**Celestial Mechanics on a Graphing Calculator**

## 3. The Runge-Kutta algorithm

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 `w`_{0} to produce
more eccentric elliptical orbits, even this powerful
method starts to strain.

For `w`_{0}=.2, step
sizes of .1 and .05 lead to non-physical solutions.

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@ Copyright 2001, American Mathematical Society.