Solution of Two Dimensional Poisson Equation Using Finite Difference Method with Uniform and Non-uniform Mesh Size

Abstract

This study focus on the finite difference approximation of two dimensional Poisson equation with uniform and non-uniform mesh size. The Poisson equation with uniform and non-uniform mesh size is a very powerful tool for modeling the behavior of electro-static systems, but unfortunately may not be solved analytically for very simplified models. Consequently, numerical simulation must be utilized in order to model the behavior of complex geometries with practical value. In most engineering problems are also coming from steady reaction-diffusion and heat transfer equation, in elasticity, fluid mechanics, electrostatics etc. the solution of meshing grid is non-uniform and uniform where fine grid is identified at the sensitive area of the simulation and coarse grid at the normal area.The discretization of non-uniform grid is done using Taylor expansion series. The purpose of such discretization is to transform the calculus problem to numerical form (as discrete equation). Therefore, in this study the two dimensional Poisson equation is discretazi with uniform and non-uniform mesh size using finite difference method for the comparison purpose. More over we also examine the ways that the two dimensional Poisson equation can be approximated by finite difference over non-uniform meshes, As result we obtain that for uniformly distributed gird point the finite difference method is very simple and sufficiently stable and converge to the exact solution whereas in non-uniformly distributed grid point the finite difference method is less stable, convergent and time consuming than the uniformly distributed grid points.

Keywords: Finite difference method, two dimensional Poisson equations, Uniform mesh size, Non-uniform mesh size, Convergence, Stability, Consistence.

DOI: 10.7176/APTA/79-01

Publication date:September 30^th 2019