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.
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 [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
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
ReferencesLai, 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.2002.7172Mittal R.C and Gahlaut, S. ,et al. (1987)High Order Finite Difference Schemes to Solve Poisson’s Equation in Cylindrical Symmetry Communications in Applied Numerical Methods 3, 457-461.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/11.2.261Alemayehu S. and Mittal, R.C. ,et al. (2013)Fast Finite Difference Solutions of the Three Dimensional Poisson’s Equation in Cylindrical Coordinates American Journal of Computational Mathematics 3, 356-361.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)90108-1Iyengar, 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)90176-3Iyengar, 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)90366-8Lai, 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.2006.02.011Smith, G.D. (1985) Numerical Solutions of Partial Differential Equations: Finite Difference Methods. Third Edition. Oxford University Press, New York.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/10.1145/360767.360777Hockney, 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
