The close encounters will cause issues even in integrators more sophisticated than Euler. One body of research used regularizing integrators to change variables

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/ http://nbodyphysics.com/blog/2015/12/08/threebody-2-0/