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完整後設資料紀錄
DC 欄位 | 值 | 語言 |
---|---|---|
dc.contributor.advisor | 崔茂培(Mao-Pei Tsui) | |
dc.contributor.author | Sing-Yuan Yeh | en |
dc.contributor.author | 葉行遠 | zh_TW |
dc.date.accessioned | 2021-06-17T00:45:10Z | - |
dc.date.available | 2020-02-10 | |
dc.date.copyright | 2020-02-10 | |
dc.date.issued | 2020 | |
dc.date.submitted | 2020-02-05 | |
dc.identifier.citation | [1] M. ALCUBIERRE, B. BRUEGMANN, T. DRAMLITSCH, J. A. FONT, P. PAPADOPOULOS, E. SEIDEL, N. STERGIOULAS AND R. TAKAHASHI, Towards a Stable Numerical Evolution of Strongly Gravitating Systems in General Relativity: The Conformal Treatments
[2] R. ARNOWITT, S. DESER AND C. W. MISNER, The Dynamics of General Relativity. Gravitation, (1962). [3] J. M. BARDEEN AND T. PIRAN, General Relativistic Axisymmetric Rotating Systems: Coordinates and Equations, Phys. Rep. 96 (1983). [4] G. CALABRESE, J. PULLIN, O. REULA, O. SARBACHV ANS M. TIGLIO, Well posed constraintpreserving boundary conditions for the linearized Einstein equations, Commun. Math. Phys. (2002). [5] J. M. CENTRELLA, Dynamical Spacetimes and Numerical Relativity, Cambridge University Press (1986). [6] M. W. CHOPTUIK, Consistency of finitedifference solutions of Einstein s equations, Phys. Rev. Lett., 17 (1991) pp.912. [7] M. W. CHOPTUIK, Universality and Scaling in Gravitational Collapse of a Massless Scalar Field, Phys. Rev. Lett., 10 (1993). [8] M. W. CHOPTUIK, A Study of Numerical Techniques for Radiative Problem in General Relativity. [9] D. CHRISTODOULOU, The Formation of Black Holes and Singularities in Spherically Symmetric Gravitational Collapse, Comm. Math. Phys., 44 (1991), pp. 339-373. [10] D. CHRISTODOULOU, The Problem of a SelfGravitating Scalar Field, Comm. Math. Phys., 10 (1986), pp. 337361. [11] D. CHRISTODOULOU, A Mathematical Theory of Gravitational Collapse, Commun. Math. Phys. (1987). [12] M.W. CHOPTUIK, The (Unstable) Threshold of Black Hole Formation, General Relativity and Quantum Cosmology (1998). [13] C. R. EVANS, A Method for Numerical Relativity, (1984). [14] H. FRIEDRICH AND G. NAGY, The Initial Boundary Value Problem for Einstein's Vacuum Field Equations, Commun. Math. Phys., (1999). [15] C. GUNDLACH AND J. M. MARTINGARCIA, Symmetric Hyperbolicity and Consistent Boundary Conditions for Second-Order Einstein Equations, Phys.Rev. (2004). [16] C. GUNDLACH AND J.M. MARTÍNGARCÍA, Critical Phenomena in Gravitational Collapse, Living Rev. Relativ. 10, 5 (2007). [17] S. G. HAHN AND R. W. LINDQUIST, The two-body problem in geometrodynamics, Ann. Phys. 29 (2) pp 304–331 (1964). [18] S. HAWKING AND G. ELLIS HAWKING, The Large Scale Structure of SpaceTime, Cambridge: Cambridge University Press, 1973. [19] W. JAMES AND JR. YORK, The Kinematics and dynamics of general relativity, (1979). [20] L. E. KIDDER, M. A. SCHEEL AND S. A. TEUKOLSKY, Extending the lifetime of 3D black hole computations with a new hyperbolic system of evolution equations Phys. Rev. (2001). [21] P. D. LAX AND R. S. PHILLIPS, Local boundary conditions for dissipative symmetric linear differential operators, Commun. Pure Appl. Math, (1960). [22] A. LIGHTMAN AND R. H. PRICE, Problem Book in Relativity and Gravitation, Princeton University Press (1975). [23] M. M. MAY AND R. H. WHITE, Hydrodynamic Calculations of GeneralRelativistic Collapse, Phys. Rev. 141, 1232 (1966). [24] O. C. SCHNURER, geometric evolution equations, Universität Konstanz Lecture note (2007). [25] L. SMARR, Spacetimes generated by computers: Black Holes with Gavitational Radiation,N. Y. Acad. Sci. 302 (1977). [26] S. L. SHAPIRO AND S. A. TEUKOLSKY, Relativistic Stellar Dynamics on the Computer, Astrophysical Journal, 298 (1985). [27] S. TEUKOLSKY, On the Stability of the Iterated CrankNicholson Method in Numerical relativity, Phys. Rev. D, (2000). [28] R. M. WALD AND V. IYER, Trapped Surfaces in the Schwartzchild Geometry and Cosmic Censorship, Phys. Rev. (1991). | |
dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/66593 | - |
dc.description.abstract | 在文獻[7]中,Choptuik 提出了球對稱時空中愛因斯坦純量場方程塌縮的數值模擬,他發現存在一個臨界值p*,可以把愛因斯坦方程的解 S[p] 被分成兩組,一組會產生黑洞,另一組則否。他另外發現所產生的黑洞質量會滿足此關係 M_{BH}∝|p-p*|^γ,其中 γ≈0.37。這篇論文將會整理出研究上述問題的數值方法,而我們將用這些方法來實現愛因斯坦純量場的演化模擬,其中我們會使用3+1的方法將四維時空拆成時間和空間,再使用iterative Crank-Nicolson方法實現模擬。除此之外,還有一些模擬的細節也會呈現在此論文中。這篇論文將不僅確認了Choptuik的數值結果,也整理了一些有用的數值方法。 | zh_TW |
