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.
Tsega, E. G (2025). Numerical solution of the two-dimensional nonlinear schrödinger equation using an alternating direction implicit method. Computational Methods for Differential Equations, 13(3), 1012-1021. doi: 10.22034/cmde.2024.61040.2619
MLA
Tsega, E. G. "Numerical solution of the two-dimensional nonlinear schrödinger equation using an alternating direction implicit method", Computational Methods for Differential Equations, 13, 3, 2025, 1012-1021. doi: 10.22034/cmde.2024.61040.2619
HARVARD
Tsega, E. G (2025). 'Numerical solution of the two-dimensional nonlinear schrödinger equation using an alternating direction implicit method', Computational Methods for Differential Equations, 13(3), pp. 1012-1021. doi: 10.22034/cmde.2024.61040.2619
CHICAGO
E. G Tsega, "Numerical solution of the two-dimensional nonlinear schrödinger equation using an alternating direction implicit method," Computational Methods for Differential Equations, 13 3 (2025): 1012-1021, doi: 10.22034/cmde.2024.61040.2619
VANCOUVER
Tsega, E. G Numerical solution of the two-dimensional nonlinear schrödinger equation using an alternating direction implicit method. Computational Methods for Differential Equations, 2025; 13(3): 1012-1021. doi: 10.22034/cmde.2024.61040.2619
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