求解非线性薛定谔方程的一类数值解法
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摘要
利用非标准有限差分方法构造了求解非线性薛定谔方程的两个非标准有限差分格式。对于离散后的差分格式,把关于时间和空间的步长函数作为分母逼近导数项。对于非线性项,通过非局部的离散方法计算了这两个非标准有限差分格式的局部截断误差。数值实验结果验证了非标准有限差分格式的有效性。
Two nonstandard finite difference schemes for solving the nonlinear Schrodinger equation were constructed by using the nonstandard finite difference method. For discrete difference schemes, the step-size function of time and space was taken as the denominator approximation derivative term. For the non-linear terms, the local truncation errors of these two non-standard finite difference schemes were calculated by non local discrete mode. The numerical results verified the effectiveness of the nonstandard finite difference scheme.
Two nonstandard finite difference schemes for solving the nonlinear Schrodinger equation were constructed by using the nonstandard finite difference method. For discrete difference schemes, the step-size function of time and space was taken as the denominator approximation derivative term. For the non-linear terms, the local truncation errors of these two non-standard finite difference schemes were calculated by non local discrete mode. The numerical results verified the effectiveness of the nonstandard finite difference scheme.
引文
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[14] FORSYTHE G E, WASOW W R. Finite-difference methods for partial differential equations[M]. New York:Wiley Publishing, 1960:102-104.
[2]李景美,张金良,王飞.柱(球)非线性薛定谔方程的精确解[J].河南科技大学学报(自然科学版),2018,39(2):83-86.
[3]员保云,庞晶.求解非线性薛定谔方程的几种方法[J].激光与光电子学进展,2014,51(4):040604-1.
[4]张凯瑞,姜子文.一维薛定谔方程的紧致有限体积方法[J].山东师范大学学报(自然科学版),2018,33(2):150-153.
[5]郑春雄.薛定谔方程的整体几何光学近似[J].计算数学,2018,40(2):241-226.
[6]刘荣华.非线性薛定谔方程的配置谱方法研究[D].昆明:云南财经大学,2018.
[7]胡馨文.非线性薛定谔方程保守恒性质间断有限元方法[D].湘潭:湘潭大学,2016.
[8]蒋朝龙,黄蓉芳,孙建强.耦合非线性薛定谔方程的平均离散梯度法[J].工程数学学报,2014,31(5):707-718.
[9]秦雯娣.几类偏微分方程的动力学相容的非标准有限差分方法[D].哈尔滨:哈尔滨工业大学,2015.
[10] MICKENS R E. A nonstandard finite-difference scheme for the Lotka-Volterra system[J]. Applied numerical mathematics, 2003, 45(2):309-314.
[11] MICKENS R E. Construction of a novel finite-difference scheme for a nonlinear diffusion equation[J]. Numerical methods for partial differential equations, 1991, 7(3):299-302.
[12] LARSSON S, THOMEE V. Partial differential quations with numerical methods[M]. Berlin:Springer Science&Business, 2008:131.
[13] LI C P,CHEN A,YE J J. Numerical approaches to fractional calculus and fractional ordinary differential equation[J].Journal of computational physics, 2011, 230(9):3352-3368.
[14] FORSYTHE G E, WASOW W R. Finite-difference methods for partial differential equations[M]. New York:Wiley Publishing, 1960:102-104.