Numerical Analysis of Computing Quasiperiodic Systems
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Quasiperiodic systems, related to irrational numbers, are important space-filling ordered structures, without decay and translational invariance. There are some efficient numerical algorithms, such as the projection method (PM) [J. Comput. Phys., 256: 428, 2014], have been proposed to compute quasiperiodic systems. However, there is also a lack of theoretical analysis of these numerical methods. In this paper, we first establish a mathematical framework for the quasiperiodic function and its high-dimensional periodic function based on Birkhoff's ergodic theorem. Then we give a theoretical analysis of PM and quasiperiodic spectral method (QSM). Results demonstrate that PM and QSM both have exponential decay. Further, we find that QSM (PM) is a generalization of the conventional Fourier (pseudo) spectral method. And the PM can use fast Fourier transform to treat nonlinear problems and cross terms with an almost optimal computational amount. Finally, we use the quasiperiodic Schrödinger equation as an example to verify our theoretical results.
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