A Phase- and Amplification-Fitted 5(4) Diagonally Implicit Runge-Kutta-Nystrom Pair for Oscillatory Systems

In this work, a phase- and amplification-fitted 5(4) diagonally implicit pair of Runge-Kutta-Nystrom methods with four stages is developed to solve special second-order ordinary differential equations (ODEs), whose solutions are oscillatory. The local truncation errors of the formulas of the new pair are given, showing that the orders are maintained, with respect to the original formulas. The stability of the new method is studied, showing that the developed formulas present non-vanishing intervals of absolute stability. The numerical examples considered clearly show the robustness of the developed embedded pair over other diagonally implicit Runge-Kutta-Nystrom (DIRKN) pairs in the literature.

Automated summary

In this study, a phase- and amplification-fitted 5(4) diagonally implicit pair of Runge-Kutta-Nystrom methods with four stages is developed for solving special second-order ordinary differential equations (ODEs) with oscillatory solutions. The local truncation errors of the new pair's formulas are given, showing that the orders are maintained compared to the original formulas. The stability of the new method is investigated, demonstrating the existence of non-vanishing intervals of absolute stability. The numerical examples considered clearly demonstrate the robustness of the developed embedded pair over other diagonally implicit Runge-Kutta-Nystrom (DIRKN) pairs in existing literature.