Multi-grid Methods for Steady State Diffusion in Random Media

Abstract

We apply multi-grid methods to study steady state densities for hopping diffusion in two-dimensional random media. We show that these methods are very efficient. For our fastest algorithm, which also uses overrelaxation, CPU times increase as L2.2 in order to reach equilibrium on square lattices of size L × L with a prescribed accuracy. Compared to standard Gauss-Seidel methods, this is at least an improvement by a factor ∝ L For L = 128, the improvement is about a factor of 10 and even more than 80 if overrelaxation is added.