對流佔優擴散問題一直是一個俱有挑戰性的問題。眾所週知，解可能在邊界或者內部表現出局部斷層。傳統的數值解法在解決對流佔優擴散問題時不夠穩定和精確。因此有很多空間穩定技巧被提出和學習。 / 在這篇論文中，我們設計了一個全變分規則化方法來解對流佔優擴散方程。首先，選取一個較小的測試函數空間構造一個欠定模型。然後，通過附加條件最小化全變分來計算唯一解。這個想法是基於離散化解雙曲形偏微分方程的全變分遞減原理。為了實現這個全變分規則化方法，我們需要找出相應的拉格郎日方程的最優條件，並當做時間依賴方程進行求解。論文中涉及了三個運算符拆分方法，Peaceman-Rachford方法，Douglas-Rachford方法和Theta方法。數值實驗證明我們的方法可以準確的描繪出對流佔優擴散問題的邊界和內部斷層現象，因此可以在擴散系數很小的時候提供可靠的數值結果。 / The convection-dominated diffusion problem has been a challenge problem for a long time. It is well known that the solution may exhibit localized layer on the boundary or in the interior. Conventional numerical schemes for the convection-dominated diffusion problem are lacking in both stability and accuracy. Therefore many spatial stabilization techniques have been proposed and studied. / In this thesis, we devise a TV regularization method for solving convection-dominated diffusion equations. First, we form an underdetermined system by choosing a small test function space. Then, we find the unique solution by minimizing the total variation. The idea is based on the fact that total variation diminishing (TVD) is a property of certain discretization schemes used to solve hyperbolic partial differential equation. To implement this TV regularization approach, we find the optimality condition of the corresponding Lagrange function for the minimization problem, and then solve it as a time dependent problem. Three operator splitting methods, the Peaceman-Rachford method, the Douglas-Rachford method and the Theta scheme are studied. The numerical experiments demonstrate that our scheme can accurately capture the boundary and internal layer for the convection-dominated problem, and therefore provide reliable numerical result for small diffusivity. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Zhang, Qi. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 56-[58]). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- TV scheme --- p.9 / Chapter 2.1 --- TV regularization --- p.9 / Chapter 2.1.1 --- The underdetermined system --- p.10 / Chapter 2.1.2 --- The minimization problem --- p.10 / Chapter 2.2 --- The optimality conditions --- p.11 / Chapter 2.3 --- Discrete version --- p.14 / Chapter 3 --- Numerical schemes --- p.17 / Chapter 3.1 --- Peachment-Rachford method --- p.17 / Chapter 3.2 --- Douglas-Rachford method --- p.18 / Chapter 3.3 --- Theta scheme --- p.19 / Chapter 3.4 --- Two systems --- p.22 / Chapter 3.4.1 --- System 1 --- p.23 / Chapter 3.4.2 --- System 2 --- p.25 / Chapter 3.5 --- Matrix representation --- p.26 / Chapter 4 --- Numerical examples --- p.29 / Chapter 4.1 --- Scenario --- p.30 / Chapter 4.1.1 --- Initial condition --- p.30 / Chapter 4.1.2 --- Notation --- p.30 / Chapter 4.1.3 --- Parameter --- p.31 / Chapter 4.2 --- Experiment 1 --- p.32 / Chapter 4.3 --- Experiment 2 --- p.38 / Chapter 4.4 --- Experiment 3 --- p.42 / Chapter 4.5 --- Experiment 4 --- p.48 / Chapter 5 --- Conclusion --- p.55
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_328083 |
| Date | January 2013 |
| Contributors | Zhang, Qi., Chinese University of Hong Kong Graduate School. Division of Mathematics. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, bibliography |
| Format | electronic resource, electronic resource, remote, 1 online resource (56, [2] leaves) : ill. (chiefly col.) |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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