薛定谔方程散射解的预处理迭代求解器
A preconditioned iterative solver for the scattering solutions of the Schrödinger equation
AI总结：
通过将薛定谔方程转化为耦合线性系统，提出一种谱限制在复平面象限的预处理器，结合多重网格和Krylov子空间方法高效求解变波数不定亥姆霍兹模型问题。
摘要：
薛定谔方程定义了量子粒子的动力学，一直是物理学中持续关注的领域。我们展示了如何通过简单的变换将薛定谔方程转化为一个耦合线性系统，其中每个对角块是一个高频亥姆霍兹问题。基于该模型，我们推导出具有强变化波数的不定亥姆霍兹模型问题。我们采用迭代方法求解这些问题。特别地，我们开发了一种预处理器，其谱被限制在复平面的一个象限内，从而可以通过标准组件的多重网格方法轻松求逆。该多重网格预处理器与合适的Krylov子空间方法结合，用于求解不定亥姆霍兹模型问题。本研究的目的是报告该预处理器对于模型问题的可行性。我们将此想法与其他流行的预处理想法进行比较，并讨论其优点。给出了数值实验的结果，这些结果补充了所提出的想法，并表明该预处理器可以在自动设置中使用。
英文摘要：
The Schrödinger equation defines the dynamics of quantum particles which has been an area of unabated interest in physics. We demonstrate how simple transformations of the Schrödinger equation leads to a coupled linear system, whereby each diagonal block is a high frequency Helmholtz problem. Based on this model, we derive indefinite Helmholtz model problems with strongly varying wavenumbers. We employ the iterative approach for their solution. In particular, we develop a preconditioner that has its spectrum restricted to a quadrant (of the complex plane) thereby making it easily invertible by multigrid methods with standard components. This multigrid preconditioner is used in conjuction with suitable Krylov-subspace methods for solving the indefinite Helmholtz model problems. The aim of this study is to report the feasbility of this preconditioner for the model problems. We compare this idea with the other prevalent preconditioning ideas, and discuss its merits. Results of numerical experiments are presented, which complement the proposed ideas, and show that this preconditioner may be used in an automatic setting.
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