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

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.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Shiferaw, A. and Mittal, R. (2014) High Accurate Fourth-Order Finite Difference Solutions of the Three Dimensional Poisson’s Equation in Cylindrical Coordinate. American Journal of Computational Mathematics, 4, 73-86. doi: 10.4236/ajcm.2014.42007.

 [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