二维线性薛定谔方程的一种高效数值解法  

An Efficient Numerical Solution for the Two-Dimensional Linear Schrödinger Equation

作　　者：王建云 钟子新 田智鲲 WANG Jianyun;ZHONG Zixin;TIAN Zhikun(College of Science,Hunan University of Technology,Zhuzhou Hunan 412007,China;School of Computational Science and Electronics,Hunan Institute of Engineering,Xiangtan Hunan 411104,China)

机构地区：[1]湖南工业大学理学院,湖南株洲412007 [2]湖南工程学院计算科学与电子学院,湖南湘潭411104

出　　处：《湖南工业大学学报》2025年第5期98-102,共5页Journal of Hunan University of Technology

基　　金：湖南省普通高校教学改革基金资助项目(HNJG-2022-0834,HNJG-2023-0951);湖南省普通高校教学改革基金资助项目(HNJG-20230951);湖南省教育厅科学研究基金资助重点项目(20240518,21A0361)。

摘　　要：针对二维线性薛定谔方程,构造了一种新的半离散有限元两网格算法。将原本在细网格上求解薛定谔方程的有限元解,简化为先在粗网格上求解原方程的有限元解,然后利用在粗网格上求得的数值解,在细网格上将方程的实部和虚部进行解耦,求解两个椭圆方程的有限元解。分析了两网格有限元解与精确解在H1范数下的误差,并进行了数值计算实验,得到的数值结果与文献[10]中的误差具有相同的误差阶,且在不同时刻的误差更小。In view of a solution of the two-dimensional linear Schrödinger equation,a new semi-discrete finite element two-grid algorithm has thus been constructed.The finite element solution of the Schrödinger equation originally solved on a fine grid is simplified to first solve the finite element solution of the original equation on a coarse grid,followed by the adoption of the numerical solution obtained on the coarse grid to decouple the real and imaginary parts of the equation on the fine grid,thus solving the finite element solutions of two elliptical equations.An analysis has been made of the error between the finite element solution of two grids and the exact solution under the H1 norm,with numerical experiments conducted.The numerical results show that the error order remains the same as that in reference[10],with the error smaller at different times.

关 键 词：两网格算法 薛定谔方程 有限元方法 

分 类 号：O241.82[理学—计算数学]