微分方程和特征值问题的高阶差分格式探索
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摘要
我们在本文中采用任意形式差分模板系数的快速算法生成高阶差分格式,探索求解偏微分方程和特征值问题。在本文中我们主要求解3D问题,通过在每个方向上独立做一维长格式差分的方案来近似Laplace算子,从而获得3D问题的高阶差分格式,并用于求解Poisson方程,Poisson-Boltzmann方程以及特征值问题。在求解Poisson-Boltzmann方程时,我们成功地采用局部作ILU(0)预条件的并行MINRES算法,获得了很高的并行效率。在数值实验中,我们可以看到长格式的高阶差分具有很好的数值效果,并且可以达到紧格式无法比拟的高精度。
In the thesis, we adopt a fast algorithm to define the coefficients of difference stencil, which is used to produce the high-order difference schemes for the numerical solution of PDEs and eigenvalue problems. For the 3D problems in the thesis, we get the high-order difference scheme in terms of approximating the Laplace operator by 1D long-stencil difference independently in every direction, and solve the Poisson equation, Poisson-Boltzmann equation and eigenvalue problems. A parallel MINRES algorithm with the local ILU(0) factorization as a preconditioner is used successfully to solve the Poisson-Boltzmann equation, and get the high parallel efficiency. high performance of high-order difference schemes is shown in numerical experiments, and the long-stencil difference schemes could attach the very high accuracy compared with compact difference schemes.
In the thesis, we adopt a fast algorithm to define the coefficients of difference stencil, which is used to produce the high-order difference schemes for the numerical solution of PDEs and eigenvalue problems. For the 3D problems in the thesis, we get the high-order difference scheme in terms of approximating the Laplace operator by 1D long-stencil difference independently in every direction, and solve the Poisson equation, Poisson-Boltzmann equation and eigenvalue problems. A parallel MINRES algorithm with the local ILU(0) factorization as a preconditioner is used successfully to solve the Poisson-Boltzmann equation, and get the high parallel efficiency. high performance of high-order difference schemes is shown in numerical experiments, and the long-stencil difference schemes could attach the very high accuracy compared with compact difference schemes.
引文
[1]U. Ananthakrishnaiah, R.P. Manohar, J.W. Stephenson. Fourth-order finite difference methods for three-dimensional general elliptic problems with variable coefficients [J]. Numer. Methods Partial Differen. Equat,1987,3:229-240.
[2]S. Abarbanel and A. Kumar. Compact higher-order schemes for the Euler equations[J]. J. Sci. Comput.1988,3:275-288.
[3]L.M. Adams, R.J. Leveque, and D.M. Young. Analysis of the SOR iteration for the 9-point Lapla-cian[J]. SIAM J.Numer. Anal.1988,25(5):1156-1180.
[4]S. Avdiaj and J. Setina. Numerical Solving of Poisson Equation in 3D Using Finite Difference Method[J]. Engineering and Applied Sciences.2010,5(1):14-18.
[5]M.H. Carpenter, D. Gottlieb, and S. Abarbanel. The stability of numerical boundary treatments for compact high-order finite-difference schemes[J]. J. Comput. Phys.1993,108:272-295.
[6]P. Concus, G.H. Golub. Use of fast direct methods for the efficient numerical solution of nonsepa-rable elliptic equations[J]. SIAM J. Numer Anal.,1973,10:1103-1120.
[7]J.J. Dongarra, I.S. Duff, D.C. Sorensen, and H.A. van der Vorst Numerical linear algebra for high-performance computers[M]. Society for Industrial Mathematics.
[8]B. Fornberg and D.M. Sloan. A review of pseudospectral methods for solving partial differential equations [J]. Acta Numerica,1994,3:203-267.
[9]B. Fornberg. Generation of finite difference formulas on arbitrarily spaced grids [J]. Math. Comput., 1988,51:699-706.
[10]B. Fornbcrg. High order finite differences and the pseudospectral method on staggered grids [J]. SIAM J. Num. Anal,1990,27:904-918.
[11]B. Fornberg. Fast generation of weighs in finite difference formulas. in Recent Developments in Nu-merical Methods and Software for ODEs/DAEs/PDEs(G.D. Byrne and W.E. Schiesser, eds),World Scientific(Singapore),1992,97-123.
