Numerical solution of the two-dimensional nonlinear schrödinger equation using an alternating direction implicit method

Document Type : Research Paper

Author

Department of Mathematics, College of Science, Bahir Dar University, Bahir Dar, Ethiopia.

Abstract

In this paper, an alternating direction implicit (ADI) finite difference scheme is proposed for solving the two-dimensional time-dependent nonlinear Schrödinger equation. In the proposed scheme, the nonlinear term is linearized by using the values of the wave function from the previous time level at each iteration step. The resulting block tridiagonal system of algebraic equations is solved using the Gauss-Seidel method in conjunction with sparse matrix computation. The stability of the scheme is analyzed using matrix analysis and is found to be conditionally stable. Numerical examples are presented to demonstrate the efficiency, stability, and accuracy of the proposed scheme. The numerical results show good agreement with exact solutions.

Keywords

Main Subjects


  • [1] G. Arora, V. Joshi, and R. C. Mittal, Numerical simulation of nonlinear Schrödinger equation in one and two dimensions, Mathematical Models and Computer Simulations,11(4) (2019), 634-648.
  • [2] A. G. Bratsos, A linearized finite-difference scheme for the numerical solution of the nonlinear cubic Schrödinger equation, Korean Journal of Computational and Applied Mathematics, 8(2001), 459-467.
  • [3] M. M. Cavalcanti, W. J. Corrêa, M. A. Sepùlveda C, and R. Vêjar-Asem, Finite difference scheme for a higher order nonlinear Schrödinger equation, Calcolo,56(4) (2019), 40.
  • [4] M. Dehghan and A. Taleei, Numerical solution of nonlinear Schrödinger equation by using time-space pseudospectral method, Numerical Methods for Partial Differential Equations: An International Journal, 26(4) (2010), 979-992.
  • [5] R. K Dodd, J. C. Eilbeck, and J. D. Gibbon, Solitons and Nonlinear Wave Equations, New York: Academic Press, 1982.
  • [6] R. Eskar, P. Huang, and X. Feng, A new high-order compact ADI finite difference scheme for solving 3D nonlinear Schrödinger equation, Advances in Difference Equations, 2018:286 (2018).
  • [7] L. Guo, Y. Guo, S. Billings, D. Coca, and Z. Lang, The use of Volterra series in the analysis of the non-linear Schrödinger equation, Nonlinear Dynamics, 73(3) (2013), 1587-1599.
  • [8] Q. Guo and J. Liu, New exact solutions to the nonlinear Schrödinger equation with variable coefficients, Results in Physics, 16 (2020), 102857.
  • [9] Y. He and X. Lin, Numerical analysis and simulations for coupled nonlinear Schrödinger equations based on lattice Boltzmann method, Applied Mathematics Letters, 106 (2020), 106391.
  • [10] H. Hu, Two-grid method for two-dimensional nonlinear Schrödinger equation by finite element method, Numerical Methods for Partial Differential Equations, 34(2) (2018), 385-400.
  • [11] A. Iqbal, N. N. Abd Hamid, and A. I. M. Ismail, Numerical solution of nonlinear Schrödinger equation with Neumann boundary conditions using quintic B-spline Galerkin method, Symmetry, 11(4) (2019), 469.
  • [12] A. Iqbal, N. N. Abd Hamid, and A. I. M. Ismail, Cubic B-spline Galerkin method for numerical solution of the coupled nonlinear Schrödinger equation, Mathematics and Computers in Simulation, 174 (2020), 32-44.
  • [13] M. S. Ismail, Numerical solution of coupled nonlinear Schrödinger equation by Galerkin method, Mathematics and Computers in Simulation, 78(4) (2008), 532-547.
  • [14] R. Jiwari, S. Kumar, R. C. Mittal, and J. Awrejcewicz, A meshfree approach for analysis and computational modeling of non-linear Schrödinger equation, Computational and Applied Mathematics, 39 (2020), 1-25.
  • [15] N. I. Karabaş, S. Ö. Korkut, G. Tanoğlu, and I. Aziz, An efficient approach for solving nonlinear multidimensional Schrödinger equations, Engineering Analysis with Boundary Elements, 132 (2021), 263-270.
  • [16] J. Lin, Y. Hong, L. Ku, and C. Liu, Numerical simulation of 3D nonlinear Schrödinger equations by using the localized method of approximate particular solutions, Engineering Analysis with Boundary Elements, 78 (2017), 20-25.
  • [17] X. Liu, M. Ahsan, M. Ahmad, I. Hussian, M. M. Alqarniand, and E. E. Mahmoud, Haar wavelets multi-resolution collocation procedures for two-dimensional nonlinear Schrödinger equation, Alexandria Engineering Journal, 60(3) (2021), 3057-3071.
  • [18] W. X. Ma and M. Chen, Direct search for exact solutions to the non-linear Schrödinger equation, Applied Mathematics and Computation, 215(8) (2009), 2835-2842.
  • [19] G. Q. Meng, Y. T. Gao, X. Yu, Y. J. Shen, and Y. Qin, Multi-soliton solutions for the coupled non-linear Schrödinger-type equations, Nonlinear Dynmics, 70(1) (2012), 609-617.
  • [20] M. Pathak, P. Joshi, and K. S. Nisar, Numerical study of generalized 2-D nonlinear Schrödinger equation using Kansa method, Mathematics and Computers in Simulation, 200 (2022), 186-198.
  • [21] E. Shivanian and A. Jafarabadi, An efficient numerical technique for solution of two-dimensional cubic nonlinear Schrödinger equation with error analysis, Engineering Analysis with Boundary Elements, 83 (2017), 74-86.
  • [22] C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, New York: Springe, 1999.
  • [23] L. Wang and M. Li, Galerkin finite element method for damped nonlinear Schrödinger equation, Applied Numerical Mathematics, 178 (2022), 216-247.
  • [24] Y. Xu and L. Zhang, Alternating direction implicit method for solving two-dimensional cubic nonlinear Schrödinger equation, Computer Physics Communications, 183 (2012), 1082-1093.