A Parallelized 3d Floating Random-walk Algorithm for the Solution of the Nonlinear Poisson-boltzmann Equation

This paper presents a new three-dimensional floating random-walk (FRW) algorithm for the solution of the Nonlinear Poisson-Boltzmann (NPB) equation. The FRW method has not been previously used in the numerical solution of the NPB equation (and other nonlinear equations) because of the non-availability of analytical expressions for volumetric Green’s functions. In the past, numerical studies using the FRW method have examined only the linearized Poisson-Boltzmann equation, producing solutions that are only accurate for small values of the potential. No such linearization is required for this algorithm. An approximate expression for a volumetric Green’s functions has been calculated with the help of a novel use of perturbation theory, and the resultant integral form has been incorporated within the FRW framework. The algorithm requires no discretization of either the volume or the surface of the problem domains, and hence the memory requirements are expected to be lower than approaches based on spatial discretization, such as finitedifference methods. Another advantage of this algorithm is that each random walk is independent, so that the computational procedure is inherently parallelizable and an almost linear increase in computational speed is expected with increase in the number of processors. We have † Adjunct Scientist at Laboratory for Electromagnetic and Electronic Systems, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA 238 Chatterjee and Poggie recently published the preliminary results for benchmark problems in one and two dimensions. In this work, we present our results for benchmark problems in three dimensions and demonstrate excellent agreement between the FRWand finite-difference based algorithms. We also present the results of parallelization of the newly developed FRW algorithm. The solution of the NPB equation has applications in diverse branches of science and engineering including (but not limited to) the modeling of plasma discharges, semiconductor device modeling and the modeling of biomolecular structures and dynamics.