A High-Order Fast Direct Solver for Singular Poisson Equations

Journal of Computational Physics

Volumn 171, Issue 1, 2001, Pages 79-94

  Zhuang, Yu ^a   Sun, Xian He ^a  

^a Illinois Institute of Technology   (United States)

Author keywords

fast Fourier transform (FFT); high order discretization; Neumann boundary condition; Poisson equation; SVD

Indexed keywords

BOUNDARY CONDITIONS; FAST FOURIER TRANSFORMS; NAVIER STOKES EQUATIONS; NUMERICAL METHODS; POISSON EQUATION; WAVE PROPAGATION;

DIRECT SOLVERS; DISCRETE EQUATIONS; FAST FOURIER TRANSFORM; FOURTH-ORDER; HIGH-ORDER; HIGHER ORDER DISCRETIZATION; HIGHER-ORDER; NEUMANN BOUNDARY CONDITION; NULL SPACE; SINGULAR EQUATIONS;

SINGULAR VALUE DECOMPOSITION;

EID: 0002346969     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcph.2001.6771     Document Type: Article

References (26)

1. H. M. Atassi, Unsteady aerodynamics of vortical flows: Early and recent developments, in Aerodynamics and Aeroacoustics, edited by K. Y. Fung (World Scientific, Singapore 1994), pp. 119-169.
2. B. Bialecki and G. Fairweather, Matrix decomposition algorithms in orthogonal spline collocation for separable elliptic boundary value problems, SIAM J. Sci. Comput. 16, 330 (1995).
3. B. Bialecki and K. A. Remington, Fourier matrix decomposition methods for the least squares solution of singular Neumann and periodic Hermite bicubic collocation problems, SIAM J. Sci. Comput. 16, 431 (1995).
4. G. Birkhoff and R. Lynch, Numerical Solution of Elliptic Problems (SIAM, Philadelphia, 1984).
5. R. F. Boisvert, Families of high order accurate discretizations of some elliptic problems, SIAM J. Sci. Stat. Comput. 2, 268 (1981).
6. R. F. Boisvert, A fourth-order-accurate Fourier method for the Helmholtz equation in three dimensions, ACM Trans. Math. Software 13, 221 (1987).
7. A. Brandt and S. Ta’asan, Multigrid methods for nearly singular and slightly indefinite problems, in Lecture Notes in Mathematics 1228: Multigrid Methods II, edited by W. Hackbusch and U. Trottenberg (Springer-Verlag, Berlin/New York, 1985) pp. 100-122.
8. X. Cai, M. A. Cassarin, F. W. Elliott, and J. O. B. Widlund, Overlapping Schwarz algorithms for solving Helmholtz equations, Contemp. Math. 218, 437 (1998).
9. A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comput. 22, 745 (1968).
10. L. Collatz, The Numerical Treatment of Differential Equation, (Springer-Verlag, New York, 1960).
11. G. H. Golub, L. C. Huang, H. Simon, and W. Tang, A Fast Solver for Incompressible Navier-Stokes Equations with Finite Difference Methods. Stanford University SCCM Technical Report, Stanford University Scientific Computing and Computational Mathematics Program (1994).
12. I. Harari and E. Turkel, Accurate finite difference methods for time-harmonic wave propogation, J. Comput. Phys. 119, 252 (1995).
13. R. Hockney, A fast direct solution of Poisson’s equation using Fourier analysis, J. ACM 12, 95 (1965).
14. L. Kaufman and D. Warner, A program for solving separable elliptic equations, ACM Trans Math. Software 16, 325 (1990).
15. S. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys. 103, 16 (1992).
16. R. E. Lynch and J. R. Rice, High accuracy finite difference approximation to solutions of elliptic partial differential equations, Proc. Nat. Acad. Sci. 75, 2541 (1978).
17. L. C. McInnes, R. Susan-Resiga, D. E. Keyes, and H. M. Atassi, Additive schwarz methods with nonreflecting boundary conditions for the parallel computation of Helmholtz problems, Contemp. Math. 218, 349 (1998).
18. J. R. Rice and R. F. Boisvert, Solving Elliptic Problems Using ELLPACK (Springer-Verlag, New York, 1985).
19. U. Schumann and R. Sweet, A direct method for the solution of Poisson equation with Neumann boundary conditions on a staggered grid of arbitrary sizes, J. Comput. Phys. 20, 171 (1976).
20. Y. Shapira, Multigrid methods for 3-D definite and indefinite problems, Appl. Numer. Math. 26, 165 (1998).
21. I. Singer and E. Turkel, High order finite differnce methods for the Helmholtz equation, Comput. Meth. Appl. Mech. Eng. 163, 533 (1998).
22. X.-H. Sun and Y. Zhuang, A high-order direct solver for Helmholtz equations with Neumann boundary conditions. NAS AICASE Technical Report No. 97-11, NASA Langley Research Center Hampton, VA 23681-0001 (1997).
23. X.-H. Sun and Y. Zhuang, A highly accurate fast solver for helmholtz equations, in Proc. ACM International Conference on Supercomputing (July 1997).
24. K. Tanabe, Projection methods for solving a singular system of linear, Numer. Math. 17, 203 (1971).
25. R. Temem, Navier-Stokes Equations, Theory and Numerical Analysis (Elsevier, New York, 1984).
26. Y. Zhuang and X.-H. Sun, A high-order ADI solver for separable generalized Helmholtz equations, Advances in Engineering Software 31, 585 (2000).