A new Green's function Monte Carlo algorithm for the solution of the three-dimensional nonlinear Poisson–Boltzmann equation: Application to the modeling of plasma sheath layers

The objective of this paper is the exposition of a recently-developed Green's function Monte Carlo (GFMC) algorithm for the solution of nonlinear partial differential equations and its application to the modeling of the plasma sheath region around a spherical conducting object, carrying a potential and moving at low speeds through a partially ionized medium. The plasma sheath is modeled in equilibrium through the GFMC solution of the nonlinear Poisson–Boltzmann (NPB) equation. The traditional GFMC-based approaches for the solution of nonlinear equations involve the iterative solution of a series of linear problems. Over the last several years, one of the authors of this paper, K. Chatterjee, has been developing a philosophically different approach, where the linearization of the equation of interest is not required and hence there is no need for iteration. Instead, an approximate expression for the Green's function is obtained using perturbation theory, which is used to formulate the random walk equations within the problem sub-domains, where the random-walker makes its walks. On the other hand, as a trade-off, the dimensions of these sub-domains have to be restricted by the limitations imposed by perturbation theory. The greatest advantage of this approach is the ease and simplicity of parallelization stemming from the lack of the need for iteration, as a result of which the parallelization procedure is identical to the parallelization procedure for the GFMC solution of a linear problem. However, the premise of the algorithm is novel and rigorous mathematical justification has to be established in the future. The application areas of interest include the communication blackout problem during a space vehicle's re-entry into the atmosphere and electromagnetic propagation through the atmosphere/ionosphere in UHF/GPS applications.