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研究生:蔡佩珊
研究生(外文):Pei-Shan Tsai
論文名稱:利用裁縫有限點法來解擴散對流反應問題
論文名稱(外文):A Tailored Finite Point Method for Convection-Diffusion-Reaction Problems
指導教授:施因澤
指導教授(外文):Yin-Tzer Shih
學位類別:碩士
校院名稱:國立中興大學
系所名稱:應用數學系所
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:2009
畢業學年度:97
語文別:英文
論文頁數:34
中文關鍵詞:裁縫有限點擴散對流反應有限單元邊界層內部層
外文關鍵詞:Tailored finite pointconvection-diffusion-reactionfinite elementboundary layerinternal layer
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在這篇論文裡,我們研究裁縫有限點法來解擴散對流反應方程式。裁縫有限點法的解空間的基底函數是由指數函數、修正的貝塞耳函數及三角函數相乘所組成。在理論分析上我們可以求得裁縫有限點法在對流控制問題的數值截尾誤差為O(e^-Pe),這裡的Pe是一個網格貝克勒數。當擴散係數很小時,由我們的數值測試證明了在邊界層的誤差不管網格大小都會在機器誤差的精準之下,而內部層的誤差則在網格大小之下。這描述了在對流控制問題上,裁縫有限點法會有相當高的準確度。
In this thesis, we study a tailored finite point method (TFP) for solving the convection-diffusion-reaction equation. The basis functions of solution space of the TFP are composed from the product of an exponential function, modified Bessel functions and trigonometric functions. The numerical truncation error for the TFP is in the order of O(e^-Pe) for the convection dominated problem, where Pe is the mesh Péclet number. Our numerical tests show that for small diffusion coefficient the numerical boundary layer is under the machine precision regardless the mesh size, and the internal layer is also within the mesh size. This depicts that the TFP method has high accuracy for convection-dominated problem.
1 Introduction ........................................ 1
2 The Tailored Finite Point Methods......... 3
2.1Introduction....................................... 3
2.2 The five-point TFP scheme................. 5
2.3 The seven-point TFP scheme............. 8
2.4 The nine-point TFP scheme............... 11
3 Truncation Errors................................. 14
4 Numerical Experiments........................ 19
5 Conclusions........................................ 23
Bibliography........................................... 33
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[2] C. Johnson, A.H. Schatz and L.B. Wahlbin, Crosswing smear and pointwise errors in streamline diffusion finite element methods, Math. Comp., 49 (1987), 25-38.
[3] Y. Shih and H.C. Elman, Modified streamline diffusion schemes for convectiondiffusion problems, Computer Methods in Applied Mechanics and Engineering, 174 (1999), 137-151.
[4] H.G. Roos, M. Stynes and L. Tobiska, Numerical methods for singular perturbed differential equations, Springer-Verlag, New York, 1996.
[5] H. Han, Z. Huang and R.B. Kellogg, The tailored finite point method and a problem of P. Hemker, Proceedings of the International Conference on Boundary and Interior Layers - Computational and Asymptotic Methods, Limerick, July 2008.
[6] H. Han, Z. Huang and R.B. Kellogg, A tailored finite point method for a singular perturbation problem on an unbounded domain, J.Sci. Comp., 36 (2008), 243-261.
[7] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, 1964.
[8] H. Han and Z. Huang, Tailored Finite point method for a singular perturbation problem with variable coefficients in two dimensions, J.Sci. Comp., 2009.
[9] M. Stynes and L. Tobiska, Necessary L2-uniform convergence conditions for difference schemes for two dimensional convection-diffusion problems, Computers Math. Applic., 29 (1998), 45-53.
[10] H. Eckhaus, Boundary layers in linear elliptic singular pertubation problems, SIAM Rev., 14 (1972), 225-270.
[11] C. Johnson, U. N¨avert, An analysis of some finite element methods for advectiondiffusion problems.Analytical and Numerical Approaches to Asymptotic Problem in Analysis (O. Axelsson, L.S. Frank, A. ver der Sluis, eds.), North-Holland, Amsterdam, 1981.
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