A numerical procedure to solve Poisson’s equation in spherical coordinates

Adilson Costa Da Silva,Vladimir Santos Da Costa,José Abdalla Helayël Neto

doi:10.1590/1806-9126-rbef-2021-0019

Abstract

This paper sets out to present a numerical procedure that solves Poisson’s equation in a spherical coordinate system. To discretize this equation, integration techniques at the interfaces between different regions have been carried out allowing the calculation of both the potential and the corresponding field inside and outside a charge distribution. The Gauss-Seidel method is adopted to determine the potential in each region and the results, whenever compared with the analytical solutions found in the literature, come out very satisfactory, with errors less than 1% for distances of the order of 1 × 10−14 m and, for larger distances, they never reach 4%.

Highlights

Poisson’s equation is an elliptic partial differential equation with a known non-trivial source term. This equation has a wide application in several areas of Physics and Engineering, such as Electrodynamics, Mechanics, Fluid Dynamics and the study of topological deffects
It is important to mention that, in other coordinate systems, such as cylindrical or spherical coordinates, analytic solutions may not be found for generic source distributions
An analytic solution may be obtained to Eq (1), if it is assumed that the potential is only r-dependent, that is, the potential is a function V (r)

Summary

Introduction

Poisson’s equation is an elliptic partial differential equation with a known non-trivial source term. In this method, we can perform an integration on an interface region between the distinct media to have a complete answer This technique, known as point- (or interface-) centered scheme [4], allows an arbitrary number of regions to be considered, as long as they are physically acceptable. This is possible by using Uniqueness theorem [5], which states that the potential and its derivative are both continuous at the interface between two different media.

Poisson’s Equation

The analytic solution

Getting the numerical solution

Boundary condition

Results

Concluding Comments