Symplectic Integration with Variable Stepsize

There is considerable evidence suggesting that for Hamiltonian systems of ordinary differential equations it is better to use numerical integrators that preserve the symplectic property of the ow of the system, at least for long-time integrations. We present what we believe is a practical way of doing symplectic integration with variable stepsize. Another idea, orthogonal to variable stepsize, is the use of diierent stepsizes for diierent processes. In applications like classical molecular dynamics it has been found advantageous, because of the wide range of timescales present, to use multiple time steps, in which diierent terms of the right-hand side are integrated with diierent time steps. It is possible to use multiple time steps so as to preserve the symplectic property. Our proposal for variable stepsize makes use of the machinery of multiple time stepping. The idea is to decompose an interaction into a sum of interactions: shorter-range interactions to be sampled at frequent intervals and smoother longer-range interactions to be sampled at less frequent intervals.