A fast direct solver for a fourth order finite difference scheme for Poisson’s equation on the unit disc in polar coordinates

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• 作者：Bernard Bialecki ; Lyndsey Wright
• 关键词：Poisson’s equation ; Polar coordinates ; Finite difference scheme ; Local truncation error ; Matrix decomposition algorithm ; Fast Fourier transforms ; 65M06 ; 65M12 ; 65M15 ; 65M22
• 刊名：Numerical Algorithms
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：70
• 期：4
• 页码：727-751
• 全文大小：444 KB
• 参考文献：1.Bialecki, B., Fairweather, G., Karageorghis, A.: Matrix decomposition algorithms for modified spline collocation for Helmholtz problems. SIAM J. Sci. Comput. 24, 1733-753 (2003)MathSciNet CrossRef MATH
  2.Documentation Center: Fast Fourier Transform (FFT). The MathWorks, Inc (2013). http://?www.?mathworks.?com/?help/?matlab/?math/?fast-fourier-transform-fft.?html
  3.Documentation Center: fft. The MathWorks, Inc (2013). http://?www.?mathworks.?com/?help/?matlab/?ref/?fft.?html
  4.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, 791-97 (1994)MathSciNet CrossRef MATH
  5.Lai, M.C.: A simple compact fourth-order Poisson solver on polar geometry. J. Comput. Phys. 182, 337-45 (2002)CrossRef MATH
  6.Lai, M.C., Wang, W.C.: Fast direct solvers for Poisson equation on 2D polar and spherical geometries. Numer. Methods Partial Differ. Equ. 18, 56-8 (2002)MathSciNet CrossRef MATH
  7.Mittal, R.C., Gahlaut, S.: High-order finite-difference schemes to solve Poisson’s equation in polar coordinates, IMA. J. Numer. Anal. 11, 261-70 (1991)MathSciNet CrossRef MATH
  8.Samarskii, A.A., Andreev, W.B.: Difference Methods for Elliptic Equations, Nauka, Moskow. (In Russian) (1976)
  9.Samarskii, A.A., Nikolaev, E.S.: Numerical Methods for Grid Equations I. Direct Methods, Birkh?user Verlag, Basel, Boston, Berlin (1989)CrossRef MATH
  10.Van Loan, C.: Computational Frameworks for the Fast Fourier Transform, Frontiers in Applied Mathematics. SIAM, Philadelphia (1992)CrossRef
  
• 作者单位：Bernard Bialecki (1)
  Lyndsey Wright (2)
  
  1. Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO, 80401, USA
  2. Instar Engineering, 6901 S. Pierce St. Suite 200, Littleton, CO, 80128, USA
  
• 刊物类别：Computer Science
• 刊物主题：Numeric Computing
  Algorithms
  Mathematics
  Algebra
  Theory of Computation
  
• 出版者：Springer U.S.
• ISSN：1572-9265

文摘

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. Keywords Poisson’s equation Polar coordinates Finite difference scheme Local truncation error Matrix decomposition algorithm Fast Fourier transforms