(untitled) (Advances in Applied Mathematics and Mechanics, 2019)
The aim of this paper is to propose a fast meshless numerical scheme for the simulation of non-linear Schrödinger equations. In the proposed scheme, the implicit-Euler scheme is used for the temporal discretization and the localized method of approximate particular solution (LMAPS) is utilized for the spatial discretization. The multiple-scale technique is introduced to obtain the shape parameters of the multi-quadric radial basis function for 2D problems and the Gaussian radial basis function for 3D problems. Six numerical examples are carried out to verify the accuracy and efficiency of the proposed scheme. Compared with well-known techniques, numerical results illustrate that the proposed scheme is of merits being easy-to-program, high accuracy, and rapid convergence even for long-term problems. These results also indicate that the proposed scheme has great potential in large scale problems and real-world applications.