High Accurate Fourth-Order Finite Difference Solutions of the Three Dimensional Poisson’s Equation in Cylindrical Coordinate ()
1. Introduction
The three-dimensional Poisson’s equation in cylindrical coordinates 
is given by
(1)
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 [1] 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 [2] 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 [3] developed a secondand fourth-order finite difference scheme to solve Poisson’s equation in the case of cylindrical symmetry; Alemayehu and Mittal [4] 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 [5] 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 [6] derived fourth-order difference schemes for the solution of the Poisson equation which occurs in problems of heat transfer; Iyengar and Goyal [7] developed a multigrid method in cylindrical coordinates system; Lai and Tseng [8] 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 [9] [10] and extend the Hockney’s method [9] [11] 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
(2)
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
(3)
(4)