COMPACT EXPONENTIAL CONSERVATIVE APPROACHES FOR THE SCHRO"\DINGER EQUATION IN THE SEMICLASSICAL REGIMES

In this paper we propose several efficient conservative prediction-correction methods for studying the Schro"\dinger equation in the semiclassical regimes with small parameter E. In the prediction step, the linear part of the equation is integrated exactly, which allows the E-oscillatory solution to be captured effectively, while only a local nonlinear system needs to be solved at each spatial grid point. Besides, the compact representation saves the storage requirement and computational cost. In the correction step, the prediction solution is modified to admit the mass/energy conservation law or both of them by adding supplementary variables to the original equation. This procedure is inexpensive since only algebraic nonlinear equations need to be solved. Extensive numerical tests suggest the meshing stategies for obtaining correct physical quantities: \tau = O(E) and h = O(E) for the equation with defocusing nonlinearities or weak O(E)--focusing/defocusing nonlinearities, and \tau independent of E and h = O(E) for the linear equation. Some numerical experiments for the equation in two and three dimensions are also carried out to demonstrate the power of the present methods in simulation of Bose-Einstein condensation.