Please use this identifier to cite or link to this item: http://hdl.handle.net/11455/18149
標題: 利用有限差分與多層譜方法和延續法解非線性薛丁格方程式
Finite Differences and Multi-level Spectral-Galerkin and Continuation Methods for Nonlinear Schrödinger Equations
作者: 陳慧霜
Chen, Huei-Shuang
關鍵字: 多層譜方法
Multi-level spectral-Galerkin method
超譜方法
延續法
玻色-愛因斯坦凝體
pseudo-spectral method
continuation method
Bose-Einstein condensate
出版社: 應用數學系所
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摘要: 在這篇博士論文中, 我們研究三種數值方法來處理具物理參數的非線性特徵值問題, 如創新的多重網格-延續法, 多層譜-延續法與超譜-延續法, 其中譜方法和超譜方法的基底是由Fourier sine 函數為基底函數。我們運用這些演算法計算玻色-愛因斯坦凝體(BEC) 在雷射光晶格照射下的能階與超流密度。此物理系統可用非線性的薛丁格方程NLS 描述。在此正與負的散色長度都被考慮, 若散色長度為正且化學位能足夠大時, 則選擇適當的定義域時, 我們的數值結果顯示BEC 前幾個能階的高峰解的個數與週期位能的波數有關。 首先, 我們提出一種創新的多重網格-延續法處理具物理參數的非線性特徵值問題, 此種演算法是[1]以前所提的雙重網格演算法的推, 但可被有彈性的執行, 即所求的近似解(點) 可以不用位於同一層細網格上。我們的數值結果顯示, 對亮的孤粒子而言, 若週期位能分別用sine 或cosine 函數表示, 則基態解的高峰解個數分別為(1/d−1)^2與(1/d)^2, 在此d為光晶格距離, 但這兩個公式對暗的孤粒子, 亦即散色長度為負時並不成立。我們把對單一方程的數值計算堆廣到一維與兩維且兩個一組的NLS 做研究。 其次, 我們研究以Fourier sine 函數為基底函數的多層譜-延續法來解具物理參數的非線性特徵值問題。此種方法的優點是比只做單一層的計算量來的少。我們的數值結果顯示在此區間(0, 1)^n, n = 1, 2下, 當化學位能夠大時, 玻色-愛因斯坦凝體(BEC) 在雷射光晶 格照射下的基態高峰解個數為Π^n_{j=1}(1/dj − 1), 其中dj表示為x方向的光晶格距離(或是在二維中分別為x方向和y方向)。當區間(−l, l)^n, l > 1變得更大時, 對於玻色-愛因斯坦凝體基態高峰解的個數, 我們可以得到此通式Π^n{j=1}( 2l/dj − 1)。 最後, 我們研究以Fourier sine 函數為基底函數的超譜-延續法來解具物理參數的非線性特徵值問題。對於Gross-Pitaevskii equation(GPE) 解在週期位能下的能階, 這種演算法為一種快速又有效率的方法。我們的數值結果顯示在一維和二維下的週期位能能階。
In this dissertation we study three methods such as a novel multigrid-continuation method, multi-level spectral-Galerkin continuation method and pseudo-spectral continuation method where the Fourier sine functions are used as the basis functions for treating parameter-dependent problems. The proposed algorithms are exploited to compute energy levels and superfluid densities of Bose-Einstein condensation (BEC) in a periodic potential, which is governed by the nonlinear Schr¨odinger equation (NLS). Both positive and negative scattering lengths are onsidered in our numerical experiments. For positive scattering length, if the chemical potential is large enough, and the domain is properly chosen, the results show that the number of peaks of the first few energy states of the 2D BEC in a periodic potential depends on the wave number of the periodic potential. First, we present a novel multigrid-continuation method for treating parameterdependent problems. The proposed algorithm which can be flexibly implemented is a generalization of the two-grid discretization schemes [1]. That is, approximating points on a solution curve do not necessarily lie on the same fine grid. Our numerical results show that for bright solitons the number of peaks of the ground state solutions is (1/d−1)^2 and (1/d)^2, where the periodic potential is expressed in terms of the sine or the cosine functions, respectively. However, these formulae do not hold if the scattering length is negative. The numerical study is extended to the twocomponent, 1D and 2D BEC in a periodic potential. Next, we study multi-level spectral-Galerkin continuation methods for parameterdependent problems, where the Fourier sine functions are used as the basis functions. The algorithm which has the advantage over the single level method is that certain amounts of computational cost can be saved. Our numerical results show that if the chemical potential is large enough and the domain Ω = (0, 1)^n, n = 1, 2, the number of peaks of the ground state solution of the BEC in a periodic potential is Π^n_{j=1}(1/dj − 1), where dj is/are the distance of neighbor wells in the x-coordinate (respectively, x- and y-coordinate). For a large domain Ω = (−l, l)^n, l > 1, we obtain the more general formula Π^n_{j=1}( 2l/dj − 1) for the ground state solution of the BEC. Finally, we study pseudo-spectral continuation method for treating parameterdependent problems, where the Fourier sine functions are used as the basis functions. This algorithm was advocated as a fast and efficient method for computing the energy levels of the Gross-Pitaevskii quation (GPE) in a periodic potential. The numerical results of 1D and 2D GPE in a periodic potential are reported.
URI: http://hdl.handle.net/11455/18149
其他識別: U0005-2810200914132000
文章連結: http://www.airitilibrary.com/Publication/alDetailedMesh1?DocID=U0005-2810200914132000
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