臺灣博碩士論文加值系統

GAMER 是一個利用繪圖處理單元(Graphic-processing-unit, GPU) 加
速及自適網格解析度調整(Adaptive-Mesh-Refinement, AMR) 的天文物理應用模擬程式。此程式已順利擴展成可以數值模擬磁流體動力學
(Magnetohydrodynamics, MHD)。磁流體動力學的數值模擬演算法是
具有限制傳輸(Constraint transport, CT) 技術的逆流邊角傳輸(Cornertransport-upwind, CTU) 演算法。另一方面，散度保持算符(Divergentpreserving operator) 保證了自適網格解析度調整中磁場的零散度條件。模擬結果顯示磁流體GAMER 程式跟其他高解析度均勻網格模擬一樣準確。我們引進一個新的三維測試問題。在此問題中磁場滿足阿諾德-貝爾特拉米-柴爾德里斯(Arnold-Beltrami-Childress) 樣態。這樣的電漿組態一開始會變成有片狀電流密度的紊流，但最後將收斂到最低能量的平衡態。此測試問題很適合檢驗本程式的效能。本程式在單一K20X 的繪圖處理單元下每秒可以演化2×10^7個網格，比藍水(Blue Waters) 超級電腦上的一顆擁有16個核心的中央處理單元(Center-processing-unit, CPU) 快25倍。我們同時發現當使用1024 個藍水超級電腦的計算節點，此程式的平行效率可高達70個百分點。
我們分析在輻射主宰時期的波形態暗物質，或者稱作 ψ 暗物質之線性微擾。在此分析中，暗物質微擾的演化可分成四個階段。其中在質量震盪(Mass oscillation) 之後的晚期階段，長波的 ψ 暗物質微擾幾乎和冷暗物質(Cold dark matter, CDM) 模型雷同。然而對於中短波的情況，在整個演化過程中，沒有一個階段跟冷暗物質一樣。我們同時討論餘弦型態純量場勢能的軸子模型(Axion model)。軸子模型的演化幾乎與 ψ 暗物質相同。但當軸子初始角度非常靠近勢能的頂端，有三個新特徵會顯現。其中最新穎的特徵是在某些波數的範圍內，頻譜會比冷暗物質高。這樣的差異可能會非常大以致於造成紅移10 以上的高紅移宇宙有顯著的改變。亞視界(Sub-horizon) 的擾動可以被馬蒂厄(Mathieu) 方程式準確地描述且受到參數不穩定(Parametric instability)的影響。這解釋了此新穎的特徵。

GAMER, a Graphic-processing-unit-accelerated Adaptive-MEsh-Refinement Astrophysical code, is extended to support magnetohydrodynamics (MHD), where the solver features the corner-transport-upwind (CTU) scheme with the constraint transport (CT) technique. The divergent preserving operator for adaptive mesh refinement (AMR) is applied to reinforce the divergence-free constraint on the magnetic field. Numerical results show GAMER-MHD is as robust as those given by high-resolution uniform-grid runs. We explore a new 3D MHD test, where the magnetic field assumes the Arnold-Beltrami-
Childress (ABC) configuration, temporarily becomes turbulent with current sheets and finally settles to a lowest-energy equilibrium state. This 3D problem is adopted for the performance test of GAMER-MHD. The single-GPU performance can reach 2×10^7 cell-updates/sec for K20X and is 25 times faster than a single 16-core CPU on the Blue Waters supercomputer. We also demonstrate a parallel efficiency of 70% using 1024 nodes on Blue Waters.
Linear perturbations of the wave dark matter, or ψ dark matter ( ψ DM), in the radiation-dominant era are analyzed. We identify four phases of evolution for ψ DM perturbations. While in late stages after mass oscillation long-wave ψ DM perturbations are almost identical to cold dark matter (CDM) perturbations except that intermediate-to-short waves that bear no resemblance with those of CDM throughout the whole evolutionary history. We also discuss the axion model with a cosine field potential. The evolution of axion models are almost identical to those of ψ DM, but three new features are found in the extreme case where the initial axion angle is near the field potential top. A particularly novel new feature is the spectral excess relative to the CDM model in some wave number range, where the excess may be so large that landscapes of high-redshift universe beyond z = 10 can be significantly altered. The sub-horizon perturbations are accurately described by Mathieu''s equation and subjected to parametric instability, which explains this novel feature.

誌謝v
摘要vii
Abstract ix
1 Introduction 1
I MAGNETOHYDRODYNAMICS WITH PARALLEL GRAPHICPROCESSING-
UNIT-ACCERALATED ADAPTIVE MESH REFINEMENT
CODE 3
2 GAMER-MHD 5
3 MHD Equations 9
4 Numerical Algorithm 13
4.1 MHD Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Adaptive Mesh Refinement . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3 Hybrid MPI/OpenMP/GPUs Parallelization . . . . . . . . . . . . . . . . 17
5 Numerical Results 19
5.1 Accuracy Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.1.1 Linear Wave Test . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.1.2 Shock Tube Test . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.1.3 Orszag and Tang Vortex Test . . . . . . . . . . . . . . . . . . . . 24
5.1.4 Blast Wave Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.1.5 Magnetic field with ABC pattern . . . . . . . . . . . . . . . . . . 34
5.2 Performance Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2.1 GPU Performance . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2.2 Overall Performance . . . . . . . . . . . . . . . . . . . . . . . . 43
6 Conclusions for Part I 51
II LINEAR WAVE DARK MATTER PERTURBATIONS 53
7 Brief Review of Wave Dark Matter 55
8 Governing Equations 59
9 Free Particle Model 63
9.1 Passive Evolution and Asymptotic Solutions . . . . . . . . . . . . . . . . 63
9.1.1 Phase (i): After mass oscillation 2H; k ≪ ma . . . . . . . . . . . 64
9.1.2 Phase (ii): Before mass oscillation . . . . . . . . . . . . . . . . . 70
9.2 Evolution of Perturbations in Full Treatment . . . . . . . . . . . . . . . . 72
9.2.1 Neutrino Decoupling . . . . . . . . . . . . . . . . . . . . . . . . 73
9.2.2 Photon Fluid Equation of State . . . . . . . . . . . . . . . . . . . 73
9.2.3 Baryon-Photon Drag . . . . . . . . . . . . . . . . . . . . . . . . 74
9.2.4 Matter Self-Gravity . . . . . . . . . . . . . . . . . . . . . . . . . 75
9.3 Critical Mode, Matter Power Spectrum and Sub-Horizon Dynamics . . . 76
9.4 Adiabatic Perturbations for Superhorizon Modes . . . . . . . . . . . . . 81
10 Axion Model 83
10.1 Abrupt Growth of δθ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
10.2 Parametric Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
10.3 Numerical Solution and General Nonlinear Model . . . . . . . . . . . . . 94
11 Conclusions for Part II 99
12 Discussion and Perspective 101
A Linear Stability Analysis of the ABC Magnetic Field Configuration 105
B Passive Evolution 115
C Full Treatment Evolution 131
D Particle Mass Dependence 137
E General Dispersion Relation 141
Bibliography 145

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