Monte Carlo simulation of diffusion in a spatially nonhomogeneous medium: A biased random walk on an asymmetrical lattice

Monte Carlo (MC) simulation of diffusion processes has proved to be a powerful and valuable adjunct to deterministic solutions of the diffusion equation. In its simplest one-dimensional implementation, a particle is stepped to left or right, with equal probability, a distance 2 D Δ t where D is the diffusion coefficient and Δt is the timestep. This gives accurate results if D is constant, but in the case where D is spatially dependent a systematic error occurs, as shown by comparing MC averages with deterministic solutions. Furthermore, this error does not reduce when the timestep Δt is reduced. We show that the results can be reconciled by altering both the MC stepsize and stepping probability, and give simple formulas for the correction terms that are also applicable in higher dimensions. This supplements our previous work on corrections to the Gaussian-step MC method [J. Comput. Phys. 198 (2004) 65].