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用延續法與 Galerkin 有限單元法解旋轉-玻色愛因斯坦凝聚

林祐蓁

Published Jan 1, 2011• 0 citations
Publisher

Abstract

In this thesis, we study a continuation algorithm to trace the solution of two dimensional coupled nonlinear Schr‥odinger equations (CNLSE) that governs rotating Bose-Einstein condensates (BEC) around z-axis. We discretize CNLSE by Galerkin finite element method and prove the existence of the finite element method solution. The numerical tests depict the robust of the continuation algorithm for tracing the solution of the CNLSE. The numerical superfluid densities appear similarly as in [18], and this algorithm is quite practicable for describing the votex lattices in fast rotating BEC.

DOI
10.6845/nchu.2011.00012

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In this thesis, we study a continuation algorithm to trace the solution of two dimensional coupled nonlinear Schr‥odinger equations (CNLSE) that governs rotating Bose-Einstein condensates (BEC) around z-axis. We discretize CNLSE by Galerkin finite element method and prove the existence of the finite element method solution. The numerical tests depict the robust of the continuation algorithm for tracing the solution of the CNLSE. The numerical superfluid densities appear similarly as in [18], and this algorithm is quite practicable for describing the votex lattices in fast rotating BEC.
We study a radial basis function collocation method (RBFCM) to discretize a coupled nonlinear Schr o¨\ddot{o}o¨ dinger equation (CNLSE) that governs a two dimensional rotating Bose-Einstein condensate (BEC) with an angular momentum rotation term. We exploit a RBFCM-continuation method (RBFCM-CM) to trace the solution curve of the CNLSE. We compare the performance of the RBFCM-CM with the FEM-CM. We observe that the RBFCM-CM is very robust in a coarse grid for resolving the ground state solution with many vortices when the angular momentum rotation is close to the limit. Numerical results demonstrate the efficiency and accuracy of the RBFCM-CM for computing the superfluid density of the ground level of the BEC.
This work is concerned with the construction and analysis of structure-preserving Galerkin methods for computing the dynamics of rotating Bose-Einstein condensate (BEC) based on the Gross-Pitaevskii equation with angular momentum rotation. Due to the presence of the rotation term, constructing finite element methods (FEMs) that preserve both mass and energy remains an unresolved issue, particularly in the context of nonconforming FEMs. Furthermore, in comparison to existing works, we provide a comprehensive convergence analysis, offering a thorough demonstration of the methods' optimal and high-order convergence properties. Finally, extensive numerical results are presented to check the theoretical analysis of the structure-preserving numerical method for rotating BEC, and the quantized vortex lattice's behavior is scrutinized through a series of numerical tests.
In this thesis, we consider a Gross–Pitaevskii (GP) energy functional as a model for rotating one component and two components Bose–Einstein condensates (BEC) in two dimensions. This model can be non-dimensionalized to highlight a strong confinement regime with strong interaction between the two components. We introduce a new discretization of this energy, featuring both finite difference and fast Fourier transform approaches, in a bounded domain of R 2 using Dirichlet boundary conditions. We develop an explicit gradient method algorithm with adaptive step and projection (EPG) over the constraints manifold for the minimization of the discrete energy. This method allows for the derivation of a stopping criterion. Moreover, we propose two post processing algorithms for the numerical minimizers. One is aimed for single vortices while the other is aimed for vortex sheets. Both algorithms detect these structures and compute their indices. In a recent article [6], the authors study the behaviour of a segregated two components BEC set into rotation. They are able to prove that for large rotation, the interface between the components gets long, conjecturing the possibility towards vortex sheets. They also study the vortex structures of BEC in a segregated regime. In this thesis, we produce numerical simulations using EPG, validating these recent theoretical results, supporting conjectures, and covering different physical cases (the cases of one component and two components in coexistence regime [45, 39]). We also illustrate the efficiency of EPG compared to the well-known GPELab method [11] which requires to solve a linearly implicit system at each step. Finally, we adapt a few theorems found in the literature to our discrete setting. We prove the existence of a global minimizer of the GP energy functional for the finite difference scheme and study some of its properties. We also work on symmetrical problems we encountered in some of the numerical simulations.
We study efficient spectral-collocation and continuation methods (SCCM) for rotating two-component Bose-Einstein condensates (BECs) and rotating two-component BECs in optical lattices, where the second kind Chebyshev polynomials are used as the basis functions for the trial function space. A novel two-parameter continuation algorithm is proposed for computing the ground state and first excited state solutions of the governing Gross-Pitaevskii equations (GPEs), where the classical tangent vector is split into two constraint conditions for the bordered linear systems. Numerical results on rotating two-component BECs and rotating two-component BECs in optical lattices are reported. The results on the former are consistent with the published numerical results.
We describe multi-parameter continuation methods combined with spectral collocation methods for computing numerical solutions of rotating two-component Bose–Einstein condensates (BECs), which are governed by the Gross–Pitaevskii equations (GPEs). Various types of orthogonal polynomials are used as the basis functions for the trial function space. A novel multi-parameter/multiscale continuation algorithm is proposed for computing the solutions of the governing GPEs, where the chemical potential of each component and angular velocity are treated as the continuation parameters simultaneously. The proposed algorithm can effectively compute numerical solutions with abundant physical phenomena. Numerical results on rotating two-component BECs are reported.