Bit-Reversible Version of Milne's Fourth-Order Time-Reversible Integrator for Molecular Dynamics

We point out that two of Milne's fourth-order integrators are well-suited to bit-reversible simulations. The fourth-order method improves on the accuracy of Levesque and Verlet's algorithm and simplifies the definition of the velocity $v$ and energy $e = (q^2 + v^2)/2$ . ( We use this one-dimensional oscillator problem as an illustration throughout this paper ). Milne's integrator is particularly useful for the analysis of Lyapunov ( exponential ) instability in dynamical systems, including manybody molecular dynamics. We include the details necessary to the implementation of Milne's Algorithms.