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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 林太家 | |
| dc.contributor.author | Po-Yu Lin | en |
| dc.contributor.author | 林柏宇 | zh_TW |
| dc.date.accessioned | 2021-06-17T09:10:01Z | - |
| dc.date.available | 2020-10-17 | |
| dc.date.copyright | 2019-10-17 | |
| dc.date.issued | 2019 | |
| dc.date.submitted | 2019-10-01 | |
| dc.identifier.citation | [1] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, Berlin Heidelberg, 2007.
[2] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer-Verlag, Berlin Heidelberg, 2007. [3] H. Chen, Y. Su and B. D. Shizgal, A direct spectral collocation Poisson solver in polar and cylindrical coordinates, J. Comput. Phys., 160 (2000), 453–469. [4] M. Carpenter and D. Gottlieb, Spectral methods on arbitrary grids, J. Comput. Phys., 129 (1996), 74-86. [5] E. H. Doha, On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials, J. Phys. A, 37 (2004), 657-675. [6] E. H. Doha and W. M. Abd-Elhameed, Efficient spectral-Galerkin algorithms for direct solution of second-order equations using ultraspherical polynomials, SIAM J. Sci. Comput., 24 (2002), 548-571. [7] E. H. Doha and W. M. Abd-Elhameed, Efficient spectral ultraspherical-dual-Petrov-Galerkin algorithms for direct solution of (2n+1)th-order linear differential equations, Math. Comput. Simulation, 79 (2009), 3221-3242. [8] W. Don and D. Gottlieb, The Chebyshev-Legendre method: Implementing Legendre methods on Chebyshev points, SIAM J. Numer Anal., 31 (1994), 1519-1524. [9] H. Eisen, W. Heinrichs and K. Witsch, Spectral collocation methods and polar coordinate singularities, J. Comput. Phys., 96 (1991), 241-257. [10] E. M. E. Elbarbary, Integration preconditioning matrix for ultraspherical pseudospectral operators, SIAM J. Sci. Comput., 28 (2006), 1186-1201. [11] D. Funaro and D. Gottlieb, A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations, Math. Comp., 51 (1988), 599-613. [12] D. Funaro and D. Gottlieb, Convergence results for pseudospectral approximations of hyperbolic systems by a penalty-type boundary treatment, Math. Comp., 57 (1991), 585-596. [13] F. Ghoreishi and S. M. Hosseini, A preconditioned implementation of pseudospectral methods on arbitrary grids, Appl. Math. Comput., 148 (2004), 15-34. [14] J. S. Hesthaven, Integration preconditioning of pseudospectral operators. I. basic linear operators, SIAM J. Numer. Anal., 35 (1998), 1571-1593. [15] J. S. Hesthaven, Spectral penalty methods, Appl. Numer. Math., 33 (2000), 23-41. [16] J. S. Hesthaven and D. Gottlieb, A stable penalty method for the compressible Navier–Stokes equations: I. open boundary conditions, SIAM J. Sci. Comput., 17 (1996), 579-612. [17] J. S. Hesthaven, S. Gottlieb and D. Gottlieb, Spectral Methods for Time-Dependent Problems, Cambridge University Press, Cambridge, UK, 2007. [18] W. Huang and D. M. Sloan, Pole condition for singular problems: The pseudospectral approximation, J. Comput. Phys., 107 (1993), 254–261. [19] S. Olver and A. Townsend, A fast and well-conditioned spectral method, SIAM Review, 55 (2013), 462-489. [20] S. A. Orszag, Numerical simulation of incompressible flows within simple boundaries: accuracy, J. Fluid Mech., 49 (1971), 75-112. [21] S. A. Orszag, Accurate solution of the Orr-Sommerfeld stability equation, J. Fluid Mech., 50 (1971), 689-703. [22] J. Shen, Efficient spectral-Galerkin method I. Direct solves for the second order and fourth order equations using Legendre polynomials, SIAM J. Comput., 15 (1994), 1489-1505. [23] J. Shen, Efficient spectral-Galerkin method II. Direct solvers of second and fourth order equations by using Chebyshev polynomials, SIAM J. SCI. Comput., 16 (1995), 74-87. [24] J. Shen, Efficient