fig the last of periods of oscillations by the th-order runge-kutta method the algorithm we first just state the four-stage algorithmunc un fonc fqnc fnnc ( where tnc nc nc tnc fnnc un fqnc tnc fonc ( ( ( application we can run the same simulation as in figs and for periods the last periods are shown in fig the results look as impressive as those of the euler-cromer method implementation the stages in the th-order runge-kutta method can easily be implemented as modification of the osc_heun py code alternativelyone can use the osc_odespy py code by just providing the argument odespy_methods[odespy rk to the compare function derivation the derivation of the th-order runge-kutta method can be presented in pedagogical way that brings many fundamental elements of numerical discretization techniques together and that illustrates many aspects of "numerical thinkingwhen constructing approximate solution methods