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A two‐dimensional stochastic algorithm for the solution of the non‐linear Poisson–Boltzmann equation: validation with finite‐difference benchmarks
Author(s) -
Chatterjee Kausik,
Poggie Jonathan
Publication year - 2005
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1539
Subject(s) - linearization , mathematics , parallelizable manifold , discretization , poisson's equation , boltzmann equation , algorithm , mathematical optimization , statistical physics , mathematical analysis , nonlinear system , physics , quantum mechanics
This paper presents a two‐dimensional floating random walk (FRW) algorithm for the solution of the non‐linear Poisson–Boltzmann (NPB) equation. In the past, the FRW method has not been applied to the solution of the NPB equation which can be attributed to the absence of analytical expressions for volumetric Green's functions. Previous studies using the FRW method have examined only the linearized Poisson–Boltzmann equation. No such linearization is needed for the present approach. Approximate volumetric Green's functions have been derived with the help of perturbation theory, and these expressions have been incorporated within the FRW framework. A unique advantage of this algorithm 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 our previous work, we have presented preliminary calculations for one‐dimensional and quasi‐one‐dimensional benchmark problems. In this paper, we present the detailed formulation of a two‐dimensional algorithm, along with extensive finite‐difference validation on fully two‐dimensional benchmark problems. The solution of the NPB equation has many interesting applications, including the modelling of plasma discharges, semiconductor device modelling and the modelling of biomolecular structures and dynamics. Copyright © 2005 John Wiley & Sons, Ltd.