[12]J. Fattebert, F. Gygi. Density Functional Theory for Efficient Ab Initio Molecular Dynamics Sim-ulations in Solution [J]. Comput Chem,2002,23:662-666.
[13]J. Fattebert, F. Gygi. First-Principles Molecular Dynamics Simulations in a Continuum Solvent [J], Quantum Chem,2003,93:139-147.
[14]A. Grama, G. Karypis, V. Kumar, A. Gupta. Introduction to Parallel Computing (2nd Edition) (Hardcover)[M]. Addison Wesley.
[15]M.M. Gupta, R.P. Manohar, and J.W. Stephenson. A single cell high order scheme for the convection-diffusion equation with variable coefficients[J]. Internat. J. Numer. Meth. Fluids.1984, 4:641-651.
[16]R.E. Lynch. Hodie approximation of boundary conditions, in:D.R. Kincaid, L.J. Hayes(Eds.). Iterative Methods for Large Linear Systems, Academic Press Inc., Niew York,1990:135-147.
[17]B.A. Luty, M.E. Davis and J.A. McCammon. Solving the finite-difference nonlinear Poisson-Boltzmann equation[J]. Comput. Chem.1992,13(9):1114-1118.
[18]A. K. Mitra. Finite Difference Method for the Solution of Laplace Equation. Department of Aerospace Engineering, Iowa State University.
[19]R.J. MacKinnon and R.W. Johnson. Differential equation based representation of truncation errors for accurate numerical simulation[J]. Internat. J. Numer. Meth. Fluids.1991,13:739-757.
[20]W.F. Spotz and G.F. Carey. A High-Order Compact Formulation for the 3D Poisson Equation [J]. Numerical Methods for Partial Differential Equations,1996,12:235-243.
[21]W.F. Spotz and G.F. Carey. Formulations and experiments with high-order compact schemes for nonuniform grids[J]. IMA J. Numer. Anal.1998,8(3):288-303.
[22]Y. Saad. Iterative methods for sparse linear systems [M], PWS Publishing Company, Boston,1996.
[23]G. Saldanha. High-order difference discretizations to Poisson's equation[J]. Ganita 2000,51: 107-115.
[24]G. Saldanha. Single cell discretizations of order two and four for self-adjoint elliptic equations[J]. Appl.Math.Comput 2003,134:1-8.
[25]G. Saldanha. A finite difference method for self-adjoint elliptic equations in three dimensions. Appl.Math.Comput 2003,146:803-811.
[26]A.I. van de Vooren and A.C. Vliegenthart. On the 9-point difference formular for Laplace's equa-tion[J]. Eng.Math.1967,1(3):187-202.
[27]J.A.C. Weideman and L.N. Trefethen. The Eigenvalues of Second-order Spectral Differentiation Matrices[J]. SIAM J. Nmuer. Anal.1988,25(6):1279-1297.
[28]Jie Wanga, Weijun Zhonga and Jun Zhang. A general meshsize fourth-order compact difference discretization scheme for 3D Poisson equation [J]. Applied Mathematics and Computation.2006, 183(2):804-812.
[29]Z. Zhou, P. Payne, M. Vasquez, N. Kuhn and M. Levitt. Finite-difference solution of the Poisson-Boltzmann equation:Complete elimination of self-energy[J]. Comput. Chem.1996,17:1344-1351.
[30]苏汝铿.量子力学,高等教育出版社,2006.
[2]S. Abarbanel and A. Kumar. Compact higher-order schemes for the Euler equations[J]. J. Sci. Comput.1988,3:275-288.
[3]L.M. Adams, R.J. Leveque, and D.M. Young. Analysis of the SOR iteration for the 9-point Lapla-cian[J]. SIAM J.Numer. Anal.1988,25(5):1156-1180.
[4]S. Avdiaj and J. Setina. Numerical Solving of Poisson Equation in 3D Using Finite Difference Method[J]. Engineering and Applied Sciences.2010,5(1):14-18.
[5]M.H. Carpenter, D. Gottlieb, and S. Abarbanel. The stability of numerical boundary treatments for compact high-order finite-difference schemes[J]. J. Comput. Phys.1993,108:272-295.
[6]P. Concus, G.H. Golub. Use of fast direct methods for the efficient numerical solution of nonsepa-rable elliptic equations[J]. SIAM J. Numer Anal.,1973,10:1103-1120.
