非线性Schrdinger方程的五次B-样条逼近

Numerical Approach of the Nonlinear Schrdinger Equation by the Quintic B-Spline

利用五次B-样条配点有限元方法研究了经典的三次非线性Schrdinger方程.在该格式中,关于时间方向的离散是基于Crank-Nicolson差分格式,而空间方向采用了分片五次B-样条函数逼近,其得到的刚度矩阵是一个分块五对角型矩阵.同时,利用线性稳定性分析方法证明了该格式是无条件稳定的.通过数值例子,验证了该格式保持了方程的守恒性质及具有较高的精度,最后模拟了两个孤立子的碰撞.

In this paper, the quintic B-spline collocation finite element method is implemented to find numerical solution of the classic cubic nonlinear Schr6dinger equation. The scheme is based on the Crank-Ni- eolson formulation for time discretization and quintic B-spline functions for space diseretizafion, and the stiff- ness matrix of the scheme is a block-five-diagonal matrix. The scheme is verified to be unconditionally stable by the method of linear stability analysis. By numerical examples, it is confirmed that the scheme keeps the conservative property of the equation preferably. Finally, the collision of two solitons is simulated.