Generalized explicit pseudo two-step Runge-Kutta-Nystr\"{o}m methods for solving second-order initial value problems

TL;DR

This paper introduces a new class of explicit pseudo two-step Runge-Kutta-Nyström methods (GEPTRKN) for efficiently solving second-order initial value problems, with improved accuracy and suitability for parallel computation.

Contribution

The paper develops a generalized class of GEPTRKN methods with higher order accuracy and super-convergence properties, enhancing efficiency for non-stiff problems and parallel computing.

Findings

GEPTRKN methods achieve step order of s and stage order of s.

Super-convergence to order s+2 under orthogonality conditions.

Numerical experiments show superior performance over classical methods.

Abstract

A class of explicit pseudo two-step Runge-Kutta-Nystr\"{o}m (GEPTRKN) methods for solving second-order initial value problems $y'' = f(t,y,y')$, $y(t_0) = y_0$, $y'(t_0)=y'_0$ has been studied. This new class of methods can be considered a generalized version of the class of classical explicit pseudo two-step Runge-Kutta-Nystr\"{o}m methods. %The new methods will be denoted by GEPTRKN methods. We proved that an $s$-stage GEPTRKN method has step order of accuracy $p=s$ and stage order of accuracy $r=s$ for any set of distinct collocation parameters $(c_i)_{i=1}^s$. Super-convergence for order of accuracy 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 GEPTRKN method can attain order of accuracy $p=s+2$. Numerical experiments have shown that the new methods work better than classical methods for solving non-stiff problems even on sequential computing environments. By their structures, the new methods will be much more efficient when implemented on parallel computers.