五次非线性Schr?dinger方程的一个新型守恒紧致差分格式
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摘要
本文研究了带五次项的非线性Schr?dinger方程初边值问题.利用有限差分法构造了一个四阶紧致差分格式,证明格式在离散意义下保持原问题的两个守恒性质,即质量守恒和能量守恒.引入"抬升"技巧,运用标准的能量方法和数学归纳法建立了误差的最优估计,证明数值解在空间和时间两个方向分别具有四阶和二阶精度.数值实验对理论结果进行了验证,并与已有结果进行了对比,结果表明本文格式在保持精度相当的前提下具有更高的计算效率.
In this paper, we study the nonlinear Schr?dinger equation with a quintic terma of the initial boundary value problem. By using the finite difference method to construct a fourth-order compact finite difference scheme, we prove that the scheme preserves the total mass and energy, respectively. By introducing the lifting technique, the optimal error estimate of the proposed scheme is established by using the standard energy method and the mathematical induction. It is proved that the numerical solution has accuracy of fourth-order and second-order in space and time, respectively. Numerical experiments are given to verify the theoretical results and compared with the existing results, which show that the proposed scheme has higher computational efficiency under the condition of maintaining high accuracy.
In this paper, we study the nonlinear Schr?dinger equation with a quintic terma of the initial boundary value problem. By using the finite difference method to construct a fourth-order compact finite difference scheme, we prove that the scheme preserves the total mass and energy, respectively. By introducing the lifting technique, the optimal error estimate of the proposed scheme is established by using the standard energy method and the mathematical induction. It is proved that the numerical solution has accuracy of fourth-order and second-order in space and time, respectively. Numerical experiments are given to verify the theoretical results and compared with the existing results, which show that the proposed scheme has higher computational efficiency under the condition of maintaining high accuracy.
引文
[1]张鲁明,常谦顺.带五次项的非线性Schr?dinger方程的守恒数值格式[J].应用数学, 1999, 12:65–71.
[2]张鲁明,常谦顺.带五次项的非线性Schr?dinger方程的差分解法[J].应用数学学报, 2000, 23:351–358.
[3] Bao W, Jaksch D, Markowich P A. Numerical solution of the Gross-Pitaevskii equation for BoseEinstein condensation[J]. J. Comput. Phys., 2003, 187:318–342.
[4] Bao W, Li H, Shen J. A generalized-Lagueree-Fourier-Hermite pseudospectral method for computing the dynamics of rotating Bose-Einstein condensates[J]. SIAM. J. Sci. Comput., 2009, 31:3685–3711.
[5] Bao W, Shen J. A Fourth-order time-splitting Lagueree-Hermite psectralmethod for Bose-Einstein condensates[J]. SIAM J. Sci. Comput., 2005, 26:2010–2028.
[6] Argyris J, Haase M. An engineer’s guide to solution phenomena:application of thefinite element method[J]. Comput. Meth. Appl. Mech. Engrg., 1987, 61:71–122.
[7] Akrivis G, Dougalis V, Karakashian O. On fully discrete Galerkin methods for second order temporal accuracy for the nonlinear Schr?dinger equation[J]. Numer. Math., 1991, 59:31–53.
[8] Karakashian O, Makridakis C. A space-timefinite element method for the nonlinear Schr?dinger equation:the discontinuous Galerkin method[J]. Math. Comp., 1998, 67:479–499.
[9] Xu Y, Shu C W. Local discontinuous Galerkin methods for nonlinear Schr?dinger equation[J]. J.Comput. Phys., 2005, 205:72–77.
[10] Dehghan M, Mirzaei D. Numerical solution to the unsteady two-dimensional Schr?dinger equation using meshless local boundary integral equation method[J]. Intern. J. Numer. Meth. Engin., 2008,76:501–520.
[11] Dehghan M, Mirzaei D. The meshless local Petrov-Galerkin(MLPG)method for the generalizad two-dimensional non-linear Schr?dinger equation[J]. Engin. Anal. Bound. Elem., 2008, 32:747–756.
[12] Taha T R, Ablowitz M J. Analytical and numerical aspects of certain nonlinear evolution equations[J]. J. Comput. Phys., 1984, 5:203–230.
[13] Cloot A, Herbet B M, Weideman J A C. A numerical study of the nonlinear Schr?dinger equation involving quintic terms[J]. J. Comput. Phys., 1990, 86:127–146.
[14] Dehghan M, Taleei A. A compact split-stepfinite difference method for solving the nonlinear Schr?dinger equation with constant and variable coeffcients[J]. Comput. Phys. Comm., 2010, 181:43–51.
[15]王廷春,郭柏灵.一维非线性Schr?dinger方程的两个无条件收敛的守恒紧致差分格式[J].中国科学:数学, 2011, 41:207–233.
[16] Sanz-Sema J M. Methods for the numerical solution of the nonlinear Schr?dinger equation[J]. Math.Comp., 1984, 43:21–27
[17] Chang Q, Jia E, Sun W. Di?erence schemes for solving the generalized nonlinear Schr?dinger equation[J]. J. Comput. Phys., 1999, 148:397–415.
