New scheme for nonlinear Schrödinger equations with variable coefficients

Volume 19, Issue 3, pp 151--157 http://dx.doi.org/10.22436/jmcs.019.03.02

Publication Date: May 17, 2019 Submission Date: August 04, 2017 Revision Date: April 13, 2019 Accteptance Date: April 19, 2019

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Authors

Xiu-Ling Yin - School of Mathematical Sciences, Dezhou University, Dezhou, China. Shu-Xia Kong - School of Mathematical Sciences, Dezhou University, Dezhou, China. Yan-Qin Liu - School of Mathematical Sciences, Dezhou University, Dezhou, China. Xiao-Tong Zheng - School of Statistics, Renmin University of China, Beijing, China.

Abstract

This paper proposes a numerical scheme for nonlinear Schrödinger equations with periodic variable coefficients and stochastic perturbation. The scheme is obtained by applying finite element method in spatial direction and finite difference scheme in temporal direction, respectively. The scheme is stable in the sense that it preserves discrete charge of the Schrödinger equations. The numerical examples verify the conservative property of the new scheme.

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Keywords

• Schrödinger equation
• finite element method
• finite difference scheme

MSC

•  65M06
•  65M12
•  65Z05

References

• [1] V. Barbu, M. Röckner, D. Zhang, Stochastic nonlinear Schrödinger equations with linear multiplicative noise: rescaling approach, J. Nonlinear Sci., 24 (2014), 383--409
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] C. C. Chen, J. L. Hong, L. H. Ji, A symplectic local discontinuous Galerkin method for stochastic Schrödinger equation, arXiv, 2014 (2014), 19 pages
  • View Article


• [3] C. C. Chen, J. L. Hong, A. Prohl, Convergence of a $\theta$--scheme to solve the stochastic nonlinear Schrödinger equation with Stratonovich noise, Stochastic Part. Diff. Equ. Anal. Comput., 4 (2016), 274--318
  • View Article
  • Google Scholar
  • MATH


• [4] J.-B. Chen, M.-Z. Qin, Multi--symplectic Fourier pseudospectral method for the nonlinear Schrödinger equation, Electron. Trans. Numer. Anal., 12 (2001), 193--204
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] A. de Bouard, A. Debussche, Weak and strong order of convergence of a semidiscrete scheme for the stochastic nonlinear Schrödinger equation, Appl. Math. Optim., 54 (2006), 369--399
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] M. Dehghan, A. Taleei, A compact split-step finite difference method for solving the nonlinear Schrödinger equation with constant and variable Coefficients, Comput. Phys. Commun., 181 (2010), 43--51
  • View Article
  • MathSciNet
  • Google Scholar


• [7] W. Gao, H. Li, Y. Liu, X. X. Wei, A Padé compact high-order finite volume scheme for nonlinear Schrödinger equations, Appl. Numer. Math., 85 (2014), 115--127
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] J.-H. He, M. A. Abdou, New periodic solutions for nonlinear evolution equations using Exp-function method, Chaos Soliton Fractals, 34 (2007), 1421--429
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] B. İnan, A. R. Bahadir, Numerical solutions of MRLW equation by a fully implicit finite-difference scheme, J. Math. Computer Sci., 15 (2015), 228--239
  • View Article
  • Google Scholar


• [10] S. S. Jiang, L. J. Wang, J. L. Hong, Stochastic multi-symplectic integrator for stochastic nonlinear Schrödinger equation, Commun. Comput. Phys., 14 (2013), 393--411
  • View Article
  • MathSciNet
  • Google Scholar


• [11] O. Karakashian, C. Makridakis, A space--time finite element method for the nonlinear Schrödinger equation: the continuous Galerkin method, SIAM J. Numer. Anal., 36 (2006), 1779--1807
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] L. H. Kong, P. Wei, Y. Q. Hong, P. Zhang, P. Wang, Efficient energy-preserving scheme of the three--coupled nonlinear Schrödinger equation, Math. Meth. Appl. Sci., 2019 (2019), 14 pages
  • View Article
  • Google Scholar


• [13] J. Liu, Order of convergence of splitting schemes for both deterministic and stochastic nonlinear Schrödinger equations, SIAM J. Numer. Anal., 51 (2013), 1911--1932
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] G. N. Milstein, M. V. Tretyakov, Stochastic Numerics for Mathematical Physics, Springer-Verlag, Berlin (2004)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] C. M. Mora, Numerical solution of conservative finite--dimensional stochastic Schrödinger equations, Ann. Appl. Probab., 15 (2005), 2144--2171
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] M. Sarboland, A. Aminataei, Integrated high accuracy multiquadric quasi-interpolation scheme for solving the nonlinear Klein--Gordon equation, J. Math. Computer Sci., 14 (2015), 258--273
  • View Article
  • Google Scholar


• [17] Y. F. Tang, J. W. Cao, X. T. Liu, Y. C. Sun, Symplectic methods for the Ablowitz--Ladik discrete nonlinear Schrödinger equation, J. Phys. A, 40 (2007), 2425--2437
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] J. L. Wang, A new error analysis of Crank--Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation, J. Sci. Comput., 60 (2014), 390--407
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] T. C. Wang, B. L. Guo, Q. B. Xu, Fourth--order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions, J. Comput. Phys., 243 (2013), 382--399
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] Y. S. Wang, Q. H. Li, Y. Z. Song, General: two new simple multi--Symplectic schemes for the nonlinear Schrödinger equation, Chinese Phys. Lett., 25 (2008), 1538--1540
  • View Article
  • Google Scholar


• [21] L. Wu, Two--grid mixed finite-element methods for nonlinear Schrödinger equations, Numer. Methods Partial Differential Equations, 28 (2012), 63--73
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] X.-J. Yang, D. Baleanu, Y. Khan, S. T. Mohyud-Din, Local fractional variational iteration method for diffusion and wave equations on Cantor sets, Romanian J. Phys., 59 (2014), 36--48
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] Y. Zhang, C. Cattani, X.-J. Yang, Local fractional homotopy perturbation method for solving non--homogeneous heat conduction equations in fractal domains, Entropy, 17 (2015), 6753--6764
  • View Article
  • MathSciNet
  • Google Scholar


• [24] H. J. Zhu, Y. M. Chen, S. H. Song, H. Y. Hu, Symplectic and multi--symplectic wavelet collocation methods for two--dimensional Schrödinger equations, Appl. Numer. Math., 61 (2011), 308--321
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH