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系統識別號 U0026-2008201920192500
論文名稱(中文) 質點網格法在不同數值近似法下之能量誤差分析
論文名稱(英文) Energy Error Analysis of Particle-In-Cell Simulations With Different Numerical Methods
校院名稱 成功大學
系所名稱(中) 太空與電漿科學研究所
系所名稱(英) Institute of Space and Plasma Sciences
學年度 107
學期 2
出版年 108
研究生(中文) 洪紹惟
研究生(英文) Shao-Wei Hong
學號 LA6051116
學位類別 碩士
語文別 中文
論文頁數 95頁
口試委員 指導教授-談永頤
口試委員-藍永強
口試委員-張博宇
中文關鍵字 電漿不穩定性  能量守恆  數值方法 
英文關鍵字 plasma instability  energy conservation  Particle-In-Cell (PIC) method 
學科別分類
中文摘要 電漿物理中的不穩定性過程可以使用質點網格法(Particle-In-Cell PIC)進行模擬,由於不穩定性的物理過程將伴隨著較劇烈的能量交換,在數值方法的局限之下,極有可能隨之產生更大的數值偏差,又因不穩定性的過程中必須滿足能量守恆,因此我們將紀錄系統每個時刻的粒子的動能與電場的能量,藉由分析系統的總能量,來評估不穩定性過程所產生的數值誤差是否可以忽略。
本篇論文將以不同數值近似法來進行比較,不穩定性的發生時,其所產生的誤差是否得到更好的控制,我們將動量方程式分別以Euler法、Velocity Verlet法、RK2法、RK4法進行比較,而空間電場的近似分別以二階、四階、六階的中央差分法進行比較。
而模擬的結果顯示,以Velocity Verlet法、RK2法、RK4法近似,在總能量的分析上,都維持不錯的能量守恆,然而透過模擬顯示,我們也確實觀察到,PIC模擬在不穩定性的發生過程中總能量確實存在的較大的數值偏差,雖在此時刻總能量的偏差相較於動能與電能變化,在程度上是可以忽略,但在不穩定性發生時總能量所產生的數值偏差,也會影響著後續能量變化的發展,透過模擬的比較中我們也得知,盡可能地提升PIC每個方程式的準確度,可以獲得較有效的改善,可以避免使用過小的時間步長(time step),而造成計算時間的增加。
英文摘要 Abstract

Energy Error Analysis of Particle-In-Cell Simulations With Different Numerical Methods

Author:Shao-Wei Hong
Advisor:S.W.Y.Tam
Institute of Space and Plasma Sciences, National Cheng Kung University
SUMMARY
From our simulations, the results show the Velocity Verlet method, RK2 method, and RK4 method all have a good accuracy for energy conservation. However we also observed a larger numerical error during the instability process. Although the error of total energy compared with the variation of energy change between the kinetic energy and the electrical energy is negligible, the numerical deviation would affect the subsequent numerical results. One needs to reduce the error as much as possible. Our simulation results show that using higher order numerical methods to approximate the momentum equation and reducing the time step can decrease the error during the plasma instability process.

Key word:plasma instability, energy conservation, Particle-In-Cell (PIC) method

INTRODUCTION
The Particle-In-Cell (PIC) method can be used to simulate the instability process in plasma physics. The limitations of the numerical method would cause the numerical errors to increase. Since the plasma instability process is associated with intense energy exchange and must satisfy the energy conservation, we record the total energy of the system at every time step, to evaluate the size of the numerical errors.

MATERIALS AND METHODS
We examine different numerical methods for their accuracy in PIC simulations: Euler method, Velocity Verlet method, RK2 method, and RK4 method. Spatially, we compare second-order, fourth-order, and sixth-order central finite-difference method for the Poisson equation.

