General Runge–Kutta–Nyström Methods for Linear Inhomogeneous Second-Order Initial Value Problems

Nadiyah Hussain Alharthi, Rubayyi T. Alqahtani, Theodore E. Simos () and Charalampos Tsitouras
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Nadiyah Hussain Alharthi: Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 90950, Riyadh 11623, Saudi Arabia
Rubayyi T. Alqahtani: Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 90950, Riyadh 11623, Saudi Arabia
Theodore E. Simos: Center for Applied Mathematics and Bioinformatics, Gulf University for Science and Technology, West Mishref, Hawally 32093, Kuwait
Charalampos Tsitouras: General Department, National and Kapodistrian University of Athens, Euripus Campus, GR-34400 Euboea, Greece

Mathematics, 2025, vol. 13, issue 17, 1-18

Abstract: In this paper, general Runge–Kutta–Nyström (GRKN) methods are developed and analyzed, tailored for second-order initial value problems of the form y ″ = L y ′ + M y + g ( t ) , where L , M ∈ R n × n are constant matrices with n ≥ 1 . The construction of embedded pairs of orders 6 ( 4 ) and 7 ( 5 ) , suitable for adaptive integration strategies, is emphasized. By utilizing rooted tree theory and recent simplifications for linear inhomogeneous systems, symbolic order conditions are derived, and efficient schemes are designed through algebraic and evolutionary techniques. Numerical tests verify the superiority of our new derived pairs. In particular, this work introduces novel embedded GRKN pairs with reduced-order conditions that exploit the linearity and structure of the underlying system, enabling the construction of low-stage, high-accuracy integrators. The methods incorporate FSAL (First Same As Last) formulations, making them computationally efficient. They are tested on representative physical systems in one, two, and three dimensions, demonstrating notable improvements in efficiency and accuracy over existing high-order RKN methods.

Keywords: initial value problem; linear inhomogeneous; Runge–Kutta–Nyström; order conditions; differential evolution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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