Stabilized Integration of Hamiltonian Systems with Hard-Sphere Inequality Constraints

We consider numerical methods for resolving the dynamics of a Hamiltonian N-body problem subject to hard-sphere inequality constraints. The dynamics of these mixed systems consists of smooth flow of a Hamiltonian system between collisions with an impulsive momentum exchange at the points of collision. The inclusion of these impulses makes traditional backward error analysis inappropriate since the flow is discontinuous and cannot be interpreted using a single modified smooth Hamiltonian system. We introduce two methods which respect the underlying modified smooth Hamiltonian system through the use of a modified map and collision operator at points of collision. In numerical experiments, these new methods show dramatically improved energy conservation over long time intervals.