在這篇論文我們將介紹多維變係數擴散方程的四階逼近的數值方法，在二維問題這個方法內部是基於九點格式的緊緻有限差分法，同時這個有限差分解法的係數矩陣是對稱且正定矩陣或接近對稱矩陣。此外這在邊界保持了一階微分，依此能夠把問題延伸到Neumann邊界條件的問題。

In this thesis, we introduce fourth-order numerical method for solving the multi-dimensional diffusion equations with variable coefficient. In two-dimensional problem, the scheme is developed based on the nine-grid points compact finite difference method at interior point. Also the coefficient matrix of this discretization method is symmetric and positive definite or at least is positive real with small non-symmetric terms. Moreover, it preserves the first-order derivative on the boundary, so we extended the problem with Neumann boundary condition.

Contents
Chapter 1. Introduction----------------------------------------1
Chapter 2. Preliminary-----------------------------------------5
Chapter 3. Two-dimension problem-------------------------------7
3.1. Dirichlet boundary condition------------------------------7
3.2. Neumann boundary condition-------------------------------12
Chapter 4. Three-dimension problem----------------------------23
4.1. Dirichlet boundary condition-----------------------------23
4.2. Neumann boundary condition-------------------------------29
Chapter 5. Heat conduction problem----------------------------41
5.1. Dirichlet boundary condition-----------------------------41
5.2. Neumann boundary condition-------------------------------42
Chapter 6. Error Estimate-------------------------------------49
Chapter 7. Numerical results----------------------------------53
7.1. Two-dimensional problem----------------------------------53
7.2. Three-dimensional problem--------------------------------57
7.3. Heat conduction problem----------------------------------59
Chapter 8. Conclusion-----------------------------------------63
Appendix A. Additional problem--------------------------------65
Bibliography--------------------------------------------------67

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