求解耗散Schrödinger方程的一个无条件收敛的线性化紧致差分格式

删除或更新信息，请邮件至freekaoyan#163.com(#换成@)

本站小编 Free考研考试/2021-12-27

求解耗散Schrödinger方程的一个无条件收敛的线性化紧致差分格式王廷春, 王国栋, 张雯, 何宁霞南京信息工程大学数学与统计学院信息与计算科学系, 南京 210044 Difference Scheme for the Nonlinear Schrödinger Equation with a Dissipative Term WANG Tingchun, WANG Guodong, ZHANG Wen, HE NingxiaCollege of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

摘要
图/表
参考文献(0)
相关文章(10)
点击分布统计
下载分布统计
-->

全文: PDF(439 KB) HTML (1 KB)
输出: BibTeX | EndNote (RIS)

摘要本文致力于提出并分析一个求解耗散Schrödinger方程的线性化紧致差分格式.通过引入一个新的变量来消除耗散项，原方程可化为一个保持总质量和总能量的守恒系统.本文继而对这个守恒系统提出了一个高效的紧致差分格式，并证明该格式在离散意义下保持总质量和总能量守恒.运用不动点定理和标准的能量方法，新格式被证明是唯一可解的.不同于经典的基于数值解先验估计的分析方法，本文引进数学归纳法并结合H¹估计，在对网格比没有任何要求的前提下建立了格式在最大模意义下的最优误差估计.格式的收敛阶在空间和时间两个方向分别为4阶和2阶.数值结果验证了理论分析的正确性，并展示了新格式较已有格式的优越性.

服务

加入引用管理器

E-mail Alert

RSS

收稿日期: 2015-12-02

PACS:

O241.82

基金资助:国家自然科学基金（No.11571181）和‘青蓝工程’资助项目.

引用本文:

王廷春, 王国栋, 张雯, 何宁霞. 求解耗散Schrödinger方程的一个无条件收敛的线性化紧致差分格式[J]. 应用数学学报, 2017, 40(1): 1-15. WANG Tingchun, WANG Guodong, ZHANG Wen, HE Ningxia. Difference Scheme for the Nonlinear Schrödinger Equation with a Dissipative Term. Acta Mathematicae Applicatae Sinica, 2017, 40(1): 1-15.

链接本文:

http://123.57.41.99/jweb_yysxxb/CN/或 http://123.57.41.99/jweb_yysxxb/CN/Y2017/V40/I1/1