[18] Delfour M, Fortin M, Payre G. Finite-difference Solution of a Non-linear Schr?dinger Equattion[J].J. Comput. Phys., 1981, 44:277–288.
[19]王询,曹圣山.带五次项的非线性Schr?dinger方程新差分格式[J].中国海洋大学学报, 2009, 39:487–491.
[20]初日辉.带五次项的非线性Schr?dinger方程的一个紧致差分格式[J].江苏师范大学学报, 2014, 32:53–57.
[21] Samarskii A A, Andreev V B. Di?erence methods for elliptic equations[M]. Moscow:Nauka. 1976.
[22] Wang T, Zhao X. Optimal l∞error estimates offinite difference methods for the coupled GrossPitaevskii equations in high dimensions[J]. Sci. China Math., 2014, 57(10):2189–2214.
[2]张鲁明,常谦顺.带五次项的非线性Schr?dinger方程的差分解法[J].应用数学学报, 2000, 23:351–358.
[3] Bao W, Jaksch D, Markowich P A. Numerical solution of the Gross-Pitaevskii equation for BoseEinstein condensation[J]. J. Comput. Phys., 2003, 187:318–342.
[4] Bao W, Li H, Shen J. A generalized-Lagueree-Fourier-Hermite pseudospectral method for computing the dynamics of rotating Bose-Einstein condensates[J]. SIAM. J. Sci. Comput., 2009, 31:3685–3711.
[5] Bao W, Shen J. A Fourth-order time-splitting Lagueree-Hermite psectralmethod for Bose-Einstein condensates[J]. SIAM J. Sci. Comput., 2005, 26:2010–2028.
[6] Argyris J, Haase M. An engineer’s guide to solution phenomena:application of thefinite element method[J]. Comput. Meth. Appl. Mech. Engrg., 1987, 61:71–122.
[7] Akrivis G, Dougalis V, Karakashian O. On fully discrete Galerkin methods for second order temporal accuracy for the nonlinear Schr?dinger equation[J]. Numer. Math., 1991, 59:31–53.
[8] Karakashian O, Makridakis C. A space-timefinite element method for the nonlinear Schr?dinger equation:the discontinuous Galerkin method[J]. Math. Comp., 1998, 67:479–499.
[9] Xu Y, Shu C W. Local discontinuous Galerkin methods for nonlinear Schr?dinger equation[J]. J.Comput. Phys., 2005, 205:72–77.
[10] Dehghan M, Mirzaei D. Numerical solution to the unsteady two-dimensional Schr?dinger equation using meshless local boundary integral equation method[J]. Intern. J. Numer. Meth. Engin., 2008,76:501–520.
[11] Dehghan M, Mirzaei D. The meshless local Petrov-Galerkin(MLPG)method for the generalizad two-dimensional non-linear Schr?dinger equation[J]. Engin. Anal. Bound. Elem., 2008, 32:747–756.
[12] Taha T R, Ablowitz M J. Analytical and numerical aspects of certain nonlinear evolution equations[J]. J. Comput. Phys., 1984, 5:203–230.
[13] Cloot A, Herbet B M, Weideman J A C. A numerical study of the nonlinear Schr?dinger equation involving quintic terms[J]. J. Comput. Phys., 1990, 86:127–146.
[14] Dehghan M, Taleei A. A compact split-stepfinite difference method for solving the nonlinear Schr?dinger equation with constant and variable coeffcients[J]. Comput. Phys. Comm., 2010, 181:43–51.
[15]王廷春,郭柏灵.一维非线性Schr?dinger方程的两个无条件收敛的守恒紧致差分格式[J].中国科学:数学, 2011, 41:207–233.
[16] Sanz-Sema J M. Methods for the numerical solution of the nonlinear Schr?dinger equation[J]. Math.Comp., 1984, 43:21–27
[17] Chang Q, Jia E, Sun W. Di?erence schemes for solving the generalized nonlinear Schr?dinger equation[J]. J. Comput. Phys., 1999, 148:397–415.
[18] Delfour M, Fortin M, Payre G. Finite-difference Solution of a Non-linear Schr?dinger Equattion[J].J. Comput. Phys., 1981, 44:277–288.
[19]王询,曹圣山.带五次项的非线性Schr?dinger方程新差分格式[J].中国海洋大学学报, 2009, 39:487–491.
[20]初日辉.带五次项的非线性Schr?dinger方程的一个紧致差分格式[J].江苏师范大学学报, 2014, 32:53–57.
[21] Samarskii A A, Andreev V B. Di?erence methods for elliptic equations[M]. Moscow:Nauka. 1976.
[22] Wang T, Zhao X. Optimal l∞error estimates offinite difference methods for the coupled GrossPitaevskii equations in high dimensions[J]. Sci. China Math., 2014, 57(10):2189–2214.