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Volume 30, Issue 5
Layer-Splitting Methods for Time-Dependent Schrödinger Equations of Incommensurate Systems

Ting Wang, Huajie Chen, Aihui Zhou & Yuzhi Zhou

Commun. Comput. Phys., 30 (2021), pp. 1474-1498.

Published online: 2021-10

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  • Abstract

This work considers numerical methods for the time-dependent Schrödinger 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.

  • Keywords

Incommensurate system, time-dependent Schrödinger equation, time stepping scheme.

  • AMS Subject Headings

35Q40, 35Q41, 65L05, 65L20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-30-1474, author = {Wang , TingChen , HuajieZhou , Aihui and Zhou , Yuzhi}, title = {Layer-Splitting Methods for Time-Dependent Schrödinger Equations of Incommensurate Systems}, journal = {Communications in Computational Physics}, year = {2021}, volume = {30}, number = {5}, pages = {1474--1498}, abstract = {

This work considers numerical methods for the time-dependent Schrödinger 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0070}, url = {http://global-sci.org/intro/article_detail/cicp/19937.html} }
TY - JOUR T1 - Layer-Splitting Methods for Time-Dependent Schrödinger Equations of Incommensurate Systems AU - Wang , Ting AU - Chen , Huajie AU - Zhou , Aihui AU - Zhou , Yuzhi JO - Communications in Computational Physics VL - 5 SP - 1474 EP - 1498 PY - 2021 DA - 2021/10 SN - 30 DO - http://doi.org/10.4208/cicp.OA-2021-0070 UR - https://global-sci.org/intro/article_detail/cicp/19937.html KW - Incommensurate system, time-dependent Schrödinger equation, time stepping scheme. AB -

This work considers numerical methods for the time-dependent Schrödinger 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.

Ting Wang, Huajie Chen, Aihui Zhou & Yuzhi Zhou. (2021). Layer-Splitting Methods for Time-Dependent Schrödinger Equations of Incommensurate Systems. Communications in Computational Physics. 30 (5). 1474-1498. doi:10.4208/cicp.OA-2021-0070
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