AJCMAmerican Journal of Computational Mathematics2161-1203Scientific Research Publishing10.4236/ajcm.2014.42007AJCM-43980ArticlesPhysics&Mathematics High Accurate Fourth-Order Finite Difference Solutions of the Three Dimensional Poisson’s Equation in Cylindrical Coordinate lemayehuShiferaw1*RameshChand Mittal1Department of Mathematics, Indian Institute of Technology, Roorkee, India* E-mail:abelhaim@gmail.com(LS);20032014040273865 November 20135 December 2013 16 December 2013© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

In this work, by extending the method of Hockney into three dimensions, the Poisson’s equation in cylindrical coordinates system with the Dirichlet’s boundary conditions in a portion of a cylinder for is solved directly. The Poisson equation is approximated by fourth-order finite differences and the resulting large algebraic system of linear equations is treated systematically in order to get a block tri-diagonal system. The accuracy of this method is tested for some Poisson’s equations with known analytical solutions and the numerical results obtained show that the method produces accurate results.

Poisson’s Equation; Tri-Diagonal Matrix; Fourth-Order Finite Difference Approximation; Hockney’s Method; Thomas Algorithm
1. Introduction

The three-dimensional Poisson’s equation in cylindrical coordinates is given by

has a wide range of application in engineering and science fields (especially in physics).

In physical problems that involve a cylindrical surface (for example, the problem of evaluating the temperature in a cylindrical rod), it will be convenient to make use of cylindrical coordinates. For the numerical solution of the three dimensional Poisson’s equation in cylindrical coordinates system, several attempts have been made in particular for physical problems that are related directly or indirectly to this equation. For instance, Lai  developed a simple compact fourth-order Poisson solver on polar geometry based on the truncated Fourier series expansion, where the differential equations of the Fourier coefficients are solved by the compact fourth-order finite difference scheme; Mittal and Gahlaut  have developed high order finite difference schemes of secondand fourthorder in polar coordinates using a direct method similar to Hockney’s method; Mittal and Gahlaut  developed a secondand fourth-order finite difference scheme to solve Poisson’s equation in the case of cylindrical symmetry; Alemayehu and Mittal  have derived a second-order finite difference approximation scheme to solve the three dimensional Poisson’s equation in cylindrical coordinates by extending Hockney’s method; Tan  developed a spectrally accurate solution for the three dimensional Poisson’s equation and Helmholtz’s equation using Chebyshev series and Fourier series for a simple domain in a cylindrical coordinate system; Iyengar and Manohar  derived fourth-order difference schemes for the solution of the Poisson equation which occurs in problems of heat transfer; Iyengar and Goyal  developed a multigrid method in cylindrical coordinates system; Lai and Tseng  have developed a fourth-order compact scheme, and their scheme relies on the truncated Fourier series expansion, where the partial differential equations of Fourier coefficients are solved by a formally fourth-order accurate compact difference discretization. The need to obtain the best solution for the three dimensional Poisson’s equation in cylindrical coordinates system is still in progress.

In this paper, we develop a fourth-order finite difference approximation scheme and solve the resulting large algebraic system of linear equations systematically using block tridiagonal system   and extend the Hockney’s method   to solve the three dimensional Poisson’s equation on Cylindrical coordinates system.

2. Finite Difference Approximation

Consider the three dimensional Poisson’s equation in cylindrical coordinates given by and the boundary condition

where is the boundary of and is

and

Consider figure 1 as the geometry of the problem. Let be discretized at the point and for simplicity write a point as and as.

Assume that there are M points in the direction of, N points in and P points in the directions to form the mesh, and let the step size along the direction of be, of be and be.

Here

Where and.

When is an interior or a boundary point of (2), then the Poisson’s equation becomes singular and to take care of the singularity a different approach will be taken. Thus in this paper we consider only for the case.

Using the approximations that

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