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AIMS Mathematics

2022, Volume 7, Issue 9: 16349-16365. doi: 10.3934/math.2022893
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Stability analysis of the implicit finite difference schemes for nonlinear Schrödinger equation

  • 1.
    School of Mathematics and Computing, Yonsei University, Seoul, 03722, Korea
  • 2.
    Department of Mathematics, Dongguk University, Seoul, 04620, Korea
  • Received: 06 May 2022 Revised: 23 June 2022 Accepted: 29 June 2022 Published: 05 July 2022

  • MSC : 65M06, 65M12

  • This paper analyzes the stability of numerical solutions for a nonlinear Schrödinger equation that is widely used in several applications in quantum physics, optical business, etc. One of the most popular approaches to solving nonlinear problems is the application of a linearization scheme. In this paper, two linearization schemes—Newton and Picard methods were utilized to construct systems of linear equations and finite difference methods. Crank-Nicolson and backward Euler methods were used to establish numerical solutions to the corresponding linearized problems. We investigated the stability of each system when a finite difference discretization is applied, and the convergence of the suggested approximation was evaluated to verify theoretical analysis.

    Citation: Eunjung Lee, Dojin Kim. Stability analysis of the implicit finite difference schemes for nonlinear Schrödinger equation[J]. AIMS Mathematics, 2022, 7(9): 16349-16365. doi: 10.3934/math.2022893

    Related Papers:

  • This paper analyzes the stability of numerical solutions for a nonlinear Schrödinger equation that is widely used in several applications in quantum physics, optical business, etc. One of the most popular approaches to solving nonlinear problems is the application of a linearization scheme. In this paper, two linearization schemes—Newton and Picard methods were utilized to construct systems of linear equations and finite difference methods. Crank-Nicolson and backward Euler methods were used to establish numerical solutions to the corresponding linearized problems. We investigated the stability of each system when a finite difference discretization is applied, and the convergence of the suggested approximation was evaluated to verify theoretical analysis.


