﻿ Fast Finite Difference Solutions of the Three Dimensional Poisson’s Equation in Cylindrical Coordinates

American Journal of Computational Mathematics
Vol.3 No.4(2013), Article ID:40966,7 pages DOI:10.4236/ajcm.2013.34045

Fast Finite Difference Solutions of the Three Dimensional Poisson’s Equation in Cylindrical Coordinates

Alemayehu Shiferaw, R. C. Mittal

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

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

Copyright © 2013 Alemayehu Shiferaw, R. C. Mittal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received November 1, 2013; revised December 1, 2013; accepted December 8, 2013

Keywords: Poisson’s Equation; Hockney’s Method; Thomas Algorithm

ABSTRACT

In this work, the three-dimensional Poisson’s equation in cylindrical coordinates system with the Dirichlet’s boundary conditions in a portion of a cylinder for is solved directly, by extending the method of Hockney. The Poisson equation is approximated by second-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.

1. Introduction

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

(1)

which is often encountered in heat and mass transfer theory, fluid mechanics, elasticity, electrostatics, and other areas of mechanics and physics. In particular, the Poisson equation describes stationary temperature distribution in the presence of thermal sources or sinks in the domain under consideration.

The analytic solution for the three-dimensional Poisson’s equation in cylindrical coordinate system is much more complicated and tedious because of the complexity of the nature of the problems and their geometry, and the availability of appropriate methods. To solve Poisson’s equation in polar and cylindrical coordinates geometry, different approaches and numerical methods using finite difference approximation have been developed. For instance, Chao [1] developed a direct solver method for the electrostatic potential in a cylindrical region; Chen [2] developed a direct spectral collocation Poisson solver for several different domains including a part of a disk, an annulus, a unit disk, and a cylinder using the weighted interpolation technique and non-classical collocation points;

Christopher [3] developed a solution method in an annulus using conformal mapping and Fast Fourier Transform; Kalita and Ray [4] have developed a high order compact scheme on a circular cylinder to solve their problem on incompressible viscous flows; Lai and Wang [5] developed a fast direct solvers for Poisson’s equation on 2D polar and spherical coordinates based on FFT; Swarztrauber and Sweet [6] developed a direct solution of the discrete Poisson equation on a disk in the sense of least squares; Mittal and Gahlaut [7,8] developed high order finite difference schemes to solve Poisson’s equation in cylindrical symmetry; Tan [9] 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 [10] derived fourthorder difference schemes for the solution of the Poisson equation which occurs in problems of heat transfer; Iyengar and Goyal [11] developed a multigrid method in cylindrical coordinates system; Lai and Tseng [12] 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; Xu et al. [13] developed a parallel three-dimensional Poisson solver in cylindrical coordinate system for the electrostatic potential of a charged particle beam in a circular, which used Fourier expansions in the longitudinal and azimuthal directions, and spectral element discretization in the radial direction, and some other developments had also been observed. The need to obtain the best solution for the Poisson’s equation is still in progress.

In this paper, we develop a second-order finite difference approximation scheme and solve the resulting large algebraic system of linear equations systematically using block tridiagonal system [14] and extend the Hockney’s method [15] 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.

Figure 1. Portion of a Cylinder.

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

Here, and where and For, using the central difference approximation scheme that

(3)

Truncating higher order differences of (3) and substituting (3) in (2), we have

(4)

Let, , and.

Multiplying both sides of (4) by, rearranging and simplifying further, we get

(5)

When there are two or more space dimensions the band width is larger and the number of operations goes up and thus the computation for the solution is not such an easy task.

The system of Equations in (5) is a linear sparse system, and thereby saving on 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 a few variables. Now we use these advantages.

Consider Equation (5) first in the direction, next in the direction and lastly in the direction, and hence Equation (5) can be put in matrix form as

(6)

where

it has M blocks and each block is of order NP.

(7)

For,

(8)

For,

(9)

is a circulant matrix;

Both matrices (8) and (9) are of order N; and is of order N.

has P blocks and

is of order N

has P blocks and

is of order N.

,

and

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

and

We write (6) a

(10)

3. Extended Hockney’s Method

Observe that the matrix is a real symmetric matrix and hence its eigenvalues and eigenvectors can easily be obtained.

for and for ,

.

Let be an eigenvector of corresponding to the eigenvalue and the matrix

be a modal matrix of,

such that and

The modal matrix Q is defined by

for and

, where for

and,

Let be a matrix of order NP.

Thus satisfy the property that since and due to the matrix is symmetric, we have (say);

and since both and are diagonal matrices.

Let

(11)

where,

;

and

Pre-multiplying Equation (10) by and applying (11), we get

(12)

Now from each Equation of (14) we collect the first equations and put them as one group of equation

(13)

Collect as the first set of equations by putting in Equation (15), for and and

(14a)

Again consider the second equations by putting, and get

and

(14b)

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

and

(14c)

All these set of Equations (14a)-(14c) are tri-diagonal ones and hence we solve for by using Thomas algorithm. With the help of (11) again we get all and this solves (5) 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 any values of M, N, and P. Here results are reported at some randomly taken points from Tables 1 to 7.

Example 1. Consider with the boundary conditions

The analytical solution is

and the results of this example are shown in Table 1.

Example 2. Consider with the boundary conditions

The analytical solution is and results of this example are shown in Table 2

Example 3. Consider

with the boundary conditions

and

The analytical solution is

Table 1. Maximum absolute error of example 1.

