Symplectic scheme for the Schrödinger equation with fractional Laplacian

Abstract

In this paper, the symplectic scheme is presented for solving the space fractional Schrödinger equation (SFSE) with one dimension. First, the symplectic conservation laws are investigated for space semi-discretization systems of the SFSE based on the existing second-order central difference scheme and the existing fourth-order compact scheme. Then, a fourth-order central difference scheme is developed in space discretization, and the resulting semi-discretization system is shown to be a finite dimension Hamiltonian system of ordinary differential equations. Moreover, we get the full discretization scheme for the Hamiltonian system by symplectic midpoint scheme in time direction. In particular, the space semi-discretization and the full discretization are shown to preserve some properties of the SFSE. At last, numerical experiments are given to verify the efficiency of the scheme, and show that these symplectic difference schemes can be applied to long time simulation for one-dimensional SFSEs.