A novel class of Runge-Kutta-Nyström pairs sharing orders 8(6)

Yu He; Jianing Yang; Theodore E. Simos; Charalampos Tsitouras; Yu He; Jianing Yang; Theodore E. Simos; Charalampos Tsitouras

doi:10.3934/math.2024237

AIMS Mathematics

2024, Volume 9, Issue 2: 4882-4895. doi: 10.3934/math.2024237

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A novel class of Runge-Kutta-Nyström pairs sharing orders 8(6)

1.
School of Computer Sci. and Artificial Intelligence, Huanghuai Univ., Zhumadian, China, 463000
2.
Henan Key Laboratory of Smart Lighting, Zhumadian, China, 463000
3.
Henan Int. Joint Lab. of Behavior Optimization Control for Smart Robots, Henan, China, 463000
4.
Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
5.
Center for Applied Mathematics and Bioinformatics, Gulf University for Science and Technology, West Mishref 32093, Kuwait
6.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung City 40402, Taiwan
7.
Data Recovery Key Laboratory of Sichun Province, Neijiang Normal University, Neijiang 641100, China
8.
Section of Mathematics, Department of Civil Engineering, Democritus University of Thrace, 67100 Xanthi, Greece
9.
Department of Mathematics, University of Western Macedonia, 50100 Kozani, Greece
10.
General Department, National & Kapodistrian University of Athens, 34400 Euripus Campus, Greece

Received: 23 October 2023 Revised: 28 December 2023 Accepted: 10 January 2024 Published: 23 January 2024
MSC : 65L06

We are examining a second-order system of non-stiff Initial Value Problems (IVP), focusing on a scenario where the first derivatives are not present. In the realm of solving IVPs, Runge-Kutta-Nyström (RKN) pairs have proven to be highly effective. In order to achieve a pair with eighth and sixth order accuracy, we need to find a solution to a well-defined set of equations regarding the coefficients. Traditionally, pairs are constructed to go through eight stages per step. However, we propose a novel approach with nine stages per step, which enables the creation of pairs with orders $ 8 $ and $ 6 $ that have notably smaller truncation errors. Our paper introduces a new pair that, as expected, outperforms existing pairs of the same orders in a range of important problems.
- initial Value Problem,
- second order,
- Runge-Kutta-Nyström
Citation: Yu He, Jianing Yang, Theodore E. Simos, Charalampos Tsitouras. A novel class of Runge-Kutta-Nyström pairs sharing orders 8(6)[J]. AIMS Mathematics, 2024, 9(2): 4882-4895. doi: 10.3934/math.2024237

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Abstract

We are examining a second-order system of non-stiff Initial Value Problems (IVP), focusing on a scenario where the first derivatives are not present. In the realm of solving IVPs, Runge-Kutta-Nyström (RKN) pairs have proven to be highly effective. In order to achieve a pair with eighth and sixth order accuracy, we need to find a solution to a well-defined set of equations regarding the coefficients. Traditionally, pairs are constructed to go through eight stages per step. However, we propose a novel approach with nine stages per step, which enables the creation of pairs with orders $ 8 $ and $ 6 $ that have notably smaller truncation errors. Our paper introduces a new pair that, as expected, outperforms existing pairs of the same orders in a range of important problems.

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