A Global Random Walk on Spheres algorithm for calculating the solution and its derivatives of the drift‐diffusion‐reaction equations

摘要

A Global Random Walk on Spheres (GRWS) algorithm for calculating the solution and its derivatives of drift‐diffusion‐reaction equations in any desired set of points is suggested. The GRWS algorithm is able to find the solution and derivative fields in any family of points simultaneously, using only one ensemble of random walks which drastically decreases the computer time compared to the standard random walk‐based methods. The method in its nature is stochastic and meshless; the cost is of the order |log(ε)|/ε2 independent of the space dimension and complexity of the boundary shape, where ε is the desired accuracy. The variance, accuracy, and the cost of the method suggested are given. For illustration, we present simulation results for exciton drift‐diffusion‐recombination transport in a 3D prism and compare them with the exact solution. Calculations are also given for derivatives of a 2D diffusion problem which show the same accuracy as for the solution itself; the simulations are compared against the exact results.