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研究生:伍興儀
研究生(外文):Hsing-Yi Wu
論文名稱:流體力學計算數學方法之研究
論文名稱(外文):A Study of Numerical Methods Used in Fluids Computation
指導教授:李天佑李天佑引用關係
指導教授(外文):Daniel Lee
學位類別:碩士
校院名稱:輔仁大學
系所名稱:數學系研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2001
畢業學年度:89
語文別:中文
論文頁數:117
中文關鍵詞:類牛頓法迭代法多層網格計算
外文關鍵詞:Newton-like MethodsIterative MethodsMultigrid Computation
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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
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Computational Methods for Fluid Dynamics.
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Baltimore,1996.
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