BUILDING A BETTER LEAPFROG

In stellar dynamical computer simulations, as well as other types of s imulations using particles, time step size is often held constant in o rder to guarantee a high degree of energy conservation. In many applic ations, allowing the time step size to change in time can offer a grea t saving in computational cost, but variable-size time steps usually i mply a substantial degradation in energy conservation. We present a '' meta-algorithm'' for choosing time steps in such a way as to guarantee time symmetry in any integration scheme, thus allowing vastly improve d energy conservation for orbital calculations with variable time step s. We apply the algorithm to the familiar leapfrog scheme, and general ize to higher order integration schemes, showing how the stability pro perties of the fixed-step leapfrog scheme can be extended to higher or der, variable-step integrators such as the Hermite method. We illustra te the remarkable properties of these time-symmetric integrators for t he case of a highly eccentric elliptical Kepler orbit and discuss appl ications to more complex problems.