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Numerical solving the Schr\"odinger equation with incommensurate potentialspresents a great challenge since its solutions could be space-fillingquasiperiodic structures without translational symmetry nor decay. In thispaper, we propose two high-accuracy numerical methods to solve thetime-dependent quasiperiodic Schr\"odinger equation. Concretely, we discretizethe spatial variables by the quasiperiodic spectral method and the projectionmethod, and the time variable by the second-order operator splitting method.The corresponding convergence analysis is also presented and shows that theproposed methods both have exponential convergence rate in space and secondorder accuracy in time, respectively. Meanwhile, we analyse the computationalcomplexity of these numerical algorithms. One- and two-dimensional numericalresults verify these convergence conclusions, and demonstrate that theprojection method is more efficient.

High-accuracy numerical methods and convergence analysis for Schr\"odinger equation with incommensurate potentials