Numerical Methods and Analysis of Computing Quasiperiodic Systems

TL;DR

This paper provides a rigorous convergence analysis of the projection method (PM) and quasiperiodic spectral method (QSM) for solving quasiperiodic systems, demonstrating their exponential decay properties and computational complexities.

Contribution

It offers the first theoretical convergence analysis of PM and QSM, establishing a mathematical framework for quasiperiodic functions and analyzing their accuracy and efficiency.

Findings

Both PM and QSM exhibit exponential decay.

QSM generalizes the periodic Fourier spectral method.

PM can utilize fast Fourier transform, unlike QSM.

Abstract

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 Schr\"{o}dinger equation.