求解Gross-Pitaevskii方程正基态的牛顿方法
Newton-based methods for finding the positive ground state of Gross-Pitaevskii equations
摘要：
Gross-Pitaevskii方程（GPE）的离散化导致了一个具有特征向量非线性（NEPv）的非线性本征值问题。在本文中，我们使用两种基于牛顿的方法来计算GPE的正基态。第一种方法来自于Liu提出的求解可饱和非线性薛定谔方程的Newton-Noda迭代法，该方法可以自然地转化为广义拟和。第二种方法结合了对分法和牛顿法的思想，使得块三对角线性系的每个子问题都能很容易地求解。我们给出了它的显式收敛性和计算复杂性分析，并通过数值实验验证了理论结果。
英文摘要：
The discretization of Gross-Pitaevskii equations (GPE) leads to a nonlinear eigenvalue problem with eigenvector nonlinearity (NEPv). In this paper, we use two Newton-based methods to compute the positive ground state of GPE. The first method comes from the Newton-Noda iteration for saturable nonlinear Schrödinger equations proposed by Liu, which can be transferred to GPE naturally. The second method combines the idea of the Bisection method and the idea of Newton method, in which, each subproblem involving block tridiagonal linear systems can be solved easily. We give an explicit convergence and computational complexity analysis for it. Numerical experiments are provided to support the theoretical results.
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