zbMATH — the first resource for mathematics

A numerical scheme for nonlinear Schrödinger equation by MQ quasi-interpolation. (English) Zbl 1352.65391
Summary: Quasi-interpolation is a very powerful tool in the field of approximation theory and its applications, which can avoid solving large scale ill-conditioned linear system arising in approximating an unknown function by means of radial basis functions. In this paper, we use an univariate multi-quadrics (MQ) quasi-interpolation scheme to solve one-dimensional nonlinear Schrödinger equation. In this novel numerical scheme, the spatial derivatives are approximated by using the derivative of the quasi-interpolation and the temporal derivative is approximated by finite difference method. The main advantage of this proposed scheme is its simplicity. Two numerical examples are given and compared with the finite difference method (FDM) to verify the good accuracy and easy implementation of this method.

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
Full Text: DOI