论文标题
计算准系统的数值方法和分析
Numerical Methods and Analysis of Computing Quasiperiodic Systems
论文作者
论文摘要
Quasiperiodic系统是重要的空间填充有序结构，而无需衰减和翻译不变。如何准确有效地解决准膜系统是巨大的挑战。一种有用的方法，即投影方法（PM）[J.计算。 Phys。，256：428，2014]已提议计算准膜系统。各种研究表明，PM是求解准碘系统的准确有效方法。但是，缺乏对PM的理论分析。在本文中，我们通过建立Quasiperiodic函数的数学框架及其高维周期函数，对PM进行了严格的合并分析。我们还基于此框架对准光谱法（QSM）进行了理论分析。结果表明，PM和QSM均具有指数衰减，而QSM（PM）是周期性傅立叶光谱（伪谱）方法的概括。然后，我们在计算准碘系统中分析了PM和QSM的计算复杂性。 PM可以使用快速的傅立叶变换，而QSM不能。此外，我们研究了PM，QSM和周期性近似方法的准确性和效率，以求解线性依赖时间依赖性的icoasiperiodicschrödinger方程。
Quasiperiodic systems are important space-filling ordered structures, without decay and translational invariance. How to solve quasiperiodic systems accurately and efficiently is of great challenge. A useful approach, the projection method (PM) [J. Comput. Phys., 256: 428, 2014], has been proposed to compute quasiperiodic systems. Various studies have demonstrated that the PM is an accurate and efficient method to solve quasiperiodic systems. However, there is a lack of theoretical analysis of PM. In this paper, we present a rigorous convergence analysis of the PM by establishing a mathematical framework of quasiperiodic functions and their high-dimensional periodic functions. We also give a theoretical analysis of quasiperiodic spectral method (QSM) based on this framework. Results demonstrate that PM and QSM both have exponential decay, and the QSM (PM) is a generalization of the periodic Fourier spectral (pseudo-spectral) method. Then we analyze the computational complexity of PM and QSM in calculating quasiperiodic systems. The PM can use fast Fourier transform, while the QSM cannot. Moreover, we investigate the accuracy and efficiency of PM, QSM and periodic approximation method in solving the linear time-dependent quasiperiodic Schrödinger equation.
