Nine-Stage Runge–Kutta–Nyström Pairs Sharing Orders Eight and Six

Hadeel Alharbi, Kusum Yadav, Rabie A. Ramadan, Houssem Jerbi, Theodore E. Simos () and Charalampos Tsitouras
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Hadeel Alharbi: College of Computer Science and Engineering, University of Hail, Ha’il 81481, Saudi Arabia
Kusum Yadav: College of Computer Science and Engineering, University of Hail, Ha’il 81481, Saudi Arabia
Rabie A. Ramadan: College of Computer Science and Engineering, University of Hail, Ha’il 81481, Saudi Arabia
Houssem Jerbi: Department of Industrial Engineering, College of Engineering, University of Hail, Ha’il 81481, Saudi Arabia
Theodore E. Simos: Center for Applied Mathematics and Bioinformatics, Gulf University for Science and Technology, Mubarak Al-Abdullah 32093, Kuwait
Charalampos Tsitouras: General Department, National & Kapodistrian University of Athens, Euripus Campus, 34400 Psachna, Greece

Mathematics, 2024, vol. 12, issue 2, 1-14

Abstract: We explore second-order systems of non-stiff initial-value problems (IVPs), particularly those cases where the first derivatives are absent. These types of problems are of significant interest and have applications in various domains, such as astronomy and physics. Runge–Kutta–Nyström (RKN) pairs stand out as highly effective methods of addressing these IVPs. In order to create a pair with eighth and sixth orders, we need to address a certain known set of equations concerning the coefficients. When constructing such pairs for use in double-precision arithmetic, we often need to meet various conditions. Primarily, we aim to maintain small coefficient magnitudes to prevent a loss of accuracy. Nevertheless, in the context of quadruple precision, we can tolerate larger coefficients. This flexibility enables us to establish pairs with eighth and sixth orders that exhibit significantly reduced truncation errors. Traditionally, these pairs are constructed to go through eight stages per step. Here, we propose using nine stages per step. Then we have available more coefficients in order to further reduce truncation errors. As a result, we construct a novel pair that, as anticipated, achieves superior performance compared to equivalent-order pairs in various significant problem scenarios.

Keywords: initial value problem; Runge–Kutta–Nyström; quadruple precision (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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