  This work considers numerical methods for the time-dependent Schr\"{o}dinger
equation of incommensurate systems. By using a plane wave method for spatial
discretization, the incommensurate problem is lifted to a higher dimension that
results in semidiscrete differential equations with extremely demanding
computational cost. We propose several fully discrete time stepping schemes
based on the idea of "layer-splitting", which decompose the semidiscrete
problem into sub-problems that each corresponds to one of the periodic layers.
Then these schemes handle only some periodic systems in the original lower
dimension at each time step, which reduces the computational cost significantly
and is natural to involve stochastic methods and parallel computing. Both
theoretical analysis and numerical experiments are provided to support the
reliability and efficiency of the algorithms.
