博碩士論文 101222034 詳細資訊




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姓名 鄭俊祥(Jun-Siang Tin)  查詢紙本館藏   畢業系所 物理學系
論文名稱
(Pseudo Spectral Method for Holographic Josephson Junction)
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摘要(中) 在解愛因斯坦方程式時,數值方法是一個重要的技術,因為在大多數情況下,愛因 斯坦方程的解析解是無法得到的。當我們用數值方法解愛因斯坦方程時,方程式 的雙曲特性以及它的非線性是兩個我們必須克服的主要問題。非線性常會讓數值 方法的捨去誤差(truncated error)增大到無法接受的地步。然而,如果捨去誤差 足夠小的話,這個數值解在有限時間內還是準確的。在眾多數值方法中,當我們 要獲得高準確度的解時,波譜法(spectral method)常常是最好的工具。在這篇論 文中,我們運用特別一類稱為擬譜法(pseudo spectral method)來解 holographic Josephson junction 問題。我們考慮的這個問題是不隨時間改變的,因此雙曲特性 並不在我們的討論範圍內。
摘要(英) Numerical method has been an important technique in solving the Einstein equation, because in most cases, the analytical solution is not known. When solving the Einstein equation numerically, hyperbolic property and non-linearity are two big problems we have to overcome. Non-linearity will make the truncated error in numerical methods grows to an unacceptable value. However, if the truncated error is small enough, the solutions are still reliable in finite time. Among many different approaches of numerical methods, spectral methods are often the best tool when the problems have to be solved in high accuracy. In this thesis, we apply pseudo spectral method, which is a special class of spectral method, to solve a holographic Josephson junction problem. The problem we consider is time independent, so the hyperbolic property is not our concern.
關鍵字(中) ★ 波譜法
★ 擬譜法
★ AdS/CFT對應
關鍵字(英)
論文目次 1 Introduction 1
2 Pseudo Spectral Method 4
2.1 CollocationMethod............................ 4
2.2 Newton’sMethod............................. 5
2.3 CardinalFunction............................. 6
2.4 ChebyshevFunction ........................... 9
2.5 BoundaryCondition ........................... 11
2.6 GeneralizationtoHigherDimensions .................. 13
3 Holographic Josephson Junction 16
3.1 Introduction................................ 16 3.2 FieldEquationsandBoundaryConditions . . . . . . . . . . . . . . . 17 3.3 NumericalResults............................. 20
4 Conclusions 23
A Cardinal Function of Chebyshev Polynomials 24
B Matlab Code 26
Bibliography 38
參考文獻 [1] J. P. Boyd, Chebyshev and Fourier Spectral Methods (University of Michigan, 2000).
[2] W. S. Don and A. Solomonoff, “Accuracy Enhancement for Higher Derivatives using Chebyshev Collocation and Mapping Technieque,” SIAM J. Sci. Comp. 18, 1040-1055 (1997).
[3] B. Forngerg, A Pratical Guide to Pseudospectral Methods (University of Col- orado, 1996).
[4] S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, “Building a Holographic Su- perconductor,” Phys. Rev. Lett. 101, 031601 (2008) [arXiv:0803.3295 [hep-th]].
[5] G. T. Horowitz, J. E. Santos and B. Way, “A Holographic Josephson Junction,” Phys. Rev. Lett. 106, 221601 (2011) [arXiv:1101.3326 [hep-th]].
[6] C. Runge, “u ̈ber eimpirische Funktionen und die Interpolation zwischen a ̈quidistanten Ordinaten,” Zeitschrift fu ̈r Mathematik und Physik 46, 224-243 (1901).
[7] L. N. Trefethen, Spectral Methods in Matlab (Oxford University, 2000).
[8] J. A. C. Weideman and S. C. Reddy,
www.lmm.jussieu.fr/ hoepffner/phdcodes/chebdif.m
[9] http://en.wikipedia.org/wiki/Radius_of_convergence#Radius_of_
convergence_in_complex_analysis
指導教授 陳江梅(Chiang-Mei Chen) 審核日期 2015-5-7
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