Volume 14, Issue 1
An Implicit Solver for the Time-Dependent Kohn-Sham Equation

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 261-284.

Published online: 2020-10

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• Abstract

The implicit numerical methods have the advantages on preserving the physical properties of the quantum system when solving the time-dependent Kohn-Sham equation. However, the efficiency issue prevents the practical applications of those implicit methods. In this paper, an implicit solver based on a class of Runge-Kutta methods and the finite element method is proposed for the time-dependent Kohn-Sham equation. The efficiency issue is partially resolved by three approaches, i.e., an $h$-adaptive mesh method is proposed to effectively restrain the size of the discretized problem, a complex-valued algebraic multigrid solver is developed for efficiently solving the derived linear system from the implicit discretization, as well as the OpenMP based parallelization of the algorithm. The numerical convergence, the ability on preserving the physical properties, and the efficiency of the proposed numerical method are demonstrated by a number of numerical experiments.

• Keywords

Time-dependent Kohn-Sham equation, implicit midpoint scheme, finite element methods, h-adaptive mesh methods, complex-valued algebraic multigrid methods.

35Q41, 81Q05, 65M60, 65M55, 65M50

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• TXT
@Article{NMTMA-14-261, author = {Yang , Lei and Shen , Yedan and Hu , Zhicheng and Hu , Guanghui}, title = {An Implicit Solver for the Time-Dependent Kohn-Sham Equation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2020}, volume = {14}, number = {1}, pages = {261--284}, abstract = {

The implicit numerical methods have the advantages on preserving the physical properties of the quantum system when solving the time-dependent Kohn-Sham equation. However, the efficiency issue prevents the practical applications of those implicit methods. In this paper, an implicit solver based on a class of Runge-Kutta methods and the finite element method is proposed for the time-dependent Kohn-Sham equation. The efficiency issue is partially resolved by three approaches, i.e., an $h$-adaptive mesh method is proposed to effectively restrain the size of the discretized problem, a complex-valued algebraic multigrid solver is developed for efficiently solving the derived linear system from the implicit discretization, as well as the OpenMP based parallelization of the algorithm. The numerical convergence, the ability on preserving the physical properties, and the efficiency of the proposed numerical method are demonstrated by a number of numerical experiments.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2020-0040}, url = {http://global-sci.org/intro/article_detail/nmtma/18335.html} }
TY - JOUR T1 - An Implicit Solver for the Time-Dependent Kohn-Sham Equation AU - Yang , Lei AU - Shen , Yedan AU - Hu , Zhicheng AU - Hu , Guanghui JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 261 EP - 284 PY - 2020 DA - 2020/10 SN - 14 DO - http://doi.org/10.4208/nmtma.OA-2020-0040 UR - https://global-sci.org/intro/article_detail/nmtma/18335.html KW - Time-dependent Kohn-Sham equation, implicit midpoint scheme, finite element methods, h-adaptive mesh methods, complex-valued algebraic multigrid methods. AB -

The implicit numerical methods have the advantages on preserving the physical properties of the quantum system when solving the time-dependent Kohn-Sham equation. However, the efficiency issue prevents the practical applications of those implicit methods. In this paper, an implicit solver based on a class of Runge-Kutta methods and the finite element method is proposed for the time-dependent Kohn-Sham equation. The efficiency issue is partially resolved by three approaches, i.e., an $h$-adaptive mesh method is proposed to effectively restrain the size of the discretized problem, a complex-valued algebraic multigrid solver is developed for efficiently solving the derived linear system from the implicit discretization, as well as the OpenMP based parallelization of the algorithm. The numerical convergence, the ability on preserving the physical properties, and the efficiency of the proposed numerical method are demonstrated by a number of numerical experiments.

Lei Yang, Yedan Shen, Zhicheng Hu & Guanghui Hu. (2020). An Implicit Solver for the Time-Dependent Kohn-Sham Equation. Numerical Mathematics: Theory, Methods and Applications. 14 (1). 261-284. doi:10.4208/nmtma.OA-2020-0040
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