Comparaison of the Runge-Kutta scheme and the chosen scheme.
The undamped pendulum
So, for an undamped problem, we can see that the Runge-Kutta scheme is not accurate. And moreover, it gives non-realistic results, as seen below.
The damped pendulum
With the introduction of the damping force, all schemes become stable. Nevertheless, the results given by the Runge-Kutta scheme are very dependant of the time step. So, the precision of the calculation is dependant of the number of points of the calculation and by extension, of its duration.
Hereunder, are charts obtained with a damped pendulum (a=0.2). The calculations have been done with the Runge-Kutta and the chosen schemes for two different time steps (Dt=0.1 and Dt=0.01).
With those results, we can see that the Runge-Kutta scheme is still very dependant of the time steps, whereas it is not the case for the chosen scheme.
The scheme used is the one which allows quick and stable calculations for all the configurations of the pendulum.