Lyapunov-Sylvester Operators for Numerical Solutions of NLS Equation
This work addresses the numerical solution of a specific PDE (NLS with singular potential) for researchers in computational mathematics, but the lack of experimental validation makes it incremental.
The paper develops a numerical method for the two-dimensional NLS equation with a singular potential, using Lyapunov-Sylvester operators. The scheme is proven consistent, convergent, and stable, but no concrete numerical results or performance metrics are provided.
In the present paper a numerical method is developed to approximate the solution of two-dimensional NLS equation in the presence of a singular potential. The method leads to Lyapunov-Syslvester algebraic operators that are shown to be invertible using original topological and differential calculus issued methods. The numerical scheme is proved to be consistent, convergent and stable using the based on Lyapunov criterion, lax equivalence theorem and the properties of the Lyapunov-Syslvester operators.