An efficient spectral-collocation difference method for two-dimensional Schrödinger equation with Neumann boundary conditions

Abstract

Based on the second-order cosine spectral differentiation matrix deduced in this paper, we present an efficient and high accurate numerical scheme for solving two-dimensional Schrödinger equation with Neumann boundary conditions. The Crank–Nicolson finite difference scheme is utilized in temporal discretization and the cosine spectral-collocation method is employed for the approximation of Laplacian operator. The new scheme conserves the mass and energy in discrete level and is implemented by a fast algorithm in terms of the relations between the cosine spectral differentiation matrix and fast cosine transform. More importantly, this strategy can be extended to address the problems with high (even)-order derivatives in space. Numerical examples and applications are listed to confirm the validity and high accuracy of the method.