[1]王 兰,周媛兰,符莉丹.Schr?dinger方程的紧致修正交替方向格式[J].江西师范大学学报(自然科学版),2016,40(05):515-519.
　WANG Lan,ZHOU Yuanlan,FU Lidan.The Compact and Modified ADI Scheme for Schr?dinger Equations[J].Journal of Jiangxi Normal University:Natural Science Edition,2016,40(05):515-519.
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Schr?dinger方程的紧致修正交替方向格式()
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《江西师范大学学报》（自然科学版）[ISSN:1006-6977/CN:61-1281/TN]

卷:
40
期数:
2016年05期
页码:
515-519
栏目:
出版日期:
2016-10-01

文章信息/Info

Title:
The Compact and Modified ADI Scheme for Schr?dinger Equations
作者:
王 兰; 周媛兰; 符莉丹
江西师范大学数学与信息科学学院,江西 南昌 330022
Author(s):
WANG Lan; ZHOU Yuanlan; FU Lidan
College of Mathematics and Informatics,Jiangxi Normal University,Nanchang Jiangxi 330022,China
关键词:
Schr?dinger方程;  修正交替方向格式;  高阶紧致格式.
Keywords:
Schr?dinger equation;  modified ADI scheme;  high-order compact scheme.
分类号:
O 241.8
摘要:
研究了多维Schr?dinger方程的紧致修正交替方向格式.通过对J.Douglas等提出的交替方向格式进行误差分析可以发现其分裂误差远远大于时间离散的截断误差.为提高计算精度和效率,在格式中加入1个扰动项以提高分裂误差的阶数,使时间离散误差占优.数值实验验证了格式的优越性和扰动项的作用.
Abstract:
A compact and modified alternative direction implicit(ADI)scheme is contributed to multidimensional Schr?dinger equations.After analyzing the error of Douglas’ ADI scheme,it is discovered that the splitting error of the ADI scheme is much larger than truncation error from time approximation.A perturbation term is inserted into Douglas and Peaceman’s ADI scheme to improve the accuracy and computational efficiency.Moreover,the order of splitting error is bettered and the error from time discretization is dominant.Numerical tests verified the advantages of the new scheme and the important role of perturbation term.

参考文献/References:

[1] Douglas J,Peaceman D W.Numerical solution of two-dimensional heat flow problems [J].Am Inst Chem Eng,1955,1(4):505-512.
[2] Douglas J,Rachford H H.On the numerical solution of heat conduction problems in two and three space variables [J].Trans Am Math Soc,1956,136(82):421-439.
[3] Strang G.On the construction and comparison of difference scheme [J].SIAM J Numer Anal,1968,5(3):506-517.
[4] McLachlan R I,Quispel G R W.Splitting methods [J].Acta Numer,2002,11:341-434.
[5] Ma Yuanping,Kong Linghua,Hong Jialin.High-order compact splitting multisymplectic method for the coupled nonlinear Schr?dinger equations [J].Comput Math with Appl,2011,61(2):319-333.
[6] Weideman J A C,Herbst B M.Split-step methods for the solution of the nonlinear Schr?dinger equation [J].SIAM J Numer Anal,1986,23(3):485-507.
[7] Hong Jialin,Qin Mengzhao.Multisymplecticity of the centered box discretization for Hamiltonian PDEs with m≥2 space dimensions [J].Appl Math Letters,2002,15(8):1006-1011.
[8] Kong Linghua,Duan Yali,Wang Lan.Spectral-like resolution compact ADI finite difference method for the multi-dimensional Schr?dinger equations [J].Math Comput Model,2012,55(5/6):1798-1812.
[9] 马院萍,孔令华,王兰.2维 Schr?dinger方程的高阶ADI格式 [J].江西师范大学学报:自然科学版,2010,34(4):421-425.
[10] 符莉丹,孔令华,符芳芳.Schr?dinger方程的交替隐格式 [J].江西师范大学学报:自然科学版,2014,38(2):167-172.
[11] Douglas J,Kim S.Improved accuracy for locally one-dimensional methods for parabolic equation [J].Math Models Meth Appl Sci,2001,11(9):1563-1579.
[12] Li Jichun,Chen Yitung,Liu Guoqing.High-order compact ADI methods for the parabolic equations [J].Comput Math Appl,2006,52(8/9):1343-1356.
[13] 赵飞,蔡志权,葛永斌.1维非定向常对流扩散方程的有理型高阶紧致差分格式 [J].江西师范大学学报:自然科学版,2014,38(4):413-418.
[14] Gao Zhen,Xie Shusen.Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schr?dinger equations [J].Appl Numer Math,2011,61(4):593-614.
[15] 开依沙尔·热合曼,努尔买买提·黑力力.求解对流扩散方程的Padé逼近格式 [J].江西师范大学学报:自然科学版,2014,38(3):261-264.
[16] Lele S K.Compact finite difference schemes with spectral-like solution [J].J Comput Phys,1992,103(1):16-42.

备注/Memo

备注/Memo:
收稿日期:2016-05-20基金项目:国家自然科学基金(11301234,11211171)和江西省自然科学基金(20161ACB20006,20142BCB23009,20151BAB201012)资助项目.作者简介:王 兰(1979-),女,安徽池州人,讲师,主要从事微分方程数值方法研究.
更新日期/Last Update: 1900-01-01