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

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.