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    [1] G. Akrivis, Finite difference discretization of the cubic Schrödinger equation, IMA J. Numer. Anal., 13 (1993), 115–124.
    [2] G. Akrivis, V. A. Dougalis, O. A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math., 59 (1991), 31–53. https://doi.org/10.1007/BF01385769 doi: 10.1007/BF01385769
    [3] T. Akram, M. Abbas, A. I. M. Ismail, N. M. Ali, M. B. Riaz, Development and analysis of new approximation of extended cubic B-spline to the non-linear time fractional Klein-Gordon equation, Fractals, 2020.
    [4] T. Akram, M. Abbas, A. Ali, A numerical study on time fractional Fisher equation using an extended cubic B-spline approximation, J. Math. Comput. Sci.-JM, 22 (2020), 85–96. https://doi.org/10.22436/jmcs.022.01.08 doi: 10.22436/jmcs.022.01.08
    [5] X. Antoine, A. Arnold, C. Besse, M. Ehrhardt, A. Schaedle, A Review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. Comput. Phys., 4 (2008), 729–796.
    [6] D. S. Bernstein, Matrix mathematics: Theory, facts, and formulas, 2 Eds, Princeton University Press, 2005.
    [7] R. Balakrishnan, Soliton propagation in non-uniform media, Phys. Rev. A, 32 (1985), 1144–1149.
    [8] Q. S. Chang, E. Jia, W. Sun, Difference schemes for solving the generalized nonlinear Schrödinger equation, J. Comput. Phys., 148 (1999), 397–415. https://doi.org/10.1006/jcph.1998.6120 doi: 10.1006/jcph.1998.6120
    [9] Y. Y. Choy, Crank-Nicolson implicit method for the nonlinear Schrödinger equation with variable coefficient, AIP Conf. Pro., 2014.
    [10] R. M. Caplan, R. Carretero-Gonzalez, Numerical stability of explicit Runge-Kutta finite-difference schemes for the nonlinear Schrödinger equation, Appl. Numer. Math., 71 (2013), 24–40. https://doi.org/10.1016/j.apnum.2013.04.002 doi: 10.1016/j.apnum.2013.04.002
    [11] D. F. Griffiths, A. R. Mitchell, J. L. Morris, A numerical study of the nonlinear Schrödinger equation, Comput. Method. Appl. Mech. Eng., 45 (1984), 177–215. https://doi.org/10.1016/0045-7825(84)90156-7 doi: 10.1016/0045-7825(84)90156-7
    [12] A. V. Gurevich, Non-linear phenomena in the ionosphere, Springer, Berlin, 1978
    [13] J. Ginibre, G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal., 32 (1979), 1–32.
    [14] A. Iqbal, N. Hamid, A. Ismail, Cubic B-spline Galerkin method for numerical solution of the coupled nonlinear Schrödinger equation, Math. Comput. Simulat., 174 (2020), 32–44.
    [15] A. Iqbal, N. Hamid, A. Ismail, M. Abbas, Galerkin approximation with quintic B-spline as basis and weight functions for solving second order coupled nonlinear Schrödinger equations, Math. Comput. Simulat., 187 (2021), 1–16. https://doi.org/10.1016/j.matcom.2021.02.012 doi: 10.1016/j.matcom.2021.02.012
    [16] R. J. LeVeque, Finite difference methods for ordinary and partial differential equations: Steady state and time dependent problems, SIAM, 2007. https://doi.org/10.1137/1.9780898717839 doi: 10.1137/1.9780898717839
    [17] M. Li, J. Zhao, N. C. Wang, S. Chen, Conforming and nonconforming conservative virtual element methods for nonlinear Schrödinger equation: A unified framework, Comput. Meth. Appl. Mech. Eng., 380 (2021), 113793.
    [18] M. Li, C. Huang, W. Ming, A relaxation-type Galerkin FEM for nonlinear fractional Schrödinger equations, Numer. Algorithms, 83 (2019), 99–124. https://doi.org/10.1007/s11075-019-00672-3 doi: 10.1007/s11075-019-00672-3
    [19] E. Lee, T. A. Manteuffel, C. R. Westphal, FOSLL$^*$ for nonlinear partial differential equations, SIAM J. Sci. Comput., 37, (2020), 503–525. https://doi.org/10.1137/140974353 doi: 10.1137/140974353
    [20] E. Lee, W. Choi, H. Ha, An $L^2$ finite element approximation for the incompressible Navier-Stokes equations, Numer. Meth. Part. D. E., 36 (2020), 1389–1404. https://doi.org/10.1002/num.22478 doi: 10.1002/num.22478
    [21] B. Malomed, Nonlinear Schrödinger equations, New York, Routledge, 2005,639–643.
    [22] V. Mehrmann, H. Xu, Perturbation of purely imaginary eigenvalues of Hamiltonian matrices under structured perturbations, Electron. J. Linear Al., 17 (2008), 234–257. https://doi.org/10.13001/1081-3810.1261 doi: 10.13001/1081-3810.1261
    [23] K. R. Meyer, G. R. Hall, O. Dan, Introduction to Hamiltonian dynamical systems and the N-body problem, Appl. Math. Sci., 2009. https://doi.org/10.1007/978-0-387-09724-4 doi: 10.1007/978-0-387-09724-4
    [24] G. Strang, Introduction to linear algebra, 4Eds, Wellesley-Cambridge Press, Wellesley, MA, 2009.
    [25] P. Wang, C. Huang, An energy conservative difference scheme for the nonlinear fractional Schrödinger equations, J. Comput. Phys., 293 (2015), 238–251. https://doi.org/10.1016/j.jcp.2014.03.037 doi: 10.1016/j.jcp.2014.03.037
    [26] R. Zhang, Y. T. Zhang, Z. Wang, B. Chen, Y. Zhang, A conservative numerical method for the fractional nonlinear Schrödinger equation in two dimensions, Sci. China Math., 62 (2019), 1997–2014. https://doi.org/10.1007/s11425-018-9388-9 doi: 10.1007/s11425-018-9388-9
    [27] V. E. Zakharov, S. V. Manakov, On the complete integrability of a nonlinear Schrödinger equation, J. Theor. Math. Phys., 19 (1974), 551–559.
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