﻿ High Accurate Fourth-Order Finite Difference Solutions of the Three Dimensional Poisson’s Equation in Cylindrical Coordinate

American Journal of Computational Mathematics
Vol.4 No.2(2014), Article ID:43980,14 pages DOI:10.4236/ajcm.2014.42007

High Accurate Fourth-Order Finite Difference Solutions of the Three Dimensional Poisson’s Equation in Cylindrical Coordinate

Alemayehu Shiferaw, Ramesh Chand Mittal

Department of Mathematics, Indian Institute of Technology, Roorkee, India

Email: abelhaim@gmail.com, mittalrc@gmail.com

Received 5 November 2013; revised 5 December 2013; accepted 16 December 2013

ABSTRACT

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.

Keywords: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

(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)

Figure 1. Portion of a cylinder.

(5)

Now using (3), (4) and (5), we get (Refer the work of Mittal and Ghalaut in [2] )

From (1) consider only the approximation of the sum of the first and the third terms, that is, the sum of

and

(6)

where

Again from (1) consider only the approximation of the sum of the first and the fourth terms, that is, the sum of

and, and we get

(7)

Once again from (1) consider only the approximation of the sum of the second and the fourth terms, that is, the sum of and; to get

(8)

Again taking the approximation of the term by

(9)

Equation (9) implying that

(10)

Now letting and adding (6), (7), (8) and twice of (10), we get

(11)

Now choose and consider the following terms in (11)

(12)

Again we can write the term in (12) as

(13)

Using (12), (13), and multiplying both sides of (11) by and rearranging and simplifying further, we get

(14)

where

The system of equations in (14) is a linear sparse system, and thereby when solving we save both work and storage compared with a general system of equations. Such savings are basically true of finite difference methods: they yield sparse systems because each equation involves only few variables.

To solve equation (14), consider first in the direction, next in the direction and lastly in the direction, and thus (14) can be written in matrix form as

(15)

where

and it has blocks and each is of order.

,

are of order.

For the domain

,

For the domain,

and are the same as in the domain.

Here in, the matrices and are circulant matrices of order; and

, and

such that each represents a known boundary values of and values of, and

, and

Thus, we write (15) as

(16)

3. Extended Hockney’s Method

Observe that matrices and are real symmetric matrices and hence their eigenvalues and eigenvectors can easily be obtained as

For

, ,

,

and for

, ,

,

Let be an eigenvector of and corresponding to the eigenvalue and; and matrix be a modal matrix of and, such that

The modal matrix Q is defined by

, for; where

for

Let be a matrix of order; thus satisfy since.

Since and are symmetric matrices, we have

where

where

where

Let

, (17)

where

,;

and

Pre-multiplying equation (16) by and applying (17), we get

(18)

Now from each equation of (18) we collect the first equations and put them as one group of equation

(19)

Now put in equation (19) and collect the entire first set of equations, for and to get

and                       (20a)

Again consider the second equations by putting, and get

and                      (20b)

Continuing in this manner and finally considering the last equations for, we obtain

and                       (20c)

All these set of equations (20a)-(20c) are tri-diagonal ones and hence we solve for by using Thomas algorithm. With the help of (17) again we get all and this solves (14) as desired. By doing this we generally reduce the number of computations and computational time.

4. Numerical Results

In order to test the efficiency and adaptability of the proposed method, computational experiments are done on some selected problems that may arise in practice, for which the analytical solutions of are known to us. The computed solutions are found for all grid points for any values of and. Here results are reported at some randomly taken mesh points in terms of the absolute maximum error from Table 1 to 7.

Example 1. Consider with the boundary conditions,

, and

The analytical solution is and the computed results of this example are shown in table 1.

Example 2. Consider with the boundary conditions

,

, , and

The analytical solution is and the computed results of this example are shown in table 2.

Example 3. Consider with the boundary conditions

, ,

Table 1. Maximum absolute error of example 1.

Table 2. Maximum absolute error of example 2.

,

The analytical solution is and the computed results of this example are shown in table 3.

Example 4. Consider with the boundary conditions

, , and

The analytical solution is and the computed results of this example are shown in table 4.

Example 5 Consider, where with the boundary conditions

,

The analytical solution is and the computed results of this example are shown in Table 5.

Table 3. Maximum absolute error of example 3.

Table 4. Maximum absolute error of example 4.

Table 5. Maximum absolute error of example 5.

This example was considered by M.C. Lai [1] as a test problem and our results are better than their results in terms of accuracy. For instance, for (8,16,16) the maximum absolute error in their result is 9.1438e-004 and while ours is 3.28689e-004.

Example 6 Consider, where with the boundary conditions

,; and

The analytical solution is and the computed results are shown in table 6.

Example 5.7 Consider where with the boundary conditions

,;

The analytical solution is and the computed results of this example are shown in table 7.

Table 6. Maximum absolute error of example 6.

Table 7. Maximum absolute error of example 7.

5. Conclusions

In this work, we have transformed the three dimensional Poisson’s equation in cylindrical coordinates system into a system of algebraic linear equations using its equivalent fourth-order finite difference approximation scheme. The resulting large number of algebraic equation is, then, systematically arranged in order to get a block matrix. By extending Hockney’s method to three dimensions, we reduced the obtained matrix into a block tridiagonal matrix, and each block is solved by the help of Thomas algorithm. We have successfully implemented this method to find the solution of the three dimensional Poisson’s equation in cylindrical coordinates system and it is found that the method can easily be applied and adapted to find a solution of some related applied problems. The method produced accurate results considering double precision. This method is direct and allows considerable savings in computer storage as well as execution speed.

Therefore, the method is suitable to apply to some three dimensional Poisson’s equations.

References

1. Lai, M.C. (2002) A Simple Compact Fourth-Order Poisson Solver on Polar Geometry. Journal of Computational Physics, 182, 337-345. http://dx.doi.org/10.1006/jcph.2002.7172
2. Mittal, R.C and Gahlaut, S. (1987) High Order Finite Difference Schemes to Solve Poisson’s Equation in Cylindrical Symmetry. Communications in Applied Numerical Methods, 3, 457-461.
3. Mittal, R.C. and Gahlaut, S. (1991) High-Order Finite Differences Schemes to Solve Poisson’s Equation in Polar Coordinates. IMA Journal of Numerical Analysis, 11, 261-270. http://dx.doi.org/10.1093/imanum/11.2.261
4. Alemayehu, S. and Mittal, R.C. (2013) Fast Finite Difference Solutions of the Three Dimensional Poisson’s Equation in Cylindrical Coordinates. American Journal of Computational Mathematics, 3, 356-361.
5. Tan, C.S. (1985) Accurate Solution of Three Dimensional Poisson’s Equation in Cylindrical Coordinate by Expansion in Chebyshev Polynomials. Journal of Computational Physics, 59, 81-95. http://dx.doi.org/10.1016/0021-9991(85)90108-1
6. Iyengar, S.R.K. and Manohar, R. (1988) High Order Difference Methods for Heat Equation in Polar Cylindrical Polar Cylindrical Coordinates. Journal of Computational Physics, 77, 425-438. http://dx.doi.org/10.1016/0021-9991(88)90176-3
7. Iyengar, S.R.K. and Goyal, A. (1990) A Note on Multigrid for the Three-Dimensional Poisson Equation in Cylindrical Coordinates. Journal of Computational and Applied Mathematics, 33, 163-169. http://dx.doi.org/10.1016/0377-0427(90)90366-8
8. Lai, M.C. and Tseng, J.M. (2007) A formally Fourth-Order Accurate Compact Scheme for 3D Poisson Equation in Cylindrical and Spherical Coordinates. Journal of Computational and Applied Mathematics, 201, 175-181. http://dx.doi.org/10.1016/j.cam.2006.02.011
9. Smith, G.D. (1985) Numerical Solutions of Partial Differential Equations: Finite Difference Methods. Third Edition. Oxford University Press, New York.
10. Malcolm, M.A. and Palmer, J. (1974) A Fast Method for Solving a Class of Tri-Diagonal Linear Systems. Communications of Association for Computing Machinery, 17, 14-17. http://dx.doi.org/10.1145/360767.360777
11. Hockney, R.W. (1965) A Fast Direct Solution of Poisson Equation Using Fourier Analysis. Journal of Alternative and Complementary Medicine, 12, 95-113. http://dx.doi.org/10.1145/321250.321259