Numerical solution of the two-dimensional nonlinear Schro<spacing diaeresis>dinger equation using an alternating direction implicit method

In this paper, an alternating direction implicit (ADI) finite difference scheme is proposed for solving the twodimensional time-dependent nonlinear Schro<spacing diaeresis>dinger 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.