Time-splitting pseudo-spectral domain decomposition method for the soliton solutions of the one- and multi-dimensional nonlinear Schrödinger equations
Time-splitting pseudo-spectral domain decomposition method for the soliton solutions of the one- and multi-dimensional nonlinear Schrödinger equations
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DOI：
10.1016/j.cpc.2014.01.013
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发表时间：
2014-06
期刊：
影响因子：
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通讯作者：
Ameneh Taleei;M. Dehghan
中科院分区：
文献类型：
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作者：
Ameneh Taleei;M. Dehghan
本文研究了一维、二维和三维非线性薛定谔方程的数值模拟。所提出的方法是基于时间分裂的方法，分解成两个部分，一个线性方程和一个非线性方程的原始问题。一维线性方程在空间变量上用Chebyshev拟谱配点法逼近，在时间变量上用Crank-Nicolson方法逼近，而常系数非线性方程可以精确求解。本文的目的是研究大有限区域上的非线性薛定谔方程，我们提出了一种区域分解方法。与单域方法相比，多域方法可以产生稀疏的微分矩阵，所需存储空间少，计算量小。在这项研究中，我们选择了一个重叠的多域计划。通过应用交替方向隐式技术，我们将这种有效的方法推广到二维和三维非线性薛定谔方程的求解，而对于每个时间步的解，只需要分别求解一维线性偏微分方程组序列.一维和多维非线性薛定谔方程的几个例子来证明所提出的方法的高精度和能力。一些数值实验表明，该格式保持了电荷和能量守恒定律。
In this paper, we study the simulation of nonlinear Schrödinger equation in one, two and three dimensions. The proposed method is based on a time-splitting method that decomposes the original problem into two parts, a linear equation and a nonlinear equation. The linear equation in one dimension is approximated with the Chebyshev pseudo-spectral collocation method in space variable and the Crank–Nicolson method in time; while the nonlinear equation with constant coefficients can be solved exactly. As the goal of the present paper is to study the nonlinear Schrödinger equation in the large finite domain, we propose a domain decomposition method. In comparison with the single-domain, the multi-domain methods can produce a sparse differentiation matrix with fewer memory space and less computations. In this study, we choose an overlapping multi-domain scheme. By applying the alternating direction implicit technique, we extend this efficient method to solve the nonlinear Schrödinger equation both in two and three dimensions, while for the solution at each time step, it only needs to solve a sequence of linear partial differential equations in one dimension, respectively. Several examples for one- and multi-dimensional nonlinear Schrödinger equations are presented to demonstrate high accuracy and capability of the proposed method. Some numerical experiments are reported which show that this scheme preserves the conservation laws of charge and energy.