高阶SF-SFDTD方法在含时薛定谔方程求解中的应用研究  

Application of high-order SF-SFDTD scheme to solving a time-dependent Schrödinger equation

作　　者：谢国大 潘攀[1,2] 任信钢[1,2] 冯乃星 方明[1,2] 李迎松 黄志祥[1,2] Xie Guo-Da;Pan Pan;Ren Xin-Gang;Feng Nai-Xing;Fang Ming;Li Ying-Song;Huang Zhi-Xiang(School of Electronic and Engineering,Anhui University,Hefei 230601,China;Key Laboratory of Intelligent Computing and Signal Processing,Ministry of Education,Anhui University,Hefei 230601,China)

机构地区：[1]安徽大学电子信息工程学院,合肥230601 [2]安徽大学,智能计算与信号处理教育部重点实验室,合肥230601

出　　处：《物理学报》2024年第3期72-81,共10页Acta Physica Sinica

基　　金：国家自然科学基金(批准号:62201003,2022YFA1404003,62101002,U20A20164,61901001)资助的课题。

摘　　要：时域有限差分方法(finite-difference time-domain,FDTD(2,2))被广泛用于量子力学中薛定谔方程的求解,然而受Courant-Friedrichs-Lewy(CFL)条件的影响,计算空间中的网格尺寸会限制时间步长的取值范围,极大降低了FDTD(2,2)方法的数值计算效率.另外,FDTD(2,2)方法在时间域和空间域只具有二阶数值精度,在计算中往往会导致较大的误差累计,影响仿真结果的正确性.为了克服这些问题,结合空间滤波方法(spatial filtering,SF)和高阶辛时域有限差分(symplectic finite-difference time-domain,SFDTD(3,4))方法(3和4分别表示时间和空间数值精度),提出了一种时间稳定性条件可扩展的SF-SFDTD(3,4)方法用于求解含时薛定谔方程.SF-SFDTD(3,4)方法无需对传统SFDTD(3,4)方法的迭代公式进行进一步的推导,只需要在每一次的数值迭代过程中加入空间滤波操作,滤除因采用不满足CFL条件的时间步长而产生的不稳定空间域高频分量,保证数值方法的稳定性,因此所提方法与传统SFDTD(3,4)方法具有较高的兼容性.同时,理论分析了SF-SFDTD(3,4)方法的数值色散误差.最后,通过数值算例验证了本文所提方法的正确性和有效性.The traditional finite-difference time-domain(FDTD(2,2))method with second-order numerical accuracy in time and space has been extensively employed in the field of quantum mechanics to solve the Schrödinger equation.Nevertheless,the presence of the Courant-Friedrichs-Lewy(CFL)condition imposes limitations on the grid size in the computational space,thereby constraining the admissible range of time steps.Accordingly,the efficiency of the FDTD(2,2)method significantly decreases.In addition,the second-order numerical accuracy of the FDTD(2,2)method both in time domain and in space domain often results in significant error accumulation during calculations,thereby undermining the fidelity of the simulation results.To surmount the constraints imposed by the CFL stability conditions and enhance the accuracy of computations,a novel approach termed SF-SFDTD(3,4)method has been proposed,with 3 and 4 referring to the accuracy in space and time,respectively.This method combines spatial filtering(SF)with the high-order symplectic finitedifference time-domain(SFDTD)method.Its primary objective is to solve the time-dependent Schrödinger equation while ensuring time stability and scalability.The SF-SFDTD(3,4)method obviates the need for further deriving the iterative formula employed in the conventional SFDTD(3,4)method.Therefore,the method under consideration exhibits a remarkable degree of compatibility with its traditional counterpart.It is merely necessary to include a spatial filtering operation during each numerical iteration to eliminate spatial high-frequency components arising from the utilization of time step sizes that fail to satisfy the CFL stability condition,thereby ensuring the stability of the numerical scheme.Moreover,when the time step value satisfies the CFL stability condition,the amplitude of the high-frequency component approaches zero,thereby exerting a minimal influence on the accuracy of the computational results.The adoption of time steps that do not meet the CFL stability conditions leads to an ampl

关 键 词：辛时域有限差分方法 空间滤波方法 时间稳定性条件 薛定谔方程 

分 类 号：O413.1[理学—理论物理] O241.82[理学—物理]