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Title: | 擬譜補償微分運算子之積分預處理矩陣及於微分方程之應用 Integration Preconditioning Matrices for Pseudospectral Penalty Differentiation Operators with Applications to Differential Equations |
Authors: | Po-Yu Lin 林柏宇 |
Advisor: | 林太家 |
Keyword: | 譜/擬譜法,補償邊界條件,積分預處理,混和微分運算子,多域格式,片段連續,上風通量, spectral/pseudospectral methods,penalty boundary conditions,integration preconditioning,mixed differential operators,multidomain schemes,piecewise continuous,upwind flux, |
Publication Year : | 2019 |
Degree: | 碩士 |
Abstract: | 對於解牽涉到純粹微分運算子d^m/dx^m的微分方程式,譜與擬譜積分預處理矩陣是有效的工具。在本論文的第一部分當中,對於在Gauss-Radau-Legendre網格點上的混和微分運算子d/dr a(r) d/dr,我們採用了一種可分離的建構框架,來明確地建構條件好的逆擬譜補償矩陣。對於在極座標系統中變係數二階微分方程式,這種逆矩陣可以作為解運算子,或是有效的預處理運算子。而在本論文的第二部分當中,對於在Gauss-Lobatto-Legendre網格點上的一階微分運算子d/dx,我們明確地建構條件好的多域逆擬譜補償矩陣。對於直角坐標系統上具有分段連續係數的一階微分方程式,這些逆矩陣可以作為解運算子,或是有效的預處理運算子。 Spectral and pseudospectral integration preconditioning matrices are effective tools for solving differential equations involving pure differential operators d^m/dx^m. In the first part of this thesis we adopt a separable construction framework to explicitly construct a well-conditioned inverse pseudospectral penalty matrix for the mixed differential operator d/dr a(r) d/dr on Gauss-Radau-Legendre grid points. This inverse matrix can be used either as a solution operator or an effective preconditioner for variable coefficient second order differential equations in polar coordinate system. In the second part of this thesis we explicitly construct well-conditioned multidomain inverse pseudospectral penalty matrices for the first order differential operator d/dx on Gauss-Lobatto-Legendre grid points. These inverse matrices can be used as either solution operators or effective preconditioners for first order differential equations with a piecewise continuous coefficient in Cartesian coordinate system. |
URI: | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/74907 |
DOI: | 10.6342/NTU201904175 |
Fulltext Rights: | 有償授權 |
Appears in Collections: | 應用數學科學研究所 |
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