Monte Carlo simulation of diffusion in a spatially nonhomogeneous medium: correction to the Gaussian steplength

Monte Carlo (MC) simulation of diffusion processes has proved to be a powerful and valuable adjunct to deterministic solutions of the diffusion equation. For the case of a constant diffusion coefficient it is well established that a MC method using a steplength chosen from the appropriate Gaussian distribution gives accurate results. However, in the case where the diffusion coefficient is spatially dependent, straightforward modification of this method, involving replacing the constant diffusion coefficient by the spatially dependent one in the steplength formula, leads to a systematic error, as shown by comparing MC averages with deterministic solutions. Furthermore, reducing the timestep, and hence the average steplength, does not reduce this error. In this paper, we trace the source of the error and provide a simple and readily calculated correction to the Gaussian steplength that reconciles the MC and deterministic approaches.