Investigation of Selected Versions of Fourth Order Runge-Kutta Algorithms as Simulation Tools for Harmonically Excited Duffing Oscillator

Abstract

This study aim is to investigate the properties of selected fourth order Runge-Kutta algorithms. Fifty-five versions of fourth order Runge-Kutta (RK_1, RK_2, RK_3 …, RK_55) methods; inclusive of the classical fourth order RK version, were selected. Thereafter, these versions were used to simulate, with a constant and adaptive step-size algorithm, the dynamics of the harmonically excited Duffing Oscillator over a range of parameters and initial conditions. The simulation was carried out with a FORTRAN program developed and validated by comparing the program generated Poincaré section with literature standard. The number of successive steps taken between start and end of simulation periods was recorded for each simulation run. A total of 91809 simulations were run. The number of successive steps taken between start and end of simulation periods show that significant variations exists among different versions of the same Runge-Kutta order used for seeking solution of Duffing oscillator dynamics. Ranking results by the number of successive steps showed that RK_55 is not the fastest version available, despite its popularity, as other versions including RK_17, RK_2, RK_14, RK_20, and RK_8 outperformed it. Furthermore, the version performance was observed to be highly dependent on the excitation frequency, but not on initial conditions.