A numerical scheme for nonlinear Schrِdinger equation by MQ quasi-interpolation

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 Schrِdinger 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.