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研究生:范文鐘
研究生(外文):Wen-Chung Fan
論文名稱:以創新有限差分法配合流形指數收歛演算法求解橢圓型偏微分方程
論文名稱(外文):Solving the Elliptic type PDEs by a Combination of a Novel Finite Difference Method with the Manifold-Based Exponentially Convergent Algorithm
指導教授:劉進賢林益煌林益煌引用關係
指導教授(外文):Chein-Shan LiuYih-Hwang Lin
學位類別:碩士
校院名稱:國立臺灣海洋大學
系所名稱:機械與機電工程學系
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2010
畢業學年度:98
語文別:中文
論文頁數:71
中文關鍵詞:橢圓型偏微分方程式創新有限差分法非規則幾何形狀流形指數收歛演算法
外文關鍵詞:Elliptic-type partial differential equationsNovel finite difference methodirregular geometric shapemanifold-based exponentially convergent algorithm (MBECA)
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對於橢圓型偏微分方程式問題,本文採用創新有限差分法將問題離散為代數方程式,再使用流形指數收歛演算法,將代數方程式轉變成常微分方程式,並配合數值積分法求解。相較於傳統的有限差分法,本文提出的創新有限差分法可以處理非規則幾何形狀的問題。使用一個虛擬的矩形覆蓋於計算域上,並引入有限差分法的概念來處理非規則幾何形狀。藉由規則網格點與邊界點的關係與雙線性分佈假設,來滿足物理問題的邊界條件,因此本文提出的創新有限差分法可以有效率的處理非規則幾何形狀的問題。同時,流形指數收歛演算法對於初始猜值並不敏感性、也不需計算Jacobian matrix的反矩陣,並且殘差有著指數收歛行為,因此能獲得相當滿意的結果與快速的收斂行為。經由幾個算例的測試及驗證,可看出本文提出之數值方法在處理線性、擬線性及非線性的偏微分方程式問題效果是很穩定的,並且在多連通區域也可適用。
In this paper the elliptic-type partial differential equations (PDEs) are transformed to a system of algebraic equations by the novel finite difference method (Novel FDM). Then, by using the manifold-based exponentially convergent algorithm (MBECA), the system of algebraic equations is converted to a system of ordinary differential equations (ODEs) which will be numerically integrated by numerical integration method. In comparing with the conventional finite difference method (FDM), the Novel FDM can deal with the problems in irregular geometric shapes. The proposed method uses a virtual rectangle to embed the physical domain and then using the conventional FDM to discretize the virtual domain. Utilizing the relationship of field values on the uniform grid and on the boundary points, the boundary conditions can be satisfied under the bi-linear assumption. In addition, the MBECA is insensitive to the guessing of initial conditions and can avoid calculating the inverse of Jacobian matrix. Besides, the evolutionary process of the residual in MBECA is exponentially convergent. So, the Novel FDM, proposed in this thesis, can efficiently obtain excellent results. From the numerical tests and validation, it is proven that the Novel FDM can stably and quickly deal with linear, quasilinear and nonlinear PDEs, even the problems in multiply-connected domains.
目錄

摘要 i
Abstract ii
目錄 iv
圖目錄 vi
第一章 緒論 1
1-1 前言 1
1-2 研究目的與動機 1
1-3 文獻回顧 2
1-4 本文架構 3
第二章 理論基礎 5
2-1 非線性現象 5
2-2 偏微分方程概述 5
2-3 邊界條件的類型 6
第三章 數值方法 7
3-1創新有限差分法 8
3-2流形指數收歛演算法 13

第四章 數值結果與討論 16
4-1 例題一 16
4-2 例題二 24
4-3 例題三 31
4-4 例題四 35
4-5 例題五 41
4-6 例題六 48
4-7 例題七 52
4-8 例題八 55
4-9 例題九 60
第五章 結論與未來展望 65
5-1 結論 65
5-2 未來展望 66
參考文獻 67

