(color online) Temporal evolution of pulse radius (a) and pulse length (b) for electron per pulse simulated by the N-particle method (blue dotted) and our improved mean-field model (red solid). (c) and (d) are the enlarged view of the pulse length and the corresponding velocity spread as a function of time for the area marked in the black square in (b), which highlights Region 1 and Region 2. As a comparison, simulations by the other two methods (see context for details) that ignore the boundary region are also shown (pink dash-dot line for method A and green dashed line for method B). An N-particle simulation for electrons with no space-charge effect is also presented (black diamond), and the sharp change of velocity spread at the boundary shows the trivial part of “boundary kick”.