experiments. The numerical results show that the new method is more accurate and efficient compared with some other implicit RKN integrators. Finally, Sect. 6 involves in some conclusions.
2 Conditions for symmetry, symplecticity, exponential fitting of modified RKN methods
c  e γ  A 
1 1  \(\bar{b}^{T}\)  
1  \(b^{T}\) 
\(c_{1}\)  1  \(\gamma_{1}\)  \(a_{11}\)  ⋯  \(a_{1s}\) 
⋮  ⋮  ⋮  ⋮  ⋱  ⋮ 
\(c_{s}\)  1  \(\gamma_{s}\)  \(a_{s1}\)  ⋯  \(a_{ss}\) 
1  1  \(\bar{b}_{1}\)  ⋯  \(\bar {b}_{s}\)  
1  \(b_{1}\)  ⋯  \(b_{s}\) 
Now, we set out to derive the three corner stones to construct our method. In the following subsections, we denote a onestep method for secondorder ODEs (1) as \(\varPhi_{h}: (y_{0},y'_{0})^{\mathrm{T}}\mapsto (y_{1},y'_{1})^{\mathrm{T}}\). Here, from \(y_{0}\) to \(y_{1}\), the variable goes forward with a step h.
2.1 Symmetry conditions
The key to understanding symmetry is the concept of the adjoint method.
Definition 2.1
The adjoint method \(\varPhi_{h}^{*}\) of \(\varPhi_{h}\) is the inverse map of the original method with reversed time step −h, i.e., \(\varPhi_{h}^{*}:=\varPhi_{h}^{1}\). In other words, \(y_{1}=\varPhi_{h}^{*}(y_{0})\) is implicitly defined by \(\varPhi_{h}(y_{1})=y_{0}\). A method for which \(\varPhi_{h}^{*}=\varPhi_{h}\) is called symmetric.
In this paper we consider scheme (2) whose coefficients are functions of z, as we do for exponentially fitted type methods [32, 33]. Then the conditions for methods (2) to be symmetric are given by the following lemma.
Lemma 1
Proof
Comparing equations (7), (6), (5) with the counterpart in (2) respectively, we can obtain the symmetric conditions (3). □
Omitting the variable z, i.e., all the coefficients are constants, they reduce to symmetric conditions for the traditional RKN methods.
2.2 Symplecticity conditions
Definition 2.2
 (a)
the solutions preserve the Hamiltonian \(H(p, q)\);
 (b)
the corresponding flow is symplectic, i.e., preserves the differential 2form \(\mathrm{d}p\wedge{\mathrm{d}}q\).
As is pointed out by Feng [7], “It is natural to look forward to those discrete systems which preserve as much as possible the intrinsic properties of the continuous system.” Based on this definition, we can easily obtain the symplectic conditions for RKN formula (2).
Lemma 2
Proof
2.3 Exponential fitting conditions

for the internal stages,$$ \varphi_{i}\bigl[y(t);h;\mathbf{a}\bigr]=y(t+c_{i}h)y(t)c_{i} \gamma _{i}hy'(t)h^{2}\sum _{j=1}^{s}a_{ij}y''(t+c_{j}h), \quad i=1,2,\ldots,s; $$

for the final stages,$$ \textstyle\begin{cases} \varphi[y(t);h;\bar{\mathbf{b}}]=y(t+h)y(t)hy'(t)h^{2}\sum_{i=1}^{s}\bar{b}_{i}y''(t+c_{i}h), \\ \varphi[y(t);h;\mathbf{b}]=y'(t+h)y'(t)h\sum_{i=1}^{s}b_{i}y''(t+c_{i}h). \end{cases} $$
Lemma 3
Proof
In this paper, we say method (2) satisfies the EF conditions (14) and (15) as an exponentially fitted RKN (EFRKN) method.
3 Algebraic order conditions
4 Construction of implicit symmetric and symplectic modified EFRKN methods
Based on the above conditions, in this section, we will construct an implicit EFMRKN method under the symmetry, symplecticity, exponential fitting conditions obtained in the previous two sections. We consider a brief case \(s=2\) in which there will not be so many coefficients.
5 Numerical experiments

DIRKNRaed: The embedded diagonally implicit RKN 4(3) pair method proposed by AlKhasawneh et al. in [1].

DIRKNNora: The threestage fourthorder diagonally implicit RKN method proposed by Senu et al. in [27].

ISSRKN2: The symmetric and symplectic twostage fourthorder implicit RKN method proposed by MENGZHAO QIN et al. in [24] with \(a_{11}=\frac{13+160\theta^{2}+720\theta^{4}}{2880\theta^{2}}\) and \(\theta=\pm\frac{\sqrt{3}}{6}\), i.e., \(a_{11}=\frac{1}{45}\).

ISSEFRKN2: The symmetric and symplectic exponentially fitted twostage RKN method proposed in [35] which is of order 4.

ISSEFMRKN2: The symmetric and symplectic exponentially fitted twostage modified RKN method (25) proposed in this paper which is of order 4.
Problem 1
Problem 2
Problem 3
Problem 4
From Figs. 1–5, we can find that the implicit modified EFRKN method ISSEFMRKN2 is more efficient than ISSEFRKN2 and the symmetric and symplectic method ISSRK2. ISSEFRKN2 does not possess higher accuracy than ISSRKN2 for Problems 3 and 4. For Problem 3, its true frequency is 1. In the numerical study, we select two frequencies 1.2 (Fig. 3) and 1 (Fig. 4). From Figs. 3–4, we can see the accuracies are quite different. The accuracy of 1 is much higher than that of 1.2. In this problem, we know its true frequency. But when it comes to applications, the true frequency is often unpredictable. Therefore, we need to try some different candidates. For Problems 2 and 3, we find that ISSEFMRKN2 is much more accurate and efficient than our methods considered in this paper. As is pointed out in introduction, ISSEFRKN2 is not a complete EF method. However, ISSEFMRKN2 is completely EF. The solutions of Problems 2 and 3 are all in the form of a triangular function. This is just in line with EF methods. So, ISSEFMRKN2 performs very well.
6 Conclusions
In this paper a twostage symmetric, symplectic IEFMRKN integrator has been derived. Like the existing EFRKN integrators such as [34, 35], the coefficients of the new method depend on the product of the dominant frequency ω and the step size h. When the parameter z approaches zero, the ISSEFMRKN2 method reduces to the classical RKN method. The numerical experiments carried out show that the new method is more efficient than some implicit RKN methods. In every experiment, the method ISSEFMRKN2 is shown to be the most efficient one among the methods used for comparison. However, like the ISSEFRKN2 method in [35], we derive only one method, not a class of methods whose coefficients can be dependent on one or more parameters. In the future, we will consider deriving a class of ISSEFRKN methods.
Declarations
Funding
This work was supported in part by the Project for Youth Scholars of Higher Education of Hebei Province (QN2017402), the Project of Teaching Research and Practice of Higher Education of Hebei Province (2017GjjG364), and the Project of Teaching and Research of Beijing Jiaotong University Haibin College (HBJY16005).
Authors’ contributions
All authors contributed equally in writing this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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