Functionally fitted Runge-Kutta-Nystr\"{o}m methods

TL;DR

This paper extends the collocation framework to functionally fitted Runge-Kutta-Nyström methods, enabling exact integration of certain second-order equations and revealing superconvergence properties under specific conditions.

Contribution

The paper introduces a collocation-based analysis for FRKN methods, highlighting their exactness for specific solutions and conditions for superconvergence.

Findings

FRKN methods can integrate certain second-order equations exactly.

Superconvergence occurs when collocation points meet orthogonality conditions.

Stability analysis of FRKN methods is provided.

Abstract

We have shown previously that functionally fitted Runge-Kutta (FRK) methods can be studied using a convenient collocation framework. Here, we extend that framework to functionally fitted Runge-Kutta-Nystr\"om (FRKN) methods, shedding further light on the fact that these methods can integrate a second-order differential equation exactly if its solution is a combination of certain basis functions, and that superconvergence can be obtained when the collocation points satisfy some orthogonality conditions. An analysis of their stability is also conducted.