求解薛定谔–泊松方程组的时间分裂紧致差分格式

Time-Splitting Compact Difference Scheme for Solving Schrodinger-Poisson Equations

本文通过应用高阶紧致差分格式,时间分裂法与Crank-Nicolson法等方法求解非线性薛定谔–泊松方程组。基于快速Sine变换,我们设计了一种求解离散系统的快速算法。我们用该算法分别求解一维、二维、三维的非线性薛定谔–泊松方程组。我们列举具体的数值例子,使用MATLAB软件编写出算法程序,并且通过程序计算近似误差与画出近似解的图像。数值计算结果证实了该算法在空间方向具有谱精度,也证实了该算法的高效性与稳定性。

In this paper, we have introduced fourth-order compact finite difference, the time splitting method and the Crank-Nicolson method to solve the nonlinear Schrödinger-Poisson equations. Based on fast Sine transform, we construct a fast solver for the fully discretized system. The presented numerical algorithm has been used to solve one-dimensional, two-dimensional and three-dimensional nonlinear Schrödinger-Poisson equations. We provide specific numerical examples. Through the MATLAB software, we write matlab programs based on the presented numerical algorithm, calcu-late approximated error and draw the approximated numerical solution. The numerical results prove that the presented algorithm has spectral accuracy in space direction. They also confirm its efficiency and stability.