Derivation of Three-Derivative Two-Step Runge–Kutta Methods

Xueyu Qin, Jian Yu () and Chao Yan
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Xueyu Qin: National Key Laboratory of Computational Fluid Dynamics, Beihang University, Beijing 100191, China
Jian Yu: National Key Laboratory of Computational Fluid Dynamics, Beihang University, Beijing 100191, China
Chao Yan: National Key Laboratory of Computational Fluid Dynamics, Beihang University, Beijing 100191, China

Mathematics, 2024, vol. 12, issue 5, 1-16

Abstract: In this paper, we develop explicit three-derivative two-step Runge–Kutta (ThDTSRK) schemes, and propose a simpler general form of the order accuracy conditions ( p ≤ 7 ) by Albrecht’s approach, compared to the order conditions in terms of rooted trees. The parameters of the general high-order ThDTSRK methods are determined by utilizing the order conditions. We establish a theory for the A -stability property of ThDTSRK methods and identify optimal stability coefficients. Moreover, ThDTSRK methods can achieve the intended order of convergence using fewer stages than other schemes, making them cost-effective for solving the ordinary differential equations.

Keywords: multiderivative methods; two-step Runge–Kutta methods; A -stability property; order conditions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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