A numerical scheme based on radial basis function finite difference (RBF-FD) technique for solving the high-dimensional nonlinear Schrödinger equations using an explicit time discretization: Runge-Kutta method
A numerical scheme based on radial basis function finite difference (RBF-FD) technique for solving the high-dimensional nonlinear Schrödinger equations using an explicit time discretization: Runge-Kutta method
复制标题
DOI：
10.1016/j.cpc.2017.03.012
复制
发表时间：
2017-08
期刊：
影响因子：
--
通讯作者：
M. Dehghan;V. Mohammadi
中科院分区：
文献类型：
--
作者：
M. Dehghan;V. Mohammadi
本文研究二维和三维非线性薛定谔方程的数值解。本文采用的无网格数值方法是径向基函数-有限差分法。这种方法的主要优点是在每个局部支撑区域上基于有限差分技术近似所需的导数为Ω i。在每个Ω i处，我们需要求解一个具有1阶条件正定矩阵（插值矩阵）的小型线性代数方程组。该方案是有效的，其计算成本是相同的移动最小二乘（MLS）近似。其中一个具有挑战性的问题是如何选择合适的插值矩阵形状参数。为了克服这个问题，将应用Sarra（2012）建立的算法。该算法利用奇异值分解（SVD）计算局部插值矩阵的条件数，以获得该矩阵的最小和最大奇异值。此外，基于四阶精度的龙格-库塔公式的显式方法将用于近似时间变量。它还降低了每个时间步长的计算成本，因为我们不会求解非线性系统。另一方面，为了将RBF-FD方法与另一种无网格方法进行比较，本文考虑了移动克里金最小二乘（MKLS）方法。我们的研究结果表明，本方法的能力，解决适用的模型，在目前的研究工作。
In this research, we investigate the numerical solution of nonlinear Schrödinger equations in two and three dimensions. The numerical meshless method which will be used here is RBF-FD technique. The main advantage of this method is the approximation of the required derivatives based on finite difference technique at each local-support domain as Ω i. At each Ω i, we require to solve a small linear system of algebraic equations with a conditionally positive definite matrix of order 1 (interpolation matrix). This scheme is efficient and its computational cost is same as the moving least squares (MLS) approximation. A challengeable issue is choosing suitable shape parameter for interpolation matrix in this way. In order to overcome this matter, an algorithm which was established by Sarra (2012), will be applied. This algorithm computes the condition number of the local interpolation matrix using the singular value decomposition (SVD) for obtaining the smallest and largest singular values of that matrix. Moreover, an explicit method based on Runge–Kutta formula of fourth-order accuracy will be applied for approximating the time variable. It also decreases the computational costs at each time step since we will not solve a nonlinear system. On the other hand, to compare RBF-FD method with another meshless technique, the moving kriging least squares (MKLS) approximation is considered for the studied model. Our results demonstrate the ability of the present approach for solving the applicable model which is investigated in the current research work.