圖目錄

圖3.1物理問題計算域、邊界及虛構矩形示意圖。 8
圖3.2於矩形上建立有限差分法的網格。 8
圖3.3二維有限差分法中, 點與周圍的點。 9
圖3.4最外圍的網格格點不需強迫滿足控制方程。 10
圖3.5部分網格內會有一個或者是一個以上的邊界點於其中。
10
圖3.6 將 座標系統轉換為局部座標系統 。 11
圖4.1:例題一的計算域與所對應的邊界。 17
圖4.2︰例題一(類型一)殘差隨虛擬時間演化情形。 18
圖4.3︰例題一(類型一) (a)解析解,(b)數值解。 19
圖4.4︰例題一(類型一)相對誤差圖。 20
圖4.5︰例題一(類型二)殘差隨虛擬時間演化情形。 22
圖4.6︰例題一(類型二)(a)解析解,(b)數值解。 23
圖4.7︰例題一(類型二)相對誤差圖。 24
圖4.8:例題二的計算域與所對應的邊界。 25
圖4.9:例題二(類型一)殘差隨虛擬時間演化情形。 26
圖4.10:例題二(類型一)(a)解析解,(b)數值解。 27
圖4.11:例題二(類型一)相對誤差圖。 28
圖4.12:例題二(類型二)殘差隨虛擬時間演化情形。 29
圖4.13:例題二(類型二)(a)解析解,(b)數值解。 30
圖4.14:例題二(類型二)相對誤差圖。 31
圖4.15:例題三的計算域與所對應的邊界。 32
圖4.16:例題三殘差隨虛擬時間演化情形。 33
圖4.17:例題三(a)解析解,(b)數值解。 34
圖4.18:例題三相對誤差圖。 35
圖4.19:例題四(類型一)殘差隨虛擬時間演化情形。 36
圖4.20:例題四(類型一)(a)解析解,(b)數值解。 37
圖4.21:例題四(類型一)相對誤差圖。 38
圖4.22:例題四(類型二)殘差隨虛擬時間演化情形。 39
圖4.23:例題四(類型二) (a)解析解,(b)數值解。 40
圖4.24:例題四(類型二)相對誤差圖。 41
圖4.25:例題五的計算域與所對應的邊界。 42
圖4.26:例題五(類型一)殘差隨虛擬時間演化情形。 43
圖4.27:例題五(類型一)(a)解析解,(b)數值解。 44
圖4.28:例題五(類型一)相對誤差圖。 45
圖4.29:例題五(類型二)殘差隨虛擬時間演化情形。 46
圖4.30:例題五(類型二)(a)解析解,(b)數值解。 47
圖4.31:例題五(類型二)相對誤差圖。 48
圖4.32:例題六的計算域與對應的邊界。 49
圖4.33:例題六中殘差隨虛擬時間演化情形。 50
圖4.34:例題六中(a)解析解,(b)數值解。 51
圖4.35:例題六中相對誤差圖。 52
圖4.36:例題七殘差隨虛擬時間演化情形。 53
圖4.37:例題七 (a)解析解,(b)數值解。 54
圖4.38:例題七相對誤差圖。 55
圖4.39:例題八的計算域與所對應的邊界。 56
圖4.40:例題八殘差隨虛擬時間演化情形。 58
圖4.41:例題八(a)解析解,(b)數值解。 59
圖4.42:例題八相對誤差圖。 60
圖4.43:例題九的計算域與所對應的邊界。 61
圖4.44:例題九殘差隨虛擬時間演化情形。 62
圖4.45:例題九(a)解析解,(b)數值解。 63
圖4.46:例題九相對誤差圖。 64


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陳駿逸,「計算偏微分方程之頻譜配點法」,東海大學,應用數學研究所,碩士班論文,民國93年。
郭仲倫,「二維多連通區域的拉普拉斯內外域問題研究」,國立海洋大學,機械與機電工程學系,碩士班論文,民國96年。

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