Runge–Kutta–Nyström Pairs of Orders 8(6) for Use in Quadruple Precision Computations

Vladislav N. Kovalnogov, Alexander F. Matveev, Dmitry A. Generalov, Tamara V. Karpukhina, Theodore E. Simos () and Charalampos Tsitouras
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Vladislav N. Kovalnogov: Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
Alexander F. Matveev: Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
Dmitry A. Generalov: Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
Tamara V. Karpukhina: Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
Theodore E. Simos: Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
Charalampos Tsitouras: General Department, National & Kapodistrian University of Athens, Euripus Campus, 34400 Psachna, Greece

Mathematics, 2023, vol. 11, issue 4, 1-13

Abstract: The second-order system of non-stiff Initial Value Problems (IVP) is considered and, in particular, the case where the first derivatives are absent. This kind of problem is interesting since since it arises in many significant problems, e.g., in Celestial mechanics. Runge–Kutta–Nyström (RKN) pairs are perhaps the most successful approaches for solving such type of IVPs. To achieve a pair attaining orders eight and six, we have to solve a well-defined set of equations with respect to the coefficients. Here, we provide a simplified form of these equations in a robust algorithm. When creating such pairings for use in double precision arithmetic, numerous conditions are often satisfied. First and foremost, we strive to keep the coefficients’ magnitudes small to prevent accuracy loss. We may, however, allow greater coefficients when working with quadruple precision. Then, we may build pairs of orders eight and six with significantly smaller truncation errors. In this paper, a novel pair is generated that, as predicted, outperforms state-of-the-art pairs of the same orders in a collection of important problems.

Keywords: initial value problem; second order; Runge–Kutta–Nyström (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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