A weakly demped Schrodinger equation possessing a global attractor are considered.The dynamical properties of a class of finite difference scheme are analysed. The exsitence of global attractor is proved for the discrete system. The stability of the difference scheme and the error estimate of the difference solution are obtained in the autonomous system case. Finally, long-time stability and convergence of the class of finite difference scheme also are analysed in the nonautonomous system case.