不均勻擴散問題之A.D.I.法的研究

Abstract

微觀世界中存在著不均勻擴散現象，此現象在科學應用領域中占有重要地位。本論文應用A.D.I.法研究不均勻擴散方程式，我們除了討論A.D.I.法的收斂速度以及無條件穩定性之外，並且將其數值結果與疊代法所得數值結果互相比較。由於不均勻擴散方程式中的擴散係數可分為常係數與變係數的形式，因此離散之後所得的線性系統亦可分為常係數與變係數的型態。我們利用疊代法中常用的C.G.法與B.I.C.G.法分別處理。基於特殊的線性系統結構，A.D.I.法在計算速度上遠勝於疊代法。

Anisotropic diffusion is a kind of microscopic phenomenon. It plays an important role in a lot of scientific applications. We use A.D.I. method which is efficient to solve anisotropic diffusion problems. We study the order of accuracy and unconditionally stable onvergence of the method, and compare it with preconditioned iterative methods. Since diffusivity of anisotropic diffusion equations can be constant and variable type. We choose conjugate gradient method to deal with the constant type equation and biconjugate gradient method to solve the general type. Because of the special structure of linear systems, A.D.I. method outperforms iterative methods of CPU time.