RESULTS AND DISCUSSION
We use the Particle-In-Cell method to simulate the bump-on-tail instability process, and the results show an intense energy change during the instability process: the particles lose energy, and wave energy grows, and we can see the bump in the velocity distribution being smooth out. These results correctly correspond to plasma physics. Moreover we use the Fourier transform to get the frequency band and the wave number of the growing waves, and use the relation v=ω/k to calculate the corresponding phase velocity in the velocity distribution. We found that the range of phase velocities corresponds to the velocity range of positive slope, so the PIC method can correctly simulate plasma instability. From simulations, we know there is a larger numerical error during the instability process, and the error deviation can be reduced as we improve the accuracy of the approximation for any equation, or reduce the time step.


CONCLUSION
In our study, the Velocity Verlet method, RK2 method, and RK4 method are numerically stable, and we have to reduce all errors as much as possible to avoid the accumulation of errors affecting the subsequent numerical accuracy significantly. Improving the order of numerical method and reducing the time step are both effective ways to reduce the numerical errors. However, practically, to improve the order of numerical method is a more effective way to improve the accuracy of the entire system, because to reduce the time step increase the overall number of steps of the calculation, causing the round-off error to occur more often.
論文目次 摘要 I
Abstract II
誌謝 VI
圖目錄 IX
表目錄 XII
第一章 序論 1
1-1簡介 1
1-2研究動機 1
第二章 理論以及數學方法 3
2.1質點網格(Particle-In-Cell)的模擬方法 3
2.2 Velocity Verlet演算法 5
2.3 Runge-Kutta法 7
2.3.1經典四階 Runge-Kutta法 10
2.3.2 Runge-Kutta法聯立方程的形式 10
2.4 朗道阻尼(Landau damping)的推導 12
第三章 數值模擬的結果 17
3.1 模擬初始條件的設定 17
3.2 模擬的物理過程記錄 20
3.3 不同參數對系統能量變化的影響 28
3.3.1粒子數的比例不同對系統能量變化的影響 29
3.3.2調整兩粒子束速度偏差的距離對系統能量變化的影響 34
3.4 不同數值近似下系統能量守恆的狀況 40
第四章 結論以及未來工作 52
參考文獻 54
附錄 55
單方程式4階Runge-kutta 展開 55
多變數聯立4階Runge-kutta展開 61
常微分方程的初值問題(單步法one-step method) 92
配置法(The Collocation Method) 93
絕對穩定性(absolutely stable) 94
無因次化過程 95
參考文獻 [1] C.K. Birdsall, A.B. Langdon,Plasma physics via computer simulation. New York : Taylor & Francis, 2005
[2]https://www.physics.drexel.edu/~valliere/PHYS305/Diff_Eq_Integrators/Verlet_Methods/Diffrntleqn3.pdf
[3] P. O.J. Scherer, Computational physics : simulation of classical and quantum systems. New York : Springer, c2013.
[4] K. Hussain, F. Ismail, N, Senua., Solving directly special fourth-order ordinary differential equations using Runge-Kutta type method, Journal of Computational and Applied Mathematics 306 (2016) 179-199
[5] http://fourier.eng.hmc.edu/e176/lectures/NM/node42.html
[6] J.C.Butcher.,Coefficients for the study of Runge-Kutta integration processes. J.Austral.Math.Soc.3 (1963), 185–201.
[7] J. C. Butcher, Numerical Methods for Ordinary Differential Equations Second Edition. 2016, John Wiley & Sons, Ltd
[8] M.Tenenbaum,H.Pollard. ,Ordinary Differential Equations. Dover Pubns Chapter 12 pp. 763-765 ,1985
[9]S. Francis, Schaum’s Outlines of Theory and Problems of Numerical Analysis New York, NY: Mcgraw-Hill,Chapter 20 pp. 232-233, 1989
[10] F.F.Chen,Introduction to Plasma Physics and Controlled Fusion.New York and London:Plenum Press, second edition, 1983.
[11]R.L.Burden, ,Numerical Analysis.9th Edition Brooks/Cole 2011
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