Layer-Splitting Methods for Time-Dependent Schrödinger Equations of Incommensurate Systems (Communications in Computational Physics, 2021)
This work considers numerical methods for the time-dependent Schrödingerequation of incommensurate systems. By using a plane wave method for spatial discretization, the incommensurate problem is lifted to a higher dimension that resultsin 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 eachcorresponds 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 thecomputational cost significantly and is natural to involve stochastic methods and parallel computing. Both theoretical analysis and numerical experiments are provided tosupport the reliability and efficiency of the algorithms.