Adaptive absorbing boundary conditions for Schrodinger-type equations: application to nonlinear and multi-dimensional problems
Zhenli Xu, Houde Han, Xiaonan Wu
备注：
18 pages; 12 figures. A short movie for the 2D NLS equation with absorbing boundary conditions can be downloaded at http://home.ustc.edu.cn/~xuzl/movie.avi. To appear in Journal of Computational Physics
我们提出了一种自适应方法，用于选择Schrödinger型方程吸收边界条件的波数参数。基于捕获人工边界附近局部频率信息的Gabor变换，通过能量加权方法确定参数，从而得到准最优吸收边界条件。结果表明，即使波函数由不同群速度的波组成，该方法也能最小化反射波。我们还通过耦合自适应方法，将分裂局部吸收边界（SLAB）方法[Z. Xu and H. Han, {\it Phys. Rev. E}, 74(2006), pp. 037704]扩展到多维非线性情况。通过一维和二维非线性Schrödinger方程的数值算例，展示了所讨论吸收边界条件的性质。
英文摘要：
We propose an adaptive approach in picking the wave-number parameter of absorbing boundary conditions for Schrödinger-type equations. Based on the Gabor transform which captures local frequency information in the vicinity of artificial boundaries, the parameter is determined by an energy-weighted method and yields a quasi-optimal absorbing boundary conditions. It is shown that this approach can minimize reflected waves even when the wave function is composed of waves with different group velocities. We also extend the split local absorbing boundary (SLAB) method [Z. Xu and H. Han, {\it Phys. Rev. E}, 74(2006), pp. 037704] to problems in multidimensional nonlinear cases by coupling the adaptive approach. Numerical examples of nonlinear Schrödinger equations in one- and two dimensions are presented to demonstrate the properties of the discussed absorbing boundary conditions.
Domain decomposition algorithms for two dimensional linear Schrödinger equation
Université de Toulouse（图卢兹大学；法国国家科学研究中心，南法综合理工学院）；CNRS, UPS IMT（CEA萨克雷家园模拟中心与北部法兰西岛大学保罗·潘勒韦实验室）；Maison de la Simulation, CEA Saclay France & Laboratoire Paul Painlevé, Université Lille Nord de France（尼斯大学J.A. 迪厄多内实验室与索菲亚安蒂波利斯国家信息与自动化研究所）；Laboratoire J.A. Dieudonné, Université de Nice & Inria Sophia Antipolis
Gradient flow finite element discretisations with energy-based adaptivity for excited states of Schrödingers equation
我们提出了一种有效的数值方法，该方法基于[Heid et al.，arXiv：1906.06954]，用于薛定谔方程激发态的数值逼近。特别是，该过程采用梯度流迭代和局部网格细化的自适应相互作用，从而在算法的每个步骤中保证能量衰减。计算测试突出表明，该策略能够提供高度准确的结果，相对于自由度的数量具有最佳的收敛速度。
Efficient Spectral and Spectral Element Methods for Eigenvalue Problems of Schrödinger Equations with an Inverse Square Potential
Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所）；Beijing Computational Science Research Center（北京计算科学研究中心）；Wayne State University（韦恩州立大学）
