Real-time adaptive finite element solution of time-dependent Kohn–Sham equation

Bao, Gang; Hu, Guanghui; Liu, Di

doi:10.1016/j.jcp.2014.10.052

Cited by 13 publications

(24 citation statements)

References 32 publications

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“…It is noted that an efficient interpolation operation can be obtained based on this hierarchy geometry tree. People may also refer to [2,3] for the detail.…”

Section: The H-adaptive Mesh Methodsmentioning

confidence: 99%

“…There have been lots of numerical methods in the market to solve the TDKS equation in the time domain, people may refer to [3,7,14] and references therein for detail. People may also refer to [11,16,28] for numerical methods of Schrödinger equation.…”

Section: Introductionmentioning

confidence: 99%

“…People may also refer to [11,16,28] for numerical methods of Schrödinger equation. Among those grid-based numerical methods, the finite difference methods [1], the finite element methods [3,8,9,17,18,27], the discontinuous Galerkin methods [20], the wavelet methods [12] etc. are popular for the spatial discretization, while there are Runge-Kutta methods, commutator-free Magnus expansion methods, etc.…”

Section: Introductionmentioning

confidence: 99%

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An Implicit Solver for the Time-Dependent Kohn-Sham Equation

Yang¹

2021

NMTMA

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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.

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“…It is noted that an efficient interpolation operation can be obtained based on this hierarchy geometry tree. People may also refer to [2,3] for the detail.…”

Section: The H-adaptive Mesh Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An Implicit Solver for the Time-Dependent Kohn-Sham Equation

Yang¹

2021

NMTMA

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“…Remark 2.1. The above numerical framework has been adopted in the previous work [2][3][4][5]. On the numerical convergence of the methods, we refer to those papers for detail.…”

Section: An H-adaptive Finite Element Methods For Kohn-sham Equationmentioning

confidence: 99%

“…To resolve very singular external potential in the all-electron calculations, a nonuniform mesh can be designed according to the structure of the given molecule in advance, see [27,28]. However, when we want to study the geometry optimization, or the dynamics of a given molecule [4], the nucleus might move in the domain, and a well designed fixed mesh might not a good choice since the mesh might need to be regenerated to keep the accuracy, and it could be very CPU time demanding. The mesh adaptive methods can overcome the above issue effectively.…”

Section: Pseudopotentialmentioning

confidence: 99%

Towards Translational Invariance of Total Energy with Finite Element Methods for Kohn-Sham Equation

Bao

Liu³

2016

Commun. Comput. Phys.

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Numerical oscillation of the total energy can be observed when the Kohn- Sham equation is solved by real-space methods to simulate the translational move of an electronic system. Effectively remove or reduce the unphysical oscillation is crucial not only for the optimization of the geometry of the electronic structure, but also for the study of molecular dynamics. In this paper, we study such unphysical oscillation based on the numerical framework in [G. Bao, G. H. Hu, and D. Liu, An h-adaptive finite element solver for the calculations of the electronic structures, Journal of Computational Physics, Volume 231, Issue 14, Pages 4967–4979, 2012], and deliver some numerical methods to constrain such unphysical effect for both pseudopotential and all-electron calculations, including a stabilized cubature strategy for Hamiltonian operator, and an a posteriori error estimator of the finite element methods for Kohn-Sham equation. The numerical results demonstrate the effectiveness of our method on restraining unphysical oscillation of the total energies.

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