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Quasiperiodic systems are important space-filling ordered structures, withoutdecay and translational invariance. How to solve quasiperiodic systemsaccurately and efficiently is of great challenge. A useful approach, theprojection method (PM) [J. Comput. Phys., 256: 428, 2014], has been proposed tocompute quasiperiodic systems. Various studies have demonstrated that the PM isan accurate and efficient method to solve quasiperiodic systems. However, thereis a lack of theoretical analysis of PM. In this paper, we present a rigorousconvergence analysis of the PM by establishing a mathematical framework ofquasiperiodic functions and their high-dimensional periodic functions. We alsogive a theoretical analysis of quasiperiodic spectral method (QSM) based onthis framework. Results demonstrate that PM and QSM both have exponentialdecay, and the QSM (PM) is a generalization of the periodic Fourier spectral(pseudo-spectral) method. Then we analyze the computational complexity of PMand QSM in calculating quasiperiodic systems. The PM can use fast Fouriertransform, while the QSM cannot. Moreover, we investigate the accuracy andefficiency of PM, QSM and periodic approximation method in solving the lineartime-dependent quasiperiodic Schr\"{o}dinger equation.

Numerical Methods and Analysis of Computing Quasiperiodic Systems