THE EXISTENCE OF THE GLOBAL SOLUTION OF THE SEMI-LINEAR SCHRÖDINGER EQUATION WITH NUMERICAL COMPUTATION ON THE DIRICHLET BOUNDARY

This paper concerns the analysis of the initial boundary value problem for the semi-linear Schrödinger equation. In the paper, we design a reliable scheme coupling the nonstandard finite difference method in time with the Galerkin combined with the compactness method in the space variables to analyze the problem. The analysis begins by showing that, given initial solutions in specified space, the global solution of the Schrödinger equation exists uniquely. We further show using the a priori estimates obtained from the existence process, that the numerical solution from the designed scheme is stable and converges optimally in specified norms. Furthermore, we show that the scheme replicates or preserves the qualitative properties of the exact solution. Numerical experiments are conducted using a carefully chosen example to justify our theoretical proposition.