Symplectic integration of simple collisions : a backward error analysis

Molecular Dynamics simulations often involve the numerical integration of pair-wise particle interactions with a constant, step size method. Of primary concern in these simulations is the introduction of error in velocity statistics. We consider the simple example of the symplectic Euler method applied to two-particle collisions in one dimension governed by linear restoring force and use backward error analysis to predict, these errors. For nearly all choices of system and method parameters, the post-collision energy is not conserved and depends upon the initial conditions of the particles and the step size of the method. The analysis of individual collisions is extended to predict energy growth in systems of particles in one dimension