Similarly to Fig. 2, we continued the fiducial SIDM simulation (light blue) from t = 10.084 Gyr with a fixed time step and different values for the accuracy of the opening criterion. We showed the ‘CDM, Δ;t_cdm’ and ‘CDM, Δ;t_cdm + α’ simulations from Fig. 2 plus three simulations with self-interactions. The simulation named ‘SIDM, Δt_CDM + α’ employs the same fixed time step as the CDM simulations (Δt ≈ 3 × 10⁻⁴ Gyr) and the more accurate opening criterion (ErrTolForceAcc = 1 × 10⁻⁴). For the simulation with the label ‘SIDM, Δt_sidm’ we use a fixed time step set by the minimum time step from the fiducial SIDM simulation (Δt ≈ 2.2 × 10⁻⁷ Gyr). Here, we use the wide opening criterion (ErrTolForceAcc = 5 × 10⁻⁴). In contrast the simulation ‘SIDM, Δt_sidm + α’ employs the stricter criterion (ErrTolForceAcc = 1 × 10⁻⁴) and the same time step (Δt ≈ 2.2 × 10⁻⁷ Gyr). We note that the evolution of the total energy is not continuous as switching to the more accurate gravity also leads to a slightly different estimate of the gravitational binding energy. Also, changing the size of a time step has an impact on the evolution of total energy too.