Fifth Order Runge-Kutta-Nystr\"om Methods with Complex Coefficients

TL;DR

This paper introduces fifth order Runge-Kutta-Nyström methods with complex coefficients, emphasizing those with positive real parts to avoid negative timesteps, and highlights their improved accuracy for real-variable problems.

Contribution

It develops new fifth order methods with complex coefficients that have positive real parts, enhancing applicability and accuracy for specific differential equations.

Findings

Methods with complex coefficients have purely imaginary leading error terms.

Positive real part coefficients make methods suitable for problems prohibiting negative steps.

Error expansion indicates increased effective order for real variables.

Abstract

We present fifth order Runge-Kutta-Nystr\"om methods, where we allow the timestep coefficients to assume complex values. Among the methods with complex timesteps, we focus on the ones with the coefficients that have positive real parts. This property makes them suitable for problems where a negative coefficient is not acceptable. In addition, the leading order terms in the error expansion of these methods are purely imaginary, effectively increasing the order of the methods by one for real variables.