Hah, my high school memories - a graphical gravity simulator project for my CS class, leaking energy due to the use of a very naive integrator (the Euler method for ODEs).

You might want to try Stormer-Cowell, it's sort of a gold standard in orbit simulation.

UPDATED: Hmm, now that I think of it, this would probably profit more from a variable step method. Some of the close approaches seem to be really close for a fixed-step integrator.