基于径向基函数的局部近似特解法求解二维薛定谔方程

Numerical Simulation of 2D Schrdinger Equation by Localized Method of Approximate Particular Solution Based on Radial Basis Functions

基于径向基函数的局部近似特解法具有形式简单、易编程、精度高、收敛速度快等优点,是一种纯无网格配点方法.它通过将计算域划分为若干子区域并利用各个区域内的节点构造局部低阶矩阵,然后再将该矩阵拓展为全局形式,从而构造一个全局稀疏矩阵,以便于快速计算.本文采用局部近似特解法数值模拟二维薛定谔方程,首先采用隐式欧拉差分格式对时间项进行离散,并利用基于Multiquadric(MQ)函数的局部近似特解法对空间项进行离散.数值实验表明,局部近似特解法求解精度高、收敛速度快且计算耗时少,具有较好的工程应用前景.

The localized method of approximate particular solution（LMAPS）is a truly meshless collocation method with merits of high accuracy,rapid convergence and easy-to-program.In the LMAPS,the interest domain is divided into several subdomains;and in each subdomain,the localized low-rank matrix is formed based on the local nodes.And then a sparse system is constructed by reformulating the localized formulation into globalized form,which can be solved efficiently.In this paper,the LMAPS is applied to simulate 2D Schrdinger equation where the implicit-Euler scheme is used for time discretization;and the LMAPS is utilized for space discretization.Numerical experiments reveal the high accuracy,fast convergence and lower computational time of the LMAPS.And it has potential to engineering applications.