[1]	Agrawall G P. Nonlinear Fiber Optics (3rd ed). New York:Academic Press, 2001
[2]	Antoine X, Bao W. Christophe Besse, Computational methods for the dynamics of the nonlinear Schröoinger/Gross-pitaevskii equations. Comput. Phys. Comm., 2013, 184:2621-2633
[3]	Bao W, Cai Y. Mathematical theorey and numerical methods for Bose-Einstein condensation. Kinet. Relat. Mod., 2013, 6:1-135
[4]	Bao W, Jaksch D. An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equations with a focusing nonlinearity. SIAM J. Numer. Anal., 2003, 41:1406-1426
[5]	Browder F E. Existence and uniqueness theorems for solutions of nonlinear boundary value problems. In:Application of Nonlinear Partial Differential Equations. Proceedings of symposia in Applied Mathematics (Edited by R.Finn), AMS, Providence, 1965, 17:24-49
[6]	Chang Q, Jia E, Sun W. Difference schemes for solving the generalized nonlinear Schrödinger equation. J. Comput. Phys., 1999, 148:397-415
[7]	Dai W. An unconditionally stable three-level explicit difference scheme for the Schrödinger equation with a variable coefficient. SIAM. J. Numer. Anal., 1992, 29:174-181
[8]	Liao H, Shi H, Zhao Y. Numerical study of fourth-order linearized compact schemes for generalized NLS equations. Comput. Phys. Comm., 2014, 185:2240-2249
[9]	Liao H, Sun Z, Shi H, Wang T. Convergence of compact ADI method for solving linear Schrödinger equations. Numerical Methods for Partial Differential Equations, 2012, 28:1598-1619
[10]	Wang H. Numerical studies on the split-step finite difference method for nonlinear Schrödinger equations. Sci. Sin. Math., Applied Mathematics and Computation, 2004, 170:17-35
[11]	王廷春, 郭柏灵. 一维非线性Schrödinger方程的两个无条件收敛的守恒紧致差分格式. 中国科学:数学, 2011, 41(3):1-27(Wang T, Guo B. Unconditional convergence of two conservative compact difference schemes for nonlinear Schrödinger equation in one dimension. Sci. Sin. Math., 2011, 41:1-27)
[12]	Wang T, Zhao X. Optimal l∞ error estimates of finite difference methods for the coupled Gross-Pitaevskii equations in high dimensions. Science China:Mathematics, 2014, 57(10):2189-2214
[13]	Wang T. Optimal Point-wise Error Estimate of a Compact Difference Scheme for the Coupled Gross-Pitaevskii Equations in One Dimension. Journal of Scientific Computing, 2014, 59(1):158-186
[14]	Wang T. Optimal Point-wise Error Estimate of a Compact Difference Scheme for the Coupled Nonlinear Schrödinger Equations. Journal of Computational Mathematics, 2014, 32(1):58-74
[15]	Wang T, Guo B, Xu Q. Fourth-order compact and energy conservative difference schemes for the nonlinear Sch"odinger equation in two dimensions. Journal of Computational Physics, 2013, 243:382-399
[16]	Xie S, Li G, Yi S. Compact finite difference schemes with high accuracy for one-dimensional onlinear Schröoinger equation. Comput. Methods Appl. Mech. Engrg., 2009, 198:1052-1060
[17]	Delfour M, Fortin M, Payre G. Finite difference solutions of a nonlinear Schrödinger equation. J. Comput. Phys., 1981, 44:277-288
[18]	Peranich L S. Finite difference scheme for solving a non-linear schrödinger equation with a linear damping term. J. Comput. Phys., 1987, 68:501-505
[19]	Dai W, Nassar R. A Finite Difference Scheme for the Generalized Nonlinear Schrödinger Equation with Variable Coefficients. J. Comp. Math., 2000, 18:123-132
[20]	Zhang R, Yu X, Zhao G. A new finite difference scheme for a dissipative cubic nonlinear Schrödinger equation. Chin. Phys., 2011, 20:030204
[21]	Fei Z, Pérez-Garcia V M, Vázquez L. Numerical simulation of nonlinear Schrödinger systems:a new conservative scheme. Appl. Math. Comput., 1995, 71:165-177
[22]	Li S, Vu-Quoc L. Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation. SIAM J. Numer. Anal., 1995, 32:1839-1875
[23]	Samarskii A, Andreev V. Difference methods for elliptic equations. Moscow:Nauka, 1976(in Russian) (萨马尔斯基A, 安德列耶夫V. 椭圆型方程差分方法(译). 北京:科学出版社, 1984)

[1]	朱世辉, 张健. 带势非线性Schrödinger方程爆破解的集中性质[J]. 应用数学学报, 2016, 39(6): 938-953.
[2]	陈娟. 一类非线性Schödinger方程的Jacobi椭圆函数周期解[J]. 应用数学学报(英文版), 2014, 37(4): 656-661.
[3]	韦玉程, 刘广刚. 一类非线性Schrödinger方程正解的存在性[J]. 应用数学学报(英文版), 2013, 36(6): 1127-1140.
[4]	赵修文, 李一鸣, 舒级. 一类具非齐次项的非线性Schrödinger 方程驻波的稳定性[J]. 应用数学学报(英文版), 2010, 33(6): 1072-1077.
[5]	迟晓丽, 向新民. 非线性KdV-Schrödinger方程Fourier谱逼近的大时间性态[J]. 应用数学学报(英文版), 2010, 33(2): 348-362.
[6]	梁宗旗. 广义混合非线性Schr\"odinger方程的拟谱方法[J]. 应用数学学报(英文版), 2005, 28(4): 598-615.
[7]	邱曙熙. 平面负位势相关的Schrdinger方程的非负广义解[J]. 应用数学学报(英文版), 2003, 26(1): 176-180.
[8]	郭柏灵, 邢家省. 具调和振子的非线性Schrodinger方程[J]. 应用数学学报(英文版), 2001, 24(4): 554-560.
[9]	邱曙熙. 平面负位势相关的Schrodinger方程的广义Picard原理[J]. 应用数学学报(英文版), 2001, 24(3): 416-472.
[10]	苗长兴. 高阶Schrodinger方程的整体强解[J]. 应用数学学报(英文版), 1996, 19(2): 213-221.

PDF全文下载地址:

http://123.57.41.99/jweb_yysxxb/CN/article/downloadArticleFile.do?attachType=PDF&id=14253