[7]J.J. Dongarra, I.S. Duff, D.C. Sorensen, and H.A. van der Vorst Numerical linear algebra for high-performance computers[M]. Society for Industrial Mathematics.
[8]B. Fornberg and D.M. Sloan. A review of pseudospectral methods for solving partial differential equations [J]. Acta Numerica,1994,3:203-267.
[9]B. Fornberg. Generation of finite difference formulas on arbitrarily spaced grids [J]. Math. Comput., 1988,51:699-706.
[10]B. Fornbcrg. High order finite differences and the pseudospectral method on staggered grids [J]. SIAM J. Num. Anal,1990,27:904-918.
[11]B. Fornberg. Fast generation of weighs in finite difference formulas. in Recent Developments in Nu-merical Methods and Software for ODEs/DAEs/PDEs(G.D. Byrne and W.E. Schiesser, eds),World Scientific(Singapore),1992,97-123.
[12]J. Fattebert, F. Gygi. Density Functional Theory for Efficient Ab Initio Molecular Dynamics Sim-ulations in Solution [J]. Comput Chem,2002,23:662-666.
[13]J. Fattebert, F. Gygi. First-Principles Molecular Dynamics Simulations in a Continuum Solvent [J], Quantum Chem,2003,93:139-147.
[14]A. Grama, G. Karypis, V. Kumar, A. Gupta. Introduction to Parallel Computing (2nd Edition) (Hardcover)[M]. Addison Wesley.
[15]M.M. Gupta, R.P. Manohar, and J.W. Stephenson. A single cell high order scheme for the convection-diffusion equation with variable coefficients[J]. Internat. J. Numer. Meth. Fluids.1984, 4:641-651.
[16]R.E. Lynch. Hodie approximation of boundary conditions, in:D.R. Kincaid, L.J. Hayes(Eds.). Iterative Methods for Large Linear Systems, Academic Press Inc., Niew York,1990:135-147.
[17]B.A. Luty, M.E. Davis and J.A. McCammon. Solving the finite-difference nonlinear Poisson-Boltzmann equation[J]. Comput. Chem.1992,13(9):1114-1118.
[18]A. K. Mitra. Finite Difference Method for the Solution of Laplace Equation. Department of Aerospace Engineering, Iowa State University.
[19]R.J. MacKinnon and R.W. Johnson. Differential equation based representation of truncation errors for accurate numerical simulation[J]. Internat. J. Numer. Meth. Fluids.1991,13:739-757.
[20]W.F. Spotz and G.F. Carey. A High-Order Compact Formulation for the 3D Poisson Equation [J]. Numerical Methods for Partial Differential Equations,1996,12:235-243.
[21]W.F. Spotz and G.F. Carey. Formulations and experiments with high-order compact schemes for nonuniform grids[J]. IMA J. Numer. Anal.1998,8(3):288-303.
[22]Y. Saad. Iterative methods for sparse linear systems [M], PWS Publishing Company, Boston,1996.
[23]G. Saldanha. High-order difference discretizations to Poisson's equation[J]. Ganita 2000,51: 107-115.
[24]G. Saldanha. Single cell discretizations of order two and four for self-adjoint elliptic equations[J]. Appl.Math.Comput 2003,134:1-8.
[25]G. Saldanha. A finite difference method for self-adjoint elliptic equations in three dimensions. Appl.Math.Comput 2003,146:803-811.
[26]A.I. van de Vooren and A.C. Vliegenthart. On the 9-point difference formular for Laplace's equa-tion[J]. Eng.Math.1967,1(3):187-202.
[27]J.A.C. Weideman and L.N. Trefethen. The Eigenvalues of Second-order Spectral Differentiation Matrices[J]. SIAM J. Nmuer. Anal.1988,25(6):1279-1297.
[28]Jie Wanga, Weijun Zhonga and Jun Zhang. A general meshsize fourth-order compact difference discretization scheme for 3D Poisson equation [J]. Applied Mathematics and Computation.2006, 183(2):804-812.
[29]Z. Zhou, P. Payne, M. Vasquez, N. Kuhn and M. Levitt. Finite-difference solution of the Poisson-Boltzmann equation:Complete elimination of self-energy[J]. Comput. Chem.1996,17:1344-1351.
[30]苏汝铿.量子力学,高等教育出版社,2006.