A fourth order implicit symmetric and symplectic exponentially fitted Runge-Kutta-Nyström method for solving oscillatory problems

Wenjuan Zhai; Bingzhen Chen

doi:10.3934/naco.2019006

Article Contents

2019, Volume 9, Issue 1: 71-84. Doi: 10.3934/naco.2019006

This issue Previous Article On construction of upper and lower bounds for the HOMO-LUMO spectral gap Next Article Semi-local convergence of the Newton-HSS method under the center Lipschitz condition

A fourth order implicit symmetric and symplectic exponentially fitted Runge-Kutta-Nyström method for solving oscillatory problems

Wenjuan Zhai^1, and
Bingzhen Chen^2, ,

1.
Department of Mathematics, Beijing Jiaotong University Haibin College, Cangzhou, China
2.
School of Science, Beijing Jiaotong University, Beijing, China

* Corresponding author: chenbingzhen6026@163.com
* Corresponding author: chenbingzhen6026@163.com

This paper was presented in the First Symposium on Machine Intelligence and Data Analytics (MIDA)-2017, Beijing, China, December 15-18, 2017.

Received: January 2018

Revised: April 2018

Published: March 2019

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Abstract

In this paper, we derive an implicit symmetric, symplectic and exponentially fitted Runge-Kutta-Nyström (ISSEFRKN) method. The new integrator ISSEFRKN2 is of fourth order and integrates exactly differential systems whose solutions can be expressed as linear combinations of functions from the set $\{\exp(λ t), \exp(-λ t)|λ∈ \mathbb{C}\}$, or equivalently $\{\sin(ω t), \cos(ω t)|λ = iω, ~ω∈ \mathbb{R}\}$. We analysis the periodicity stability of the derived method ISSEFRKN2. Some the existing implicit RKN methods in the literature are used to compare with ISSEFRKN2 for several oscillatory problems. Numerical results show that the method ISSEFRKN2 possess a more accuracy among them.

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Mathematics Subject Classification: Primary: 65L05, 65L06; Secondary: 65M20, 65N40.

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Figure 1. Periodicity regions for the method ISSEFRKN2.

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Figure 2. Maximum global error in the solution for Problem 1.

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Figure 3. Maximum global error in the solution for Problem 2.

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Figure 4. Maximum global error in the solution for problem 3 with $\varepsilon = 0$.

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Figure 5. Maximum global error in the solution for problem 3 with $\varepsilon = 10^{-3}$.

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Figure 6. Maximum global error in the solution for problem 4.

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