My comments on the accuracy of the PIC method: The PIC method is more accurate than the semi-Lagrangian continuous method because the PIC method uses the Monte-Carlo method to evaluate the high-dimension phase-space integral, which is more accurate than the corresponding methods used in the semi-Lagrangian algorithm, which uses traditional regular-grids based methods to evaluate the phase space integral.
**wrong or unclear**Because of the discrete representation of continuous media (a marker representing many physical particles), the PIC method usually gives rise to considerable fluctuations in the solution**. At present, I do not fully understand why PIC approach gives rise to numerical noise while the continuum approach does not seem to have this problem.== >Update: I think now I understand the reason: The fluctuation in the number of sampling points per spatial cell gives rise to the noise in the results. However, this noisy result is not necessarily less accurate than a smooth result (a bigger error may be hidden in a smooth result).