solving ordinary differential equations fig comparison of the runge-kutta-fehlberg adaptive method against the euler-cromer scheme for long time simulation ( periodsnote that the time_intervals_per_period argument refers to the time points where we want the solution these points are also the ones used for numerical computations in the odespy eulercromer solverwhile the odespy rkfehlberg solver will use an unknown set of time points since the time intervals are adjusted as the method runs one can easily look at the points actually used by the method as these are available as an array solver t_all (but plotting or examining the points requires modifications inside the compare methodfigure shows computational example where the runge-kutta-fehlberg method is clearly superior to the euler-cromer scheme in long time simulationsbut the comparison is not really fair because the runge-kutta-fehlberg method applies about twice as many time steps in this computation and performs much more work per time step it is quite complicated task to compare two so different methods in fair way so that the computational work versus accuracy is scientifically well reported the th-order runge-kutta method the th-order runge-kutta method (rk is clearly the most widely used method to solve odes its power comes from high accuracy even with not so small time steps