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 本篇論文的主要目的是要研究線性和非線性的方法並且把他們應用在解偏微分方程題目的設計。我們使用這些方法去解決的測試題目包括非線性對流擴散方程和不可壓縮流體力學方程。這些方法的數值特性和應用方面的實際問題將在本文中討論。
 The primary object of this thesis is to investigate the linear and nonlinear methods and apply them to the design in solving partial differential equations.We use these to solve test problems including a nonlinear convection-diffusion equation and the incompressible Navier-Stokes equation. Numerical characteristics of these methods and practical issues in application are discussed.
 Conetnts 1 Introduction 4 2 The Test Problems 5 2.1 A Nonlinear Convection-Diffusion Equation ............ 5 2.2 The Poisson Equation ................................. 5 2.3 The Incompressible Navier-Stokes Equation ............ 6 3 Discretization Methods 8 3.1 Finite Volume Method and Burgers' Equation ........... 8 3.2 Finite Difference Method and Upwinding ............... 9 3.3 Application to Navier-Stokes Equation ................ 12 4 The Discrete System and Numerical Methods 23 4.1 Nonlinear Solvers .................................... 23 4.1.1 Newton's Method ................................ 23 4.1.2 Inexact Newton's Method ........................ 24 4.1.3 Global Convergence Method ...................... 25 4.2 Linear Solvers ....................................... 28 4.2.1 Conjugate Gradient Method and Its Variants ..... 28 4.2.2 Bi-Conjugate Gradient Method ................... 31 4.2.3 More Extensions to Bi-Conjugate Gradient Method 34 4.2.4 Quasi-Minimal Residual Method .................. 38 4.3 Complexity Analysis of the Linear Solvers ............ 39 4.4 Multigrid Methods .................................... 41 4.1.1 The Basic Idea and Components .................. 41 4.1.2 Various Issues in the Design ................... 43 5 Test Results and Discussions 45 5.1 Basic Accuracy Test .................................. 47 5.1.1 Results of the Nonlinear Solvers ............... 47 5.1.2 Results of the Linear Solvers .................. 66 5.2 Application Test ..................................... 93 5.2.1 The Two-dimensional Incompressible NS Equation . 93 5.2.2 A Closer Look at the Cavity Problem ............ 97 5.2.3 A Closer Look at the Channel Problem ...........105 5.3 Conclusions ..........................................114
 References[1]Chan, T., Gallopoulos, E., Simoncini, V., Szeto, T., andTong, C.A quasi-minimal residual variant of the Bi-CGSTABalgorithm for nonsymmetric systems.SIAM J. Sci. Comput., 15, p. 338, 1994.[2]Chen, Chih Hua.A Study in Domain Decomposition Method withApplications, master thesis for Department of Mathematics,Fu-Zen University, Taiwan. June 1998.[3]Chorin, A.Numerical solution of the Navier-Stokes-equations.Math. Comp., 22, 745-762, 1968.[4]Dongarra, Duff, Sorense, and van der Vorst.Numerical Linear Algebra for High-Performance Computers.Society for Industrial and Applied Mathematics,1999.[5]Ferziger, J. H. and Peric, M.Computational Methods for Fluid Dynamics.Second Edition. springer-verlag Berlin Heidelberg 1999.[6]Freund, R. W., Gutknecht, M. H., and Nachtigal, N. M.QMR: a quasi-minimal residual algorithm for non-hermitianlinear systems.Numer. Math., 60, pp. 315-339, 1991.[7]Freund, R. W.A transpose-free quasi-minimal residualalgorithm for non-Hermitian linear system.SIAM J. Sci. Comput., 14, pp. 470-482i, 1993.[8]Golub, G. H. and Van Loan, C. F.Matrix Computations.Third Edition. The Johns Hopkins University Press,Baltimore,1996.[9]Griebel Michael, Dornseifer Thomas, Neunhoeffer Tilman.Numerical Simulation in Fluid Dynamics.Society for Industrial and Applied Mathematics, 1998.[10]Hackbusch, W.Multi-Grid Methods and Applications.New York: Springer-Verlag, 1985[11]Hirt, C., Nichols, B. \& Romero, N.SOLA-A Numerical Solution Algorithm for Transient FluidFlows.Technical report LA-5852, Los Alamos, NM: Los AlamosNational Lab, 1975.[12]Kahaner David, Moler Cleve, Nash Stephen.Numerical Methods and Software.Printice-hall, 1991.[13]Kelley, C. T.Iterative Methods for Linear and Nonlinear Equations.Society for Industrial and Applied Mathematics, 1999.[14]Kelley, C. T.Iterative Methods for optimization.Society for Industrial and Applied Mathematics, 1999.[15]Martin H. Gutknecht.Variants of Bicgstable for matrices with complex spectrum.SIAM J. SCI. COMPUT. Vol. 14, No. 5, pp. 1020-1033,September 1993.[16]Meijerink, J. A. and van der Vorst, H. A.An iterative solution method for linear systems of whichthe coefficient matrix is a symmetric M-matrix.Mathematics of Computation, 31:148-162, 1997.[17]Meijerink, J. A. and van der Vorst, H. A.Guidelines for the usage of incomplete decompositions insolving sets of linear equations as they occur in practicalproblems.J. Comp. Phys., 44:134-155, 1981.[18]More, J.J., Garbow, B.S., and Hillstrom, K.E.User Guide for MINPACK-1, Report ANL-80-74,Argonne National Laboratory, Argonne, Illinois, 1980.[19]Sleijpen, G. L. G. and Fokkema, D. R.BICGSTAB(l) for linear equations involving unsymmetricmatrices with complex spectrum.ETNA, 1:11-32, 1993.[20]Stone,H. S.Iterative solution of implicit approximations ofmultidimensional partial differential equations.SIAM J. Numerical Analysis, 5:530-558, 1968.[21]Timshenko, S.Strength of Matrials, Part II.Van Nostrand, Princetion, NJ, 1956.[22]Temam, R.Sur l'approximation de la solution des equations deNavier-Stokes par la methode des pas fractionnaires.Arch. Rational Mech. Anal., 32, 135-153, 1969[23]van der Vorst, H. A.Bi-CGSTAB: A fast and smoothly converging variant to Bi-CGfor the solution of nonlinear systems.SIAM J. Sci. Statist. Comput., 13, pp.631-644, 1992.[24]Varga, R. S.Factorizations and normalized iterative methods.In R. E. Langer, editor, Boundary Problems in differentialequations, pages 121-142. University of Wisconsin Press,Madison, WI, 1960.[25]Zhou, L. and Walker, H. F.Residual smoothing techniques for iterative methods.SIAM J. Sci. Comput., 15, pp. 297-312, 1994.
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