Bose-Einstein凝聚基态解的时空自适应方法

发布时间：2018-08-08 21:20

【摘要】：偏微分方程被广泛用于描述现代科学和工程计算中的许多实际问题。对于解析解不存在的微分方程,有效地进行数值求解就尤为重要。随着数学理论的深入研究和计算方法的不断发展,有限差分法,有限体积法以及有限元方法已经能够较好得求解大部分微分方程。然而,对于数值解比较奇异的方程,如果采用均匀网格需要大量的计算资源,尤其是高维问题可能会超出计算机的计算能力。移动网格方法根据数值解的特点对网格进行重新分布,可以在不浪费计算资源的前提下有效减少计算误差。同时,在实际数值计算中,选取均匀的时间步长可能需要较长的计算时间。时间自适应方法可以在计算的过程中不断调整时间步长,从而提高数值计算的效率。1925年,Einstein预测了在极低温度下气体中的粒子会处于相同的量子态。1995年,在稀薄的碱金属气体中发现了 Bose-Einstein凝聚态(BEC)。该问题引起了物理学家和数学家的广泛关注,通常用非线性薛定谔(NLS)方程来描述Bose-Einstein凝聚态的单粒子性。大量的科研工作者在理论和数值方面对非线性薛定谔方程进行了研究,并提出了一系列的数值求解方法。在无穷势阱下,当粒子间存在强相互作用时,Bose-Einstein凝聚的基态解中会出现边界层。因此,利用均匀网格计算该基态解需要大量的计算资源。同时,求解Bose-Einstein凝聚的基态解就是在限制条件下求能量泛函的极小值点,该能量在数值计算初期变化剧烈,而在接近收敛时变化非常缓慢,因此采用均匀时间步长需要较长的计算时间。根据该问题数值解在空间和时间上的特点,在空间上利用移动网格方法,在时间上利用时间自适应方法能够有效提高数值计算的效率。本文主要介绍一种时空自适应有限元方法来求解Bose-Einstein凝聚态的基态解。首先,本文介绍了自适应方法和非线性薛定谔方程的相关理论知识。其次,介绍了一维问题基于等分布原理的移动网格方法,二维问题基于调和映射的移动网格方法以及时间自适应方法。然后,本文分析了不同势阱下Bose-Einstein凝聚态基态解的数值特点,提出了如何在空间上实现移动网格技术以及在时间上实现自适应。基于时空自适应有限元方法,本文给出了一维和二维情况下Bose-Einstein凝聚基态解的数值算例,分析比较了均匀网格和移动网格的数值结果,并指出时空自适应方法的有效性。
[Abstract]:Partial differential equations are widely used to describe many practical problems in modern science and engineering calculation. It is very important to solve the differential equations which do not exist in analytic solutions. With the development of mathematical theory and calculation methods, the finite difference method, finite volume method and finite element method have been able to solve most differential equations. However, for the equations with singular numerical solutions, if the uniform grid requires a lot of computing resources, especially the high-dimensional problem, it may be beyond the computing power of the computer. The moving mesh method redistributes the mesh according to the characteristics of the numerical solution, which can effectively reduce the calculation error without wasting computing resources. At the same time, in the actual numerical calculation, it may take a long time to select the uniform time step. The time adaptive method can continuously adjust the time step in the calculation process, thus improving the efficiency of numerical calculation. In 1925, Einstein predicted that the particles in the gas at very low temperature would be in the same quantum state. Bose-Einstein condensed state (BEC). Was found in rarefied alkali metal gases. This problem has attracted the attention of physicists and mathematicians. The nonlinear Schrodinger (NLS) equation is usually used to describe the single particle properties of Bose-Einstein condensed matter. A large number of researchers have studied the nonlinear Schrodinger equation theoretically and numerically, and put forward a series of numerical solutions. In the infinite potential well, the boundary layer will appear in the ground state solution of Bose-Einstein condensate when there is a strong interaction between the particles. Therefore, it needs a lot of computing resources to calculate the ground state solution using uniform grid. At the same time, to solve the ground state solution of Bose-Einstein condensation is to find the minimum point of the energy functional under the limited condition. The energy changes sharply in the initial stage of numerical calculation, but changes very slowly when it is near convergence. Therefore, it takes a long calculation time to adopt the uniform time step. According to the spatial and temporal characteristics of the numerical solution of the problem, using the moving grid method in space and time adaptive method in time can effectively improve the efficiency of numerical calculation. In this paper, a spatiotemporal adaptive finite element method is introduced to solve the ground state solution of Bose-Einstein condensed matter. Firstly, the adaptive method and the theory of nonlinear Schrodinger equation are introduced. Secondly, the moving grid method based on equal distribution principle for one-dimensional problem, the mobile grid method based on harmonic mapping and the time adaptive method for two-dimensional problem are introduced. Then, the numerical characteristics of ground state solutions of Bose-Einstein condensed matter under different potential wells are analyzed, and how to realize mobile grid technology in space and self-adaptation in time are proposed. Based on the spatio-temporal adaptive finite element method, a numerical example of the ground state solution of Bose-Einstein condensation in one and two dimensions is given. The numerical results of uniform mesh and moving grid are analyzed and compared, and the validity of the spatio-temporal adaptive method is pointed out.
【学位授予单位】：浙江大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

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