Figure 15

Explanation of the asymmetry at small errors. We show distributions of the positive (solid, black) and negative (dashed, red) errors in the total energy (top row), total angular momentum (second row), escaper velocity (third row), binary semimajor axis fourth row) and eccentricity (bottom row). This is shown for different algorithms: Leapfrog (left column), standard Hermite (middle column) and Hermite with \(P(EC)^{n}\) method (right column, \(n=3\)). Each method implements a shared, adaptive time-step criterion according to equation (1), with a time-step parameter \(\eta= 2^{-7}\). Each of these three integrators has a different asymmetry in the conservation of energy and angular momentum. By propagating these asymmetric errors as a small perturbation to the converged solution, we can estimate the resulting asymmetry in the derived quantities. These estimated error distributions are also given separately for the positive (dot-dash, blue) and negative (dotted, green) errors. We observe that the estimated error distributions are located at the asymmetry in the empirical error distributions. The asymmetry at small errors is caused by a bias in the integrator.