A domain-decomposition method to implement electrostatic free boundary conditions in the radial direction for electric discharges

At high pressure electric discharges typically grow as thin, elongated filaments. In a numerical simulation this large aspect ratio should ideally translate into a narrow, cylindrical computational domain that envelops the discharge as closely as possible. However, the development of the discharge is driven by electrostatic interactions and, if the computational domain is not wide enough, the boundary conditions imposed to the electrostatic potential on the external boundary have a strong effect on the discharge. Most numerical codes circumvent this problem by either using a wide computational domain or by calculating the boundary conditions by integrating the Green's function of an infinite domain. Here we describe an accurate and efficient method to impose free boundary conditions in the radial direction for an elongated electric discharge. To facilitate the use of our method we provide a sample implementation. Finally, we apply the method to solve Poisson's equation in cylindrical coordinates with free boundary conditions in both radial and longitudinal directions. This case is of particular interest for the initial stages of discharges in long gaps or natural discharges in the atmosphere, where it is not practical to extend the simulation volume to be bounded by two electrodes. Program summary: Program Title: poisson_sparse_fft.py Program Files doi: http://dx.doi.org/10.17632/x7f6czrnsh.1 Licensing provisions: CC By 4.0 Programming language: Python Nature of problem: Electric discharges are typically elongated and their optimal computational domain has a large aspect ratio. However, the electrostatic interactions within the discharge volume may be affected by the boundary conditions imposed to the Poisson equation. Computing these boundary conditions using a direct integration of Green's function involves either heavy computations or a loss of accuracy. Solution method: We use a Domain Decomposition Method to efficiently impose free boundary conditions to the Poisson equation. This code provides a stand-alone example implementation. © 2018 The Author(s)