Mesh-free stochastic algorithms for systems of drift–diffusion–reaction equations and anisotropic diffusion flux calculations

Abstract

We suggest in this paper two new random walk based stochastic algorithms for solving high-dimensional PDEs for domains with complicated geometrical structure. The first one, a Random Walk on Spheres (RWS) algorithm is developed for solving systems of coupled drift–diffusion–reaction equations where the random walk is living both on randomly sampled spheres and inside the relevant balls. The second method suggested solves transient anisotropic diffusion equations, where the random walk is carried out on random rectangular parallelepipeds inside the domain. The two methods are mesh-free both in space and time, and are well applied to solve high-dimensional problems with complicated domains. The algorithms are based on tracking the trajectories of the diffusing particles exactly in accordance with the probabilistic distributions derived from the explicit representation of the relevant Green functions for a sphere and a parallelepiped. They can be conveniently used not only for the solutions, but also for a direct calculation of fluxes to any part of the boundary without calculating the whole solution in the domain. Applications to exciton transport in semiconductors and related cathodoluminescence imaging of a set of randomly distributed threading dislocations are presented.