The Runge-Kutta algorithm (strictly speaking the fourth-order R-K algorithm; see example) allows much better accuracy than Euler's method. Their relative efficiency is like that of Simpson's method and left-hand sums for approximating integrals, algorithms to which they are closely related. It was published by Carle Runge (1856-1927) and Martin Kutta (1867-1944) in 1901.
Euler's method and 4th order Runge-Kutta, applied to the restricted 2-body problem with the same initial conditions.
The Runge-Kutta method easily accomplishes in 30 steps what Euler's method could not do in 1000. Even though every Runge-Kutta step is computationally the equivalent of 4 Euler steps, the savings are enormous.
But when we decrease w0 to produce more eccentric elliptical orbits, even this powerful method starts to strain.
For w0=.2, step sizes of .1 and .05 lead to non-physical solutions.