Functionally-fitted explicit pseudo two-step Runge-Kutta-Nystr\"om methods

TL;DR

This paper introduces a new class of functionally-fitted explicit pseudo two-step Runge-Kutta-Nyström methods for second-order problems, demonstrating improved accuracy and efficiency through theoretical proofs and numerical experiments.

Contribution

It develops the FEPTRKN methods with proven order properties and superconvergence, extending existing EPTRKN methods with enhanced performance for specific problem classes.

Findings

FEPTRKN methods achieve step and stage order equal to the number of stages.

Superconvergence occurs under certain orthogonality conditions on collocation parameters.

Numerical results show FEPTRKN outperform EPTRKN on suitable problems.

Abstract

A general class of functionally-fitted explicit pseudo two-step Runge-Kutta-Nystr\"{o}m (FEPTRKN) methods for solving second-order initial value problems has been studied. These methods can be considered generalized explicit pseudo two-step Runge-Kutta-Nystr\"{o}m (EPTRKN) methods. We proved that an $s$-stage FEPTRKN method has step order $p = s$ and stage order $r = s$ for any set of distinct collocation parameters $(c_i)_{i=1}^s$. Supperconvergence for the accuracy orders 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 FEPTRKN method can attain accuracy order $p = s + 3$. Numerical experiments have shown that the new FEPTRKN methods work better than do EPTRKN methods on problems whose solutions can be well approximated by the functions in bases on which these FEPTRKN methods are developed.