解對流擴散方程之正反算問題

河口海岸工程中，汙染問題對於環境生態影響甚大，分析汙染排放和其在流體中運動的過程，對於工程規畫和施工都能有所助益。本文將在汙染源項未知的情形下，以無網格法其中的邊界積分法及配點法求解對流擴散方程，分析一維汙染源及其汙染濃度分布。本篇論文使用邊界積分法反算汙染源；使用邊界積分法及配點法正算汙染濃度分布。使用傅立葉級數、多項式作為數值試驗解的基底外，嘗試使用指數函數及模態作為數值試驗解之基底。更進一步探討分數階指數函數基底對於精度的影響。除此之外，亦使用多尺度的技巧降低病態程度、共軛梯度法求解線性代數方程組、三角函數正交性解待定係數。

關鍵字

對流擴散方程；反算問題；無網格法；邊界積分法；配點法；指數函數基底；分數階指數函數基底；多尺度技巧

並列摘要

The pollution is a big issue in estuarine and coastal engineering, inducing a huge impact on the environment. It would be helpful for engineering practice to analyze the emission and the dynamic motion of the pollution. The subject of this thesis is to solve the convection diffusion equation without knowing the source term by the boundary integral element method (BIEM) and the collocation method which are both meshless methods to recover 1D pollution source and concentration problem. Firstly, BIEM is applied to solve the inverse source problem. Secondly, BIEM as well as the collocation method are applied to solve the direct concentration problem. Other than Fourier series and polynomial basis, we tried to use the exponential functions and mode shapes as the basis of trial solution as well. Furthermore, we tried the fractional order exponential functions as the basis to see if the accuracy can be increased. The supplementary techniques such as multi-scale method, conjugate gradient method (CGM) and orthogonality of trigonometric functions are also used in this thesis.

並列關鍵字

convection diffusion equation ； inverse problem ； meshless method ； BIEM ； collocation ； exponential function basis ； fractional exponential function basis ； multi-scale technique

參考文獻

[21] Chein-Shan Liu1 , Weichung Yeih and Satya N. Atluri: On Solving the Ill-Conditioned System Ax = b: General-Purpose Conditioners Obtained From the Boundary-Collocation Solution of the Laplace Equation, Using Trefftz Expansions With Multiple Length Scales. CMES, vol.44, no.3, pp.281-311, 2009

[6] C.-S. Liu, C.W. Chang, J.R. Chang, A quasi-boundary semi-analytical method for back-ward in time advection-dispersion equation, CMC: Comput. Mater. Contin. 9 (2009)111-135.

[1] S.M. Gorelick, B.E. Evans, I. Remson, Identifying sources of groundwater pollution: An optimization approach, Water Resour. Res. 19 (1983) 779-790.

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[4] J. Atmadja, A.C. Bagtzoglou, Pollution source identification in heterogeneous porousmedia, Water Resour. Res. 37 (2001) 2113-2125.

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解對流擴散方程之正反算問題

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