The use of Monte Carlo methods for solving the diffusion equation in the case of a constant diffusion coefficient was implemented by using particles. At every time step, a constant step size was added to, or subtracted from, the coordinates of each particle with equal probability. For a spatially dependent diffusion coefficient, the naive extension of the previous method using a spatially variable step size introduces a systematic error: particles migrate in the directions of decreasing diffusivity. A correction of step sizes and stepping probabilities had been proposed and the numerical tests had given satisfactory results. A quasi-Monte Carlo method was described here for solving the diffusion equation in a spatially inhomogeneous medium. The random samples in the corrected Monte Carlo scheme were replaced by low-discrepancy point sets. In order to make a proper use of the better uniformity of these point sets, the particles were reordered according to their successive coordinates at each time step. The method was illustrated using numerical examples. In dimensions 1 and 2, it was shown that the quantum Monte Carlo approach led to improved accuracy when compared with the original Monte Carlo method using the same number of particles.

Diffusion in a Nonhomogeneous Medium: Quasi-Random Walk on a Lattice. R.El Haddad, C.Lécot, G.Venkiteswaran: Monte Carlo Methods and Applications, 2010, 16[3-4], 211-30