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中文摘要：
考察一类带幂次非线性项的Schrodinger方程的Dirichlet初边值问题，提出了一个有效的计算格式，其中时间方向上应用了一种守恒的二阶差分隐格式，空间方向上采用Legendre谱元法．对于时间半离散格式，证职了该格式具有能量守恒性质，并给出了L^2误差估计，对于全离散格式，应用不动点原理证明了数值解的存在唯一性，并给出了L^2误差估计．最后，通过数值试验验证了结果的可信性．
英文摘要：
In this paper, an efficient numerical method, based on a second-order implicit difference scheme in time and Legendre spectral element method in space, is developed for the initial- and Dirichlet boundary-value problem of a class of nonlinear SchrSdinger equations with power nonlinear terms. For semi-discrete approximation, the conservation properties and the L^2 error bound are proved. For full- discrete approximation： the existence and uniqueness of the solution are estasblished by using the fix-point method, and the L^2 error estimates of the optimal orders are proved. Finally, some numerical experiments are performed to support the theoretical claims.
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