A class of implicit symmetric symplectic and exponentially fitted Runge\u2013Kutta\u2013Nystr\xf6m methods for solving oscillatory problems

Huai Yuan Zhai,Wen Juan Zhai,Bing Zhen Chen

doi:10.1186/s13662-018-1910-6

Huai Yuan Zhai, Wen Juan Zhai + Show 1 more

Open Access

https://doi.org/10.1186/s13662-018-1910-6

Copy DOI

Abstract

The construction of implicit Runge–Kutta–Nyström (RKN) method is considered in this paper. Based on the symmetric, symplectic, and exponentially fitted conditions, a class of implicit RKN integrators is obtained. The new integrators called ISSEFMRKN integrate exactly differential systems whose solutions are linear combinations of functions from the set {exp(lambda t), exp(-lambda t), lambdainmathbb{C}}. In addition, their final stages also preserve the quadratic invariants {exp(2lambda t), exp(-2lambda t)}. Especially, we derived two methods: ISSEFMRKNs1o2 and ISSEFMRKNs2o4 which are of order 2 and 4, respectively. And the method ISSEFMRKNs2o4 has variable nodes. The derived method ISSEFMRKNs2o4 reduces to the classical RKN method (Qin and Zhu in Comput. Math. Appl. 22(9):85–95, 1991) as lambda hrightarrow0. The numerical results show that our methods possess the efficiency and competence compared with some implicit RKN methods in the literature. Especially, ISSEFMRKNs2o4 improves the accuracy compared with unmodified method ISSEFRKNs2o4 proposed in (Zhai and Chen in Numer. Algebra Control Optim. 9(1):71–84, 2019).

Highlights

IntroductionSome symmetric and symplectic RKN methods have been proposed such as [20]
1 Introduction In this paper we focus on the initial value problems (IVP) related to systems of secondorder ODEs of the form y = f (t, y), y(t0) = y0, y (t0) = y0, t ∈ [0, tend], (1)
2 Symmetric, symplectic, exponential fitting conditions In this paper, we focus on the following s-stage modified implicit RKN method for the second-order ODEs (1):

Summary

Introduction

Some symmetric and symplectic RKN methods have been proposed such as [20] They do not consider exponential fitting conditions. In [33], the authors derived an implicit symmetric, symplectic, and exponentially fitted RKN integrator. Symplectic, and EF conditions for our modified RKN methods in Sect. Two-stage implicit symmetric symplectic exponentially fitted RKN (EFRKN) integrators. We have obtained an implicit symmetric and symplectic exponentially fitted Runge–Kutta–Nyström method whose coefficients are given as. For small values of z, the Taylor series expansions of the coefficients are given by From these Taylor series, we can verify that our method ISSEFMRKNs2o4 satisfies all the fourth-order conditions. ISSEFRKNs2o4 proposed reduces to SSRKN of order [20]

Numerical experiments

Conclusions