Comparison of the time steps used in the simulations of Figure 14 with either the explicit integration in Yang & Johansen (2014) or our algorithm in this work. Both the time step ﹩{\rm{\Delta }}t﹩ and the evolution time t are normalized by the local orbital period P. For the explicit integration, two possibilities are shown; one is limited by 100% T_drag, and the other is limited by 20% T_drag, where T_drag is the characteristic time of the mutual drag force between the gas and the particles. See Section 5.1.