【摘要】：偏微分方程被廣泛用于描述現(xiàn)代科學(xué)和工程計(jì)算中的許多實(shí)際問題。對(duì)于解析解不存在的微分方程,有效地進(jìn)行數(shù)值求解就尤為重要。隨著數(shù)學(xué)理論的深入研究和計(jì)算方法的不斷發(fā)展,有限差分法,有限體積法以及有限元方法已經(jīng)能夠較好得求解大部分微分方程。然而,對(duì)于數(shù)值解比較奇異的方程,如果采用均勻網(wǎng)格需要大量的計(jì)算資源,尤其是高維問題可能會(huì)超出計(jì)算機(jī)的計(jì)算能力。移動(dòng)網(wǎng)格方法根據(jù)數(shù)值解的特點(diǎn)對(duì)網(wǎng)格進(jìn)行重新分布,可以在不浪費(fèi)計(jì)算資源的前提下有效減少計(jì)算誤差。同時(shí),在實(shí)際數(shù)值計(jì)算中,選取均勻的時(shí)間步長可能需要較長的計(jì)算時(shí)間。時(shí)間自適應(yīng)方法可以在計(jì)算的過程中不斷調(diào)整時(shí)間步長,從而提高數(shù)值計(jì)算的效率。1925年,Einstein預(yù)測了在極低溫度下氣體中的粒子會(huì)處于相同的量子態(tài)。1995年,在稀薄的堿金屬氣體中發(fā)現(xiàn)了 Bose-Einstein凝聚態(tài)(BEC)。該問題引起了物理學(xué)家和數(shù)學(xué)家的廣泛關(guān)注,通常用非線性薛定諤(NLS)方程來描述Bose-Einstein凝聚態(tài)的單粒子性。大量的科研工作者在理論和數(shù)值方面對(duì)非線性薛定諤方程進(jìn)行了研究,并提出了一系列的數(shù)值求解方法。在無窮勢(shì)阱下,當(dāng)粒子間存在強(qiáng)相互作用時(shí),Bose-Einstein凝聚的基態(tài)解中會(huì)出現(xiàn)邊界層。因此,利用均勻網(wǎng)格計(jì)算該基態(tài)解需要大量的計(jì)算資源。同時(shí),求解Bose-Einstein凝聚的基態(tài)解就是在限制條件下求能量泛函的極小值點(diǎn),該能量在數(shù)值計(jì)算初期變化劇烈,而在接近收斂時(shí)變化非常緩慢,因此采用均勻時(shí)間步長需要較長的計(jì)算時(shí)間。根據(jù)該問題數(shù)值解在空間和時(shí)間上的特點(diǎn),在空間上利用移動(dòng)網(wǎng)格方法,在時(shí)間上利用時(shí)間自適應(yīng)方法能夠有效提高數(shù)值計(jì)算的效率。本文主要介紹一種時(shí)空自適應(yīng)有限元方法來求解Bose-Einstein凝聚態(tài)的基態(tài)解。首先,本文介紹了自適應(yīng)方法和非線性薛定諤方程的相關(guān)理論知識(shí)。其次,介紹了一維問題基于等分布原理的移動(dòng)網(wǎng)格方法,二維問題基于調(diào)和映射的移動(dòng)網(wǎng)格方法以及時(shí)間自適應(yīng)方法。然后,本文分析了不同勢(shì)阱下Bose-Einstein凝聚態(tài)基態(tài)解的數(shù)值特點(diǎn),提出了如何在空間上實(shí)現(xiàn)移動(dòng)網(wǎng)格技術(shù)以及在時(shí)間上實(shí)現(xiàn)自適應(yīng)。基于時(shí)空自適應(yīng)有限元方法,本文給出了一維和二維情況下Bose-Einstein凝聚基態(tài)解的數(shù)值算例,分析比較了均勻網(wǎng)格和移動(dòng)網(wǎng)格的數(shù)值結(jié)果,并指出時(shí)空自適應(yīng)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.
【學(xué)位授予單位】：浙江大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O241.82

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