A Parallelized Monte Carlo Algorithm for the Nonlinear Poisson-Boltzmann Equation in Two Dimensions

Abstract

This paper presents the parallelization of a previously-developed two-dimensional floating random walk (FRW) algorithm for the solution of the nonlinear Poisson-Boltzmann (NPB) equation. Historically, the FRW method has not been applied efficiently to the solution of the NPB equation which can be attributed to the absence of analytical expressions for volumetric Green’s functions. Stochastic approaches to solving nonlinear equations (in particular the NPB equation) that have been suggested in literature involve an iterative solution of a series of linear problems. As a result, previous applications of the FRW method have examined only the linearized Poisson-Boltzmann equation. In our proposed approach, an approximate (yet accurate) expression for the Green’s function for the nonlinear problem is obtained through perturbation theory, which gives rise to an integral formulation that is valid for the entire nonlinear problem. As a result, our algorithm does not have any iteration steps, and thus has a lower computational cost. A unique advantage of the FRW method is that it requires no discretization of either the volume or the surface of the problem domains. Furthermore, each random walk is independent, so that the computational procedure is highly parallelizable. In previously published work, we have presented the fundamentals of our algorithm and in this paper we report the parallelization of this algorithm in two dimensions. The solution of the NPB equation has many interesting applications, including the modeling of plasma discharges, semiconductor device modeling and the modeling of biomolecules.