Ninth-Order Two-Step Methods with Varying Step Lengths

Rubayyi T. Alqahtani, Theodore E. Simos () and Charalampos Tsitouras
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Rubayyi T. Alqahtani: Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia
Theodore E. Simos: Center for Applied Mathematics and Bioinformatics, Gulf University for Science and Technology, West Mishref 32093, Kuwait
Charalampos Tsitouras: General Department, National & Kapodistrian University of Athens, GR34-400 Euripus Campus, Greece

Mathematics, 2025, vol. 13, issue 8, 1-15

Abstract: This study investigates a widely recognized ninth-order numerical technique within the explicit two-step family of methods (a.k.a. hybrid Numerov-type methods). To boost its performance, we incorporate an economical step-size control algorithm that, after each iteration, either preserves the current step length, reduces it by half, or doubles it. Any additional off-grid points needed by this strategy are computed using a local interpolation routine. Indicative numerical experiments confirm the substantial efficiency gains realized by this method. It is particularly adept at resolving differential equations with oscillatory dynamics, delivering high precision with fewer function evaluations. Furthermore, a detailed Mathematica implementation is supplied, enhancing usability and fostering further research in the field. By simultaneously improving computational efficiency and accuracy, this work offers a significant contribution to the numerical analysis community.

Keywords: initial value problem; second-order; two-step methods; step-size change control (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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