| Volume 15, Number 2, 2025, Pages - DOI：10.11948/JAAC-2024-0161 |
| The existence of global solution and optimal convergence of the semi-linear Schr\ |
| Pius W Molo Chin |
| Keywords:Schr\ |
| Abstract: |
| This paper concerns the analysis of the initial value problem for the semi-linear Schr\"{o}dinger 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 Schr\"{o}dinger 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 proceed to show also 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. |
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