A new Global Random Walk algorithm for calculation of the solution and its derivatives of elliptic equations with constant coefficients in an arbitrary set of points

Abstract

A new random walk based stochastic algorithm for calculation of the solution and its derivatives of high-dimensional second order elliptic equations with constant coefficients in any desired set of points is suggested. In contrast to the conventional random walk methods the new Global Random Walk (GRW) algorithm is able to find the solution in many points using only one ensemble of random walks. The method is meshless, highly efficient, the cost is of the order of |log(ε)|∕ε2 independent of the complexity of the boundary shape, where ε is the desired accuracy. The method can be used to solve nonlinear equations by applying the GRW algorithm iteratively. We demonstrate this by solving a nonlinear system of semiconductor equations. The system includes drift–diffusion equations for electrons and holes, and the Poisson equation for a potential term whose gradient enters the drift–diffusion equations as a drift velocity. The nonlinear drift–diffusion–Poisson system is solved by the iteration procedure including alternating simulation of the drift–diffusion processes and solving the Poisson equation by the GRW algorithm.