一类非线性Schrdinger方程的差分/Legendre谱元法

Spectral Element Method for A Class of Nonlinear Schrdinger Equations

考察一类带幂次非线性项的Schrdinger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.证明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.

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.