Kodogo
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I've been tooling around with this, and found that the method known in some quarters as "Leapfrog" gives pretty good results even with modest computing power. The method is the same as described in Feynman Lectures volume 1 chapter 9. It seems to be the same as what some people call Verlet. There's an excellent discussion here:
http://young.physics.ucsc.edu/115/leapfrog.pdf
For a free iPad app to show the results of the method see here:
https://itunes.apple.com/nz/app/orbit-simulator/id1048345753?mt=8
The main problem I see is that highly elliptical orbits do start to precess, depending on the time step you use. Also when the speed gets very high the orbits develop sharp corners. To get round this you's need smaller steps or a method with adaptive step size, but Leapfrog is well good enough just to play around with.
http://young.physics.ucsc.edu/115/leapfrog.pdf
For a free iPad app to show the results of the method see here:
https://itunes.apple.com/nz/app/orbit-simulator/id1048345753?mt=8
The main problem I see is that highly elliptical orbits do start to precess, depending on the time step you use. Also when the speed gets very high the orbits develop sharp corners. To get round this you's need smaller steps or a method with adaptive step size, but Leapfrog is well good enough just to play around with.