The discrete-particle effects in particle-in-cell (PIC) simulations can numerically enhance the thermalization of collisionless plasmas, such that they can potentially change the dynamic properties of the simulated plasma system. The simulation results show that the numerical fluctuation induced by discrete-particle effects can be remedied by taking ensemble average over many computer runs to obtain the Landau damping rate, which is consistent with the theoretical estimation. But the nonlinear phase trapping can only be recovered from the numerical noise by using a reasonable number of macro-particle number in a Debye region. Moreover, both Krook-type and head-on collision models are implemented in the PIC simulation for studying the Landau damping in collisional plasmas. The convergence of numerical results due to discrete-particle effects in PIC simulations will be discussed in the paper.