dc.description.abstract | In [7], Choptuik presented a numerical study of spherically symmetric collapse of Einstein scalar field equations. He discovered that a family of solutions S[p] with the property that a critical parameter value, p*, separates solutions containing black holes from those which do not. He also conjectured that the masses of black holes satisfy a power law M_{BH}∝|p-p*|^γ, where γ≈0.37 is a universal exponent. This thesis is a summary of numerical techniques which could be useful for the study of such problems. We use them to simulate the evolution of Einstein scalar field equations with spherical symmetry. The 3+1 formalism for numerical simulation is used and treatment of dynamical system is iterative Crank-Nicolson method. Moreover, the details of several numerical algorithm are presented in this thesis. This thesis not only confirms the simulative results of Choptuik but also summarize useful numerical techniques. | en |
dc.description.provenance | Made available in DSpace on 2021-06-17T00:45:10Z (GMT). No. of bitstreams: 1 ntu-109-R06246002-1.pdf: 5588589 bytes, checksum: c8726d8dde9e6368a65230fcf5d72898 (MD5) Previous issue date: 2020 | en |
dc.description.tableofcontents | 口試委員會審定書 . . . . . . . . . . . . . . . . . i
誌謝 . . . . . . . . . . . . . . . . . ii 摘要 . . . . . . . . . . . . . . . . . iii Abstract . . . . . . . . . . . . . . . . . iv Table of Contents . . . . . . . . . . . . . . . . . v List of Figures . . . . . . . . . . . . . . . . . vii List of Tables . . . . . . . . . . . . . . . . . viii 1 Introduction . . . . . . . . . . . . . . . . . 1 2 Review of the 3+1 Decomposition of Einstein’s Equations . . . . . . . . . . . . 4 2.1 Preliminary . . . . . . . . . . . . . . . . . 4 2.2 Projection the Einstein’s equations . . . . . . . . . . . . . . . . . 6 2.3 Choosing coordinate conditions . . . . . . . . . . . . . . . . . 7 2.4 Description of the evolution system . . . . . . . . . . . . . . . . . 9 3 Locating Black holes Horizons . . . . . . . . . . . . . . . . . 12 3.1 Apparent horizons . . . . . . . . . . . . . . . . . 13 3.2 The mass aspect function . . . . . . . . . . . . . . . . . 15 4 Boundary Treatments . . . . . . . . . . . . . . . . . 18 4.1 Basic energy estimation . . . . . . . . . . . . . . . . . 18 4.2 Maximally dissipative boundary conditions . . . . . . . . . . . . . . . . . 22 4.3 Constraintpreserving boundary conditions . . . . . . . . . . . . . . . . . 23 4.4 Restriction to spherically symmetric spacetime . . . . . . . . . . . . . . . . . 24 5 Numerical Theoretical Analysis . . . . . . . . . . . . . . . . . 30 5.1 Characteristic analysis of the scalar field . . . . . . . . . . . . . . . . . 30 5.2 Numerical radiation boundary conditions at r = R . . . . . . . . . . . . . . . . . 31 5.3 Numerical problems at r = 0 . . . . . . . . . . . . . . . . . 33 6 Numerical Methods . . . . . . . . . . . . . . . . . 35 6.1 Iterative CrankNicolson method . . . . . . . . . . . . . . . . . 35 6.2 Artificial dissipative term . . . . . . . . . . . . . . . . . 37 6.3 Numerical test for weak scalar field . . . . . . . . . . . . . . . . . 40 7 Critical phenomenon . . . . . . . . . . . . . . . . . 45 8 Reference . . . . . . . . . . . . . . . . . 51 Appendix A Stability Analysis of the Iterative Crank-Nicolson Scheme . . . . . . 54 Appendix B Evolution Equations . . . . . . . . . . . . . . . . . 57 | |
dc.language.iso | en | |
dc.title | 球對稱時空中愛因斯坦純量場方程數值模擬之研究 | zh_TW |
dc.title | A Numerical Study of Einstein Scalar Field Equations in Spherically Symmetric Spacetime | en |
dc.type | Thesis | |
dc.date.schoolyear | 108-1 | |
dc.description.degree | 碩士 | |
dc.contributor.oralexamcommittee | 蔡忠潤(Chung-Jun Tsai),李瑩英(Yng-Ing Lee) | |
dc.subject.keyword | 數值廣義相對論,臨界現象,黑洞, | zh_TW |
dc.subject.keyword | numerical relativity,critical phenomena,black holes, | en |
dc.relation.page | 59 | |
dc.identifier.doi | 10.6342/NTU202000311 | |
dc.rights.note | 有償授權 | |
dc.date.accepted | 2020-02-06 | |
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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