Finite Difference Method for Solving Poisson’s Equation in Two Dimensions with Alternating Direction Implicit Method

Abstract

This paper presents some numerical techniques for the solution of two dimensional Poisson’s equations. The discrete approximation of Poisson’s equations is based on Finite difference method over a regular domain 16 R"> . In this research five point difference approximations is used for Poisson’s equation .To solve the resulting finite difference approximation basic iterative methods; Jacobi, Gauss-Seidal and successive over relaxation (SOR) have been used.  Several model problems of Poisson’s equations are solved by the iterative methods to identify the efficiency of the iterative methods. And we study the convergence, consistence and stability of the schemes. Alternating direction implicit (ADI) method is a finite difference method (FDM) to solve partial differential equations (PDEs). We employ ADI method to solve Poisson’s equations for dirichlet boundary conditions. The discretized equation is modified near the boundaries to incorporate arbitrary domain shapes. The ADI method performs much better than traditional iterative methods because of usage of efficient tri-diagonal solvers.

Keywords: Iteration, Discretized , Finite Difference method (FDM), Alternating direct implicit method (ADI), Poisson’s equation, stability and convergence.

DOI: 10.7176/APTA/85-01

Publication date: November 30^th 2021