Distributed Source Scheme to solve the classical form of Poisson equation using 3-D Finite-Difference Method for improved accuracy and unrestricted source position

Abstract

The Finite-Difference Method (FDM) despite being old and simple is not being used as rigorously as its counterpart Finite Element Method (FEM) for solving partial differential equations. This study aims to examine and improve the accuracy of FDM by eliminating significant sources of error. Since an expression for exact potential from the most accurate Method of Moment (MoM) is available in 3-D electrostatics, the classical form of Poisson equation is chosen for this study such that the error due to boundary conditions can be eliminated. The error due to the source term in the Poisson equation is studied with a single source and different grid densities by applying FDM in 3-D. Since the error is only present in the immediate surroundings of the source, a Distributed Source Scheme (DSS) has been proposed to reduce error due to the source term. A modified DSS i.e., Truncated Distributed Source Scheme (TDSS) is applied for the practical implementation of DSS. The maximum error in FDM when the source term is present has been reduced from 8.151% to 0.00091% with DSS. With the application of TDSS, it is shown that the maximum error can be maintained well below 0.1% for truncation values n > 15. The error due to source at off-center and off-grid positions was computed using TDSS and the maximum error is observed to be less than 0.05% and 0.01%, respectively. With off-grid error being low due to TDSS, it is shown that sources in TDSS can now take any position irrespective of grid nodes, which is forbidden in FDM with an average maximum error of 0.026%. It is also shown that DSS can also be used to find the charge distribution for a given potential distribution. While still maintaining the simplicity, improved accuracy and unrestricted source positions are achieved in FDM with exact boundary conditions using DSS.