So here's an odd thing. I made a puzzle game for Android where I use Euler, and some of the puzzles require the divergance to work. It has caused a curious phenomenon where some people think the game is wrong, yet the game at no stage mentions what the rules of motion are.

I have seen people talk about the puzzles in terms of planets, gravity, massless bodies etc. when in-fact they are just circles that move towards or away from other circles. Theoretically people who can identifiy the Euler intergrator should be able to go "aha! I know what to do now" but so many get hung up on the idea that it is wrong.

The close encounters will cause issues even in integrators more sophisticated than Euler. One body of research used regularizing integrators to change variables when the bodies get close, to avoid numerical instabilities.

This integrator is used in my three body app, to remove the issues due to close encounters. An old blog post of mine has the details: http://nbodyphysics.com/blog/2015/12/08/threebody-2-0/