Implicit symmetric and symplectic exponentially fitted modified Runge\u2013Kutta\u2013Nystr\xf6m methods for solving oscillatory problems

Bing Zhen Chen,Wen Juan Zhai

doi:10.1186/s13660-018-1915-4

Bing Zhen Chen, Wen Juan Zhai

Open Access

https://doi.org/10.1186/s13660-018-1915-4

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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)}, lambdainmathbb{C}, or equivalently {sin(omega t),cos(omega t)} when lambda=iomega, omega inmathbb{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.

Highlights

In plenty of applied sciences such as celestial mechanics, astrophysics, chemistry, electronics, molecular dynamics, and so forth, the following second-order ODEs initial value problems (IVP) often arise:y = f (t, y), y(t0) = y0, y (t0) = y0, t ∈ [0, tend], (1)whose solutions exhibit an oscillatory character
For Problems 2 and 3, we find that ISSEFMRKN2 is much more accurate and efficient than our methods considered in this paper
Like the existing exponentially fitted RKN (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

Summary

Introduction

In plenty of applied sciences such as celestial mechanics, astrophysics, chemistry, electronics, molecular dynamics, and so forth, the following second-order ODEs initial value problems (IVP) often arise:. We deduce a class of exponentially fitted RKN methods which integrate exactly second-order differential systems whose solutions can be expressed as linear combinations of the set of functions {exp(λt), exp(–λt), λ ∈ C}, or equivalently, from {sin(ωt), cos(ωt), ω ∈ R} with λ = iω, i2 = –1. 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 obtain the symplectic conditions for RKN formula (2). We have obtained an implicit symmetric and symplectic exponentially fitted Runge–Kutta–Nyström method whose coefficients are given by θ =±. When z → 0, ISSMEFRKN2 reduces to SSRKN of order in [24]

Numerical experiments

Conclusions