It was noted that the behavior of a lattice Monte Carlo algorithm had to approach that of the continuum system that it simulated as the time step and the mesh step tended to zero. It was shown for an algorithm for unbiased particle diffusion that, if one of these two parameters remained fixed, the accuracy of the algorithm was optimal for a certain finite value of the other parameter. In one dimension, the optimal algorithm with moves to the two nearest neighbor sites reproduced the correct second and fourth moments (and minimized the error for the higher moments at long times) of the particle distribution and preserved the first two moments of the first-passage time distributions. In two and three dimensions, the same level of accuracy required simultaneous moves along two axes ("diagonal" moves). Such moves attempting to cross an impenetrable boundary had to be projected along the boundary, rather than simply rejected. Also treated was the case of absorbing boundaries. The relationship between optimally accurate lattice Monte Carlo algorithms was considered, as well as a particular case of lattice Boltzmann algorithms for simulating diffusion.

Optimizing the Accuracy of Lattice Monte Carlo Algorithms for Simulating Diffusion. M.V.Chubynsky, G.W.Slater: Physical Review E, 2012, 85[1], 016709