In this paper, we discuss the uniform convergence of the simple upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh for solving a singularly perturbed Robin boundary value problem, and investigate the m...In this paper, we discuss the uniform convergence of the simple upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh for solving a singularly perturbed Robin boundary value problem, and investigate the midpoint upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh to achieve better uniform convergence. The elaborate ε-uniform pointwise estimates are proved by using the comparison principle and barrier functions. The numerical experiments support the theoretical results for the schemes on the meshes.展开更多
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 app...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.展开更多
文摘In this paper, we discuss the uniform convergence of the simple upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh for solving a singularly perturbed Robin boundary value problem, and investigate the midpoint upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh to achieve better uniform convergence. The elaborate ε-uniform pointwise estimates are proved by using the comparison principle and barrier functions. The numerical experiments support the theoretical results for the schemes on the meshes.
基金The work of the third author was partially supported by the National Natural Science Foundation of China(Grant No.11601229)the Natural Science Foundation of Jiangsu Province of China(Grant No.BK20160784)+2 种基金The work of the last author was partially supported by FDCT of Macao SAR(029/2016/A1)National Natural Science Foundation of China(Grant Nos.11922120,11871489,11401608)Multi-Year research grant(2019-00154-FST,2017-00189-FST)of University of Macao。
文摘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.
