Layer-splitting methods for time-dependent Schr\"{o}dinger equations of incommensurate systems

TL;DR

This paper introduces layer-splitting numerical methods for efficiently solving the time-dependent Schrödinger equation in incommensurate systems by decomposing the problem into periodic sub-problems, reducing computational cost.

Contribution

It proposes novel fully discrete time stepping schemes based on layer-splitting, enabling efficient computation for incommensurate systems with theoretical and numerical validation.

Findings

Significant reduction in computational cost.

Schemes handle periodic sub-problems efficiently.

Validated through theoretical analysis and numerical experiments.

Abstract

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.