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 - <p style="text-align: justify;">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.</p>