The meshless local Petrov–Galerkin (MLPG) method for the generalized two-dimensional non-linear Schrödinger equation

In this paper the meshless local Petrov–Galerkin (MLPG) method is presented for the numerical solution of the two-dimensional non-linear Schrödinger equation. The method is based on the local weak form and the moving least squares (MLS) approximation. For the MLS, nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables. A time stepping method is employed for the time derivative. To deal with the non-linearity, we use a predictor–corrector method. A very simple and efficient method is presented for evaluation the local domain integrals. Finally numerical results are presented for some examples to demonstrate the accuracy, efficiency and high rate of convergence of this method.