Numerical solving the Schr\"odinger equation with incommensurate potentials presents a great challenge since its solutions could be space-filling quasiperiodic structures without translational symmetry nor decay. In this paper, we propose two high-accuracy numerical methods to solve the time-dependent quasiperiodic Schr\"odinger equation. Concretely, we discretize the spatial variables by the quasiperiodic spectral method and the projection method, and the time variable by the second-order operator splitting method. The corresponding convergence analysis is also presented and shows that the proposed methods both have exponential convergence rate in space and second order accuracy in time, respectively. Meanwhile, we analyse the computational complexity of these numerical algorithms. One- and two-dimensional numerical results verify these convergence conclusions, and demonstrate that the projection method is more efficient.

翻译：求解含非一致势的薛定谔方程数值解极具挑战性，因其解可能呈现空间填充的准周期结构，既无平移对称性也无衰减性。本文针对含时准周期薛定谔方程提出两种高精度数值方法：空间方向分别采用准周期谱方法和投影法进行离散，时间方向采用二阶算子分裂方法。我们给出了相应的收敛性分析，证明所提方法在空间上具有指数收敛速率，在时间上具有二阶精度。同时分析了这些数值算法的计算复杂度。一维和二维数值实验验证了收敛结论，并表明投影法更具计算效率。