Collocation Runge–Kutta–Nyström methods for solving second-order initial value problems. Issue 12 (2nd December 2021)

• Record Type: Journal Article
• Authors: Hoang, N. S.
• Is Part Of: International journal of computer mathematics. Volume 98:Issue 12(2021)
• Page Start: 2423
• Page End: 2444
• Publication Date: 2021-12-02
• Subjects: Runge–Kutta–Nyström -- superconvergence -- collocation -- second-order IVP; 65L05 -- 65L06 -- 65L20 -- 65L60; Computers -- Periodicals; Numerical analysis -- Periodicals; Automation -- Periodicals; 004.0151
• DOI: 10.1080/00207160.2021.1900567
• Languages: English
• ISSNs: 0020-7160

Abstract

ABSTRACT: A class of modified collocation Runge–Kutta–Nyström (RKN) methods for solving second-order initial value problem y ″ = f ( t, y, y ′ ), y ( t 0 ) = y 0, y ′ ( t 0 ) = y 0 ′ has been formulated and studied in the paper. The new methods are applicable to a larger class of second-order initial value problems compared to the classical collocation RKN methods and they reduce to the classical methods when y ′ is absent from the equation. Superconvergence for the new methods is attained when the set of collocation points satisfies orthogonality conditions. We proved that an s -stage modified collocation RKN method is of accuracy order of at least s for any set of collocation parameters c and at most 2 s when c are Gauss points. The stability function and the stability of some modified RKN methods have also been investigated. Numerical experiments are included to demonstrate the advantage of the new methods.