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

Genet Mekonnen Assef

doi:10.7176/apta/79-01

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

Highlights

Numerous and diverse physical circumstances, and other mathematical problems are modeled in terms of partial differential equations (PDEs)
As we observe from the table the solution is done for only the first iteration at n=0 the error we get is too small but to generalize the stability, consistence and the convergence of the method it is difficult so we can check for maximum number of iteration using mat-lab code
5 Conclusion and Recommendation 5.1 Conclusion The 2-dimensional Poisson equation is discretized in order to transform the calculus partial differential equation to algebra discrete equation

Summary

1.Introduction

Numerous and diverse physical circumstances, and other mathematical problems are modeled in terms of partial differential equations (PDEs). Finite difference method is used as direct conversion of the partial differential equation from continuous function and operator into their discretely-sampled counterpart This converts the entire problem into a system of linear equations that may be readily solved by means of matrix inversion, Jacobi, Gauss-elimination, successive over-relaxation method. 1.2.2 Specific Objectives The principal objectives of this study are: To determine the convergence, stability and consistence of finite difference method for two dimensional Poisson equations with uniform and non-uniform mesh size. 3.3 Finite difference discretization of two dimensional Poisson equation with uniform mesh size finite difference method is used as direct conversion of the 2-D Poisson equation from continuous function and operator into their discretely-sampled counterpart This converts the entire problem into a system of linear equations that may be readily solved by means of iterative method. The exact solution is we have spares matrix A as flows: We can use different iterative methods to solve this matrix but this study we use matrixinverse method by using mat-lab we get the following solution

Approximate solution

For i

Conclusion and Recommendation