High-Order Special Two-Derivative Runge–Kutta Pairs

Ibraheem Alolyan, Theodore E. Simos () and Charalampos Tsitouras
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Ibraheem Alolyan: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, 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 & Kapodistrian University of Athens, Euripus Campus, 34400 Psachna, Greece

Mathematics, 2025, vol. 13, issue 22, 1-18

Abstract: This paper presents the development and analysis of novel explicit special two-derivative Runge–Kutta (STDRK) pairs for the numerical integration of ordinary differential equations (ODEs), with a focus on achieving seventh-order accuracy and embedded fifth-order error estimation. The proposed schemes utilize both the first and second derivatives of the solution, leveraging the identity y ′ ′ = f ′ ( y ) f ( y ) , to attain high-order accuracy while minimizing the number of evaluations of the primary function f . A notable feature of the constructed methods is that they require only a single evaluation of f per step, along with five evaluations of g = f ′ f , resulting in a significant reduction in computational cost compared to classical Runge–Kutta methods. The necessary order conditions are derived via an algebraic framework based on compositions with parts not exceeding 2. A supporting Mathematica package facilitates the construction of methods of arbitrary order. A new STDRK pair of orders seven and five is derived. Numerical experiments on standard benchmark problems, including the Prothero–Robinson, Kaps, and Kepler systems, highlight the efficiency and competitive performance of the proposed schemes relative to established Runge–Kutta pairs.

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