Generalized explicit pseudo two-step Runge-Kutta-Nyström methods for solving second-order initial value problems

2022/07/17 by N. S. Hoang, Hoang, Nguyen S.
Computer Science · Mathematics · #65L05 #65L06 #65L20 #65L60 #Differential Equations and Numerical Methods #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2207.08260

openalex publication_date 2022/07/17 · openalex created_date 2022/07/21 · openalex updated_date 2026/07/28

Abstract

A class of explicit pseudo two-step Runge-Kutta-Nyström (GEPTRKN) methods for solving second-order initial value problems y'' = f(t,y,y'), y(t₀) = y₀, y'(t₀)=y'₀ has been studied. This new class of methods can be considered a generalized version of the class of classical explicit pseudo two-step Runge-Kutta-Nyström methods. %The new methods will be denoted by GEPTRKN methods. We proved that an s-stage GEPTRKN method has step order of accuracy p=s and stage order of accuracy r=s for any set of distinct collocation parameters (c_i)_i=1^s. Super-convergence for order of accuracy of these methods can be obtained if the collocation parameters (c_i)_i=1^s satisfy some orthogonality conditions. We proved that an s-stage GEPTRKN method can attain order of accuracy p=s+2. Numerical experiments have shown that the new methods work better than classical methods for solving non-stiff problems even on sequential computing environments. By their structures, the new methods will be much more efficient when implemented on parallel computers.

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• Functionally-fitted explicit pseudo two-step Runge-Kutta-Nyström methods
• Multi-Order Runge-Kutta Methods or how to numerically solve initial value problems of any order
• Fifth Order Runge-Kutta-Nyström Methods with Complex Coefficients
• Functionally fitted Runge-Kutta-Nyström methods
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