A new compact difference scheme is proposed for solving the nonlinear Schr？dinger equation. The scheme is proved to conserve the total mass and the total energy and the optimal convergent rate, without any restriction on the grid ratio, at the order of in the discrete -norm with time step τ and mesh size h. In numerical analysis, beside the standard techniques of the energy method, a new technique named “regression of compactness” and some lemmas are proposed to prove the high-order convergence. For computing the nonlinear algebraical systems generated by the nonlinear compact scheme, an efficient iterative algorithm is constructed. Numerical examples are given to support the theoretical analysis. 1. Introduction The time-dependent Schr？dinger equation is one of the most important equations in quantum mechanics [1] such as the Bose-Einstein condensate (BEC), which is also used widely in many other different fields [2, 3] such as plasma physics, nonlinear optics, water waves, and bimolecular dynamics. There are many studies on numerical approaches, including finite difference [4–16], finite element [17–19], and polynomial approximation methods [20–24], of the initial or initial-boundary value problems of the Schr？dinger equations. Recently, there has been growing interest in high-order compact methods for solving partial differential equations (PDEs) [25–35]. It was shown that the high-order difference methods play an important role in the simulation of high-frequency wave phenomena. However, there is few proof of unconditional norm convergence of any compact difference scheme for nonlinear equations. In this paper, we consider the following cubic nonlinear Schr？dinger (NLS) equation: subject to periodic boundary condition and initial condition where is dimensionless constant characterizing the interaction (positive for repulsive interaction and negative for the attractive interaction) between particles in BEC. is a real function corresponding to the external trap potential and it is often chosen as a harmonic potential, that is, a quadratic polynomial, in most experiments. is the unknown periodical complex-valued function whose initial value is a given periodical complex-valued function. is the period of the . Denote ; it is easy to show that the problem (1.1)–(1.3) conserves the total mass and the global energy Fei et al. pointed out in [15] that the nonconservative schemes may easily show nonlinear blowup, and they presented a new conservative linear difference scheme for nonlinear Schr？dinger equation. In [16], Li and Vu-Quoc also said, “… in some
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