Table 2. Maximum absolute error of example 2.

and results of this example are shown in Table 3

Example 4. Consider

where

, and

The analytical solution is

This problem was considered as one test problem by Iyengar and Goyal [11] and their result and our are found to be the same for h = 1/8, but their method cannot be applied for non-uniform values, and. We have shown the results of this example in Table 4.

Table 3. Maximum absolute error of example 3.

Table 4. Maximum absolute error of example 4.

Example 5. Consider with the boundary conditions

,

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

Example 6. Consider with the boundary conditions

,

and

The analytical solution is and results of this example are shown in Table 6.

Example 7. Consider

with the boundary conditions

The analytical solution is

and results of this example are shown in Table 7. Here, we have displayed only for some points which are taken randomly, but we can show the results at any point inside the cylinder.

5. Conclusion

In this work, first we apply Hockney’s method in order to reduce (5) as a tri-diagonal matrix and after that all the computations directly rely on the Thomas Algorithm. By

Table 5. Maximum absolute error of example 5.

Table 6. Maximum absolute error of example 6.

Table 7. Maximum absolute error of example 7.

doing this, we saved the number of computation and computational time. The method is shown to produce good results.

REFERENCES

1. E. H Chao, S. F. Paul, R. C. Davidson and K. S. Fine, “Direct Numerical Solution of Poisson’s Equation in Cylindrical (r, z) Coordinates,” Princeton University, Princeton, 1997.
2. H. Chen, Y. Su and B. D. Shizgal, “A Direct Spectral Collocation Poisson Solver in Polar and Cylindrical Coordinates,” Journal of Computational Physics, Vol. 160, No. 2, 2000, pp. 453-469. http://dx.doi.org/10.1006/jcph.2000.6461
3. I. Christopher, G. Knorr, M. Shoucri and P. Bertrand, “Solution of the Poisson Equation in an Annulus,” Journal of Computational Physics, Vol. 131, No. 2, 1997, pp. 323-326. http://dx.doi.org/10.1006/jcph.1996.5598
4. J. C. Kalita and K. K. Ray, “A Transformation-Free HOC Scheme for Incompressible Viscous Flows Past an Impulsively Started Circular Cylinder,” Journal of Computational Physics, Vol. 228, No. 14, 2009, pp. 5207-5236. http://dx.doi.org/10.1016/j.jcp.2009.04.016
5. M. C. Lai and W. C. Wang, “Fast Direct Solvers for Poisson Equation on 2D Polar and Spherical Geometries,” Numerical Methods for Partial Differential Equations, Vol. 18, No. 1, 2002, pp. 56-68.
6. P. N. Swarztrauber and R. A. Sweet, “The Direct Solution of the Discrete Poisson Equation on a Disk,” SIAM Journal on Numerical Analysis, Vol. 10, No. 5, 1973, pp. 900- 907.
7. R. C. Mittal and S. Gahlaut, “High Order Finite Difference Schemes to Solve Poisson’s Equation in Cylindrical Symmetry,” Communications in Applied Numerical Methods, Vol. 3, 1987, pp. 457-461.
8. R. C. Mittal and S. Gahlaut, “High-Order Finite Differences Schemes to Solve Poisson’s Equation in Polar Coordinates,” IMA Journal of Numerical Analysis, Vol. 11, No. 2, 1991, pp. 261-270. http://dx.doi.org/10.1093/imanum/11.2.261
9. C. S. Tan, “Accurate Solution of three Dimensional Poisson’s Equation in Cylindrical Coordinate by Expansion in Chebyshev Polynomials,” Journal of Computational Physics, Vol. 59, No. 1, 1985, pp. 81-95. http://dx.doi.org/10.1016/0021-9991(85)90108-1
10. S. R. K. Iyengar and R. Manohar, “High Order Difference Methods for Heat Equation in Polar Cylindrical Polar Cylindrical Coordinates,” Journal of Computational Physics, Vol. 77, No. 2, 1988, pp. 425-438. http://dx.doi.org/10.1016/0021-9991(88)90176-3
11. S. R. K. Iyengar and A. Goyal, “A Note on Multigrid for the Three-Dimensional Poisson Equation in Cylindrical Coordinates,” Journal of Computational and Applied Mathematics, Vol. 33, No. 2, 1990, pp. 163-169. http://dx.doi.org/10.1016/0377-0427(90)90366-8
12. M. C. Lai and J. M. Tseng, “A Formally Fourth-Order Accurate Compact Scheme for 3D Poisson Equation in Cylindrical and Spherical Coordinates,” Journal of Computational and Applied Mathematics, Vol. 201, No. 1, 2007, pp. 175-181. http://dx.doi.org/10.1016/j.cam.2006.02.011
13. J. Xu, P. N. Ostroumov and J. Nolen, “A Parallel 3D Poisson Solver for Space Charge Simulation in Cylindrical Coordinates,” Computer Physics Communications, Vol. 178, No. 4, 2008, pp. 290-300. http://dx.doi.org/10.1016/j.cpc.2007.09.008
14. M. A. Malcolm and J. Palmer, “A Fast Method for Solving a Class of Tri-Diagonal Linear Systems,” Communications of the ACM, Vol. 17, No. 1, 1974, pp. 14-17. http://dx.doi.org/10.1145/360767.360777
15. R. W. Hockney, “A Fast Direct Solution of Poisson Equation Using Fourier Analysis,” JACM, Vol. 12, No. 1, 1965, pp. 95-113. http://dx.doi.org/10.1145/321250.321259