Accuracy of lattice Monte Carlo simulation of biased diffusion models

The accuracy of lattice Monte Carlo (LMC) simulation of biased diffusion models is of great importance as far as the simulation credibility is concerned. It is known that the fixed time step LMC algorithm can reproduce the mean and the variance of the particle displacement exactly for all discrete time steps. Thereby, we propose to use the skewness and other quantities to measure the accuracy. For the one-dimensional fixed time step LMC simulation, we obtain an explicit expression for the skewness and find that the algorithm always produces a negative skewness that converges to zero in the long-time limit when the velocity is positive. It is proved that the skewness is inversely proportional to the square root of the number of simulation steps and the first step error only depends on the Péclet number. We further discuss several other measures of the accuracy of the approximation based on appropriately defined mean-square errors, leading to interesting, unexpected results. The accuracy measures can exhibit complicated nonmonotonic behavior and the optimal step size may depend on the measure of accuracy used.