spectral Galerkin method III. Polar and cylindrical geometries, SIAM J. SCI. Comput., 18 (1997), 1583-1604. [25] J. Shen, A new dual-petrov-Galerkin method for third and higher odd-order differential equations: Application to the KdV equation, SIAM J. Numer. Anal., 41 (2003), 1595-1619. [26] J. Shen and L. L. Wang, Legendre and Chebyshev-dual-Petrov-Galerkin methods for hyperbolic equations, Comput. Methods Appl. Mech. Engrg., 196 (2007), 3785-3797. [27] J. Shen and L. L. Wang, Some recent advances on spectral methods for unbounded domains, Commun. Comput. Phys., 5 (2009), 195-241. [28] J. Shen, T. Tang and L. L. Wang, Spectral Methods: Algorithms, Analysis, and Applications, Springer, Heidelberg, 2009. [29] J. Shen, Y. Wang and J. Xia, Fast structured direct spectral methods for differential equations with variable coefficients, I. the one dimensional case, SIAM J. Sci. Comput., 38 (2016), A28-A54. [30] L. L. Wang, M. D. Samson and X. Zhao, A well-conditioned collocation method using a pseudospectral integration matrix, SIAM J. Sci. Comput., 36 (2014), A907-A929. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/74907 | - |
| dc.description.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。 | zh_TW |
| dc.description.abstract | 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. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-17T09:10:01Z (GMT). No. of bitstreams: 1 ntu-108-R06246011-1.pdf: 856524 bytes, checksum: a9819043a341212f1707b35da6e7adfa (MD5) Previous issue date: 2019 | en |
| dc.description.tableofcontents | 口試委員會審定書 i
誌謝 ii 摘要 iii Abstract iv Contents v List of Figures vii List of Tables ix 1 Introduction 1 2 An inverse pseudospectral penalty matrix for the mixed differential operator d/dr a(r) d/dr in polar coordinate system 6 2.1 Formulation 6 2.1.1 Basic concepts of the Legendre pseudospectral method 6 2.1.2 Inverse pseudospectral penalty matrices for the d/dx operator 11 2.1.3 Invertible pseudospectral penalty matrix for the d/dx a(x) d/dx operator 15 2.1.4 An inverse matrix for the L operator 21 2.1.5 Applications to Poisson equations 28 2.2 Numerical validations 33 2.2.1 First order differential equations 34 2.2.2 Second order differential equations 35 2.2.3 Discussions 39 3 Multidomain inverse pseudospectral penalty matrices for the first order differential operator d/dx in Cartesian coordinate system 41 3.1 Formulation 41 3.1.1 Basic concepts of the Legendre pseudospectral method 41 3.1.2 Multidomain schemes for discretizing d/dx 48 3.2 Numerical validations 56 3.2.1 First order differential equations u′ = f 57 3.2.2 First order differential equations u′ + au = f 58 3.2.3 Nonlinear first order differential equations u′ + F(u)u = f 66 4 Conclusions 67 Bibliography 68 | |
| dc.language.iso | en | |
| dc.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 | zh_TW |
| dc.title | Integration Preconditioning Matrices for Pseudospectral Penalty Differentiation Operators with Applications to Differential Equations | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 108-1 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 鄧君豪,李勇達 | |
| dc.subject.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, | zh_TW |
| dc.subject.keyword | spectral/pseudospectral methods,penalty boundary conditions,integration preconditioning,mixed differential operators,multidomain schemes,piecewise continuous,upwind flux, | en |
| dc.relation.page | 70 | |
| dc.identifier.doi | 10.6342/NTU201904175 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2019-10-02 | |
| dc.contributor.author-college | 理學院 | zh_TW |
| dc.contributor.author-dept | 應用數學科學研究所 | zh_TW |
| 顯示於系所單位： | 應用數學科學研究所 | |
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