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標題: 裁縫有線點法解擴散對流方程式
Tailored Finite Point Method for Convection-Diffusion Equations
作者: 施因澤
關鍵字: 數學類
摘要: 我們提出一個兩年的研究計畫,主要是研究裁縫有限點法對PDE解的性質,尤其是對擴散對流問題的數值解與誤差分析。最主要是研究裁縫有限點法的準確性尤其當擴散係數相當小或是對流速度是變數時。我們也將與其他常用的數值方法來比較其結果。裁縫有限點法是ㄧ個對擴散對流問題新的數值方法是2008年由韓厚德、黃忠儀與Kellogg提出。許多理論與其應用上仍有相當大的研究空間。我們將分為兩階段來作研究。在第一年的研究部分,我們已經對常數係數homogeneous問題,根據之前我與Kellogg和楊肅煜教授之前合作經驗(已經整理出兩篇論文),先建立一個格子點方法使其格子點盡量落在其特徵線(流線)上用。我們可以將問題簡化為雙曲線微分方程式或是先使用upwind有限單元法在均勻粗網格求數值解,再根據其解的等位圖決定格子點的位置然後再利用裁縫有限點法來求出準確數值解。我們將利用泰勒展開式與泛函分析的技巧研究其誤差分析。在本計畫的第二部分,我們將研究裁縫有限點法對nonhomogeneous擴散對流問題以及如何將裁縫有限點法擴展到解其他微分方程─ Stokes或Navier Stokes問題?主要著重數值解的準確性。研究是否可能在局部內部與邊界層改進一些有限單元法的誤差。
Here we propose a two-year research project to consider the tailored finite point method (TFP) for solving the partial differential equations, especially the numerical solution and error analysis for convection-diffusion equations. We will study the accuracy of the TFP when the diffusion coefficient is sufficient small or the convection velocity is variable. We also will compare the numerical results with other well-known numerical methods.The TFP is a new method proposed by Hung, Huang and Kellogg in 2008 for solving the singular perturbation problem. So there still are many problems left in both theory and application. In their numerical tests, they only solve the problem with diffusion coefficient down to 0.001, and the TFP fails as the diffusion term is smaller than 0.001. In the first year, we study a grid point generation logic based on our previous works that we have adequately studied with Prof. Kellogg and Prof. Yang for solving the convection-diffusion and steady MHD duct flow problems with constant convection coefficient and constant source function. The grid points will be generated on the characteristic lines (or streamlines) for approaching accurate numerical solution, and the characteristic can be found by the reduced hyperbolic problem. We also can use a stabliized finite element method (upwind-like method) to solve the problem to the coarse uniform grids, so the finer grid can be generated on the solution solution contour curves. Finally the TFP will resolve the interior/boundary layers. We can implement the Taylor expansion and some functional analysis skill to obtain the local truncation error analysis for the TFP.We will consider the TFP in the nonhomogeneous convection-diffusion equations, and expand the appliaction into the Stoke's equation and the Navier-Stoke's problems. It's a challange to resolve the layers in these problems. The main issue for this project is to see how robust of the tailored finite point method for those problems with interior/boundary layers.
其他識別: NSC99-2115-M005-003
Appears in Collections:應用數學系所



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