Numerical Methods and Analysis of Computing Quasiperiodic Systems

IF 2.8 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis Pub Date : 2024-02-01 DOI:10.1137/22m1524783

Kai Jiang, Shifeng Li, Pingwen Zhang

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引用次数: 0

Abstract

SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 353-375, February 2024.
Abstract. Quasiperiodic systems are important space-filling ordered structures, without decay and translational invariance. How to solve quasiperiodic systems accurately and efficiently is a great challenge. A useful approach, the projection method (PM) [J. Comput. Phys., 256 (2014), pp. 428–440], 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 the 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 the quasiperiodic spectral method (QSM) based on this framework. Results demonstrate that the PM and QSM both have exponential decay, and the QSM (PM) is a generalization of the periodic Fourier spectral (pseudospectral) method. Then, we analyze the computational complexity of the PM and QSM in calculating quasiperiodic systems. The PM can use a fast Fourier transform, while the QSM cannot. Moreover, we investigate the accuracy and efficiency of the PM, QSM, and periodic approximation method in solving the linear time-dependent quasiperiodic Schrödinger equation.

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计算准周期系统的数值方法与分析

SIAM 数值分析期刊》第 62 卷第 1 期第 353-375 页，2024 年 2 月。摘要准周期系统是重要的空间填充有序结构，不存在衰减和平移不变性。如何准确高效地求解准周期系统是一个巨大的挑战。有人提出了一种有用的方法--投影法（PM）[J. Comput. Phys., 256 (2014), pp.各种研究表明，投影法是一种精确、高效的求解准周期系统的方法。然而，目前还缺乏对 PM 的理论分析。本文通过建立准周期函数及其高维周期函数的数学框架，对 PM 进行了严格的收敛性分析。我们还基于此框架给出了准周期谱方法（QSM）的理论分析。结果表明，PM 和 QSM 都具有指数衰减，而 QSM（PM）是周期傅里叶谱（伪谱）方法的广义化。然后，我们分析了 PM 和 QSM 计算准周期系统的计算复杂性。PM 可以使用快速傅立叶变换，而 QSM 则不能。此外，我们还研究了 PM、QSM 和周期近似法在求解线性时变准周期薛定谔方程时的精度和效率。

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来源期刊

SIAM Journal on Numerical Analysis 数学-应用数学

CiteScore

4.80

自引率

6.90%

发文量

110

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Numerical Analysis (SINUM) contains research articles on the development and analysis of numerical methods. Topics include the rigorous study of convergence of algorithms, their accuracy, their stability, and their computational complexity. Also included are results in mathematical analysis that contribute to algorithm analysis, and computational results that demonstrate algorithm behavior and applicability.