Trajectory in the θ-coordinate for orbit RKA3 simulated with the four methods and Δt = 1 (left panel), compared to a high-resolution reference run (dashed line). The error trends in terms of the root mean square difference with respect to the reference simulation are reported for increasing numbers of integration steps (right panel), confirming the second-order character of the IMR, Hamiltonian, and modified Hamiltonian schemes.