In the paper it is suggested a correction of the Bird’s algorithm in the DSMC method. It takes account of real distribution of collision events inside the time steps Δt and actual trajectories for the collided particles there thus diminishing asymptotical order of the error in time evolution from O(Δt) to O((Δt) ² ). However the structure of the algorithm turned out to be more complicated and parallel implementation of it becomes a new problem. As some solution of this problem the corrected DSMC method in its domain decomposited version was applied for simulation of unsteady flow in a two-dimensional cavity with a moving bottom. The numerical results of this simulation presented in the paper show a noticeable artificial acceleration of changes for system parameters by the uncorrected version in comparison with the corrected one as the former locates all collision events from the previous collisional time step at one time point at the beginning of the space motional step. The difference between their results in calculation of the mean velocity circulation along the identical loops in its time development increases proportionally to value of the time step O(Δt) used and to mean molecular collision number on the distance from the source of perturbation to a measuring point.