Leapfrog method |
Asynchronous leapfrog method |

Runge-Kutta second order |
Direct midpoint method (StÃ¶rmer-Verlet) |

The main observation is that the * asynchronous leapfrog method *
gives much less deformation as the normal * leapfrog method *.
The system under consideration is the
Kepler oscillator
which describes the radial motion in the Kepler problem.
The simulation covers 10 revolutions in an eliptic orbit
of numerical excenticity ε = 0.22. There are 40 integration steps
per revolution, and for each step the states of 200 oscilators are updated.
The phase space locations of these oscillators mark initially an oval
curve. The dynamics of the system is discrete time integration with one of
four integrators (appearing as titles to the subfigures), combined with backward
evolution with the exact dynamics
(which in our case is available from Kepler's equation).