Abstract
We present a fourth order finite difference scheme for solving Poisson’s equation on the unit disc in polar coordinates. We use a half-point shift in the r direction to avoid approximating the solution at r = 0. We derive our scheme from analysis of the local truncation error of the standard second order finite difference scheme. The resulting linear system is solved very efficiently (with cost almost proportional to the number of unknowns) using a matrix decomposition algorithm with fast Fourier transforms.
Similar content being viewed by others
References
Bialecki, B., Fairweather, G., Karageorghis, A.: Matrix decomposition algorithms for modified spline collocation for Helmholtz problems. SIAM J. Sci. Comput. 24, 1733–1753 (2003)
Documentation Center: Fast Fourier Transform (FFT). The MathWorks, Inc (2013). http://www.mathworks.com/help/matlab/math/fast-fourier-transform-fft.html
Documentation Center: fft. The MathWorks, Inc (2013). http://www.mathworks.com/help/matlab/ref/fft.html
Jain, M.K., Jain, R.K., Krishna, M.: A fourth-order difference scheme for quasilinear Poisson equation in polar co-ordinates. Commun. Numer. Methods in Eng. 10, 791–797 (1994)
Lai, M.C.: A simple compact fourth-order Poisson solver on polar geometry. J. Comput. Phys. 182, 337–345 (2002)
Lai, M.C., Wang, W.C.: Fast direct solvers for Poisson equation on 2D polar and spherical geometries. Numer. Methods Partial Differ. Equ. 18, 56–68 (2002)
Mittal, R.C., Gahlaut, S.: High-order finite-difference schemes to solve Poisson’s equation in polar coordinates, IMA. J. Numer. Anal. 11, 261–270 (1991)
Samarskii, A.A., Andreev, W.B.: Difference Methods for Elliptic Equations, Nauka, Moskow. (In Russian) (1976)
Samarskii, A.A., Nikolaev, E.S.: Numerical Methods for Grid Equations I. Direct Methods, Birkhäuser Verlag, Basel, Boston, Berlin (1989)
Van Loan, C.: Computational Frameworks for the Fast Fourier Transform, Frontiers in Applied Mathematics. SIAM, Philadelphia (1992)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bialecki, B., Wright, L. A fast direct solver for a fourth order finite difference scheme for Poisson’s equation on the unit disc in polar coordinates. Numer Algor 70, 727–751 (2015). https://doi.org/10.1007/s11075-015-9971-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-015-9971-z
Keywords
- Poisson’s equation
- Polar coordinates
- Finite difference scheme
- Local truncation error
- Matrix decomposition algorithm
- Fast Fourier transforms