小波分析理论下无网格偏微分方程数值解方法  

NUMERICAL SOLUTION METHOD OF MESHLESS PDES BASED ON WAVELET ANALYSIS

作　　者：王慧[1] 安宗灵[1] WANG Hui;AN Zong-ling(Normal School,Nanyang Institute of Technology,Nanyang 473004,China)

机构地区：[1]南阳理工学院教师教育学院,河南南阳473004

出　　处：《南阳理工学院学报》2020年第2期119-124,共6页Journal of Nanyang Institute of Technology

摘　　要：针对传统偏微分方程数值解方法求解精度和效率不高的问题,在小波分析理论下,提出无网格偏微分方程数值解方法。首先利用拟Shannon小波配点法,获取常微分方程组,然后利用插值问题替代离散偏微分方程,逼近该偏微分方程组精确解。在此基础上,通过基函数空间求解偏微分方程的方法定义为无网格偏微分方程数值解方法,考虑加权的最小二乘法可确定较为集中的点,致使偏微分方程与边界条件在确定较为集中的点上成立。以较典型的Convection Diffusion方程为例,在不同参数值设置条件下进行两次算例验证,实验结果表明,该所得的逼近解均较为接近精确解,可提升偏微分方程数值求解精度。In view of the low accuracy and efficiency of the traditional numerical solution methods for PDEs, a meshless numerical solution method for PDEs is proposed under the theory of wavelet analysis. First, we use the quasi Shannon wavelet collocation method to obtain the ordinary differential equations. Then we use the interpolation problem to replace the discrete partial differential equations and approach the exact solution of the partial differential equations. On this basis, the method of solving partial differential equation by the basis function space is defined as the numerical solution method of meshless partial differential equation. Considering the weighted least square method, the more concentrated points can be determined, so that the partial differential equation and boundary conditions are established on the more concentrated points. Taking the typical meeting diffusion equation as an example, two numerical examples are carried out under different parameter settings. The experimental results show that the approximate solutions are close to the exact solutions, which can improve the numerical solution accuracy of PDE.

关 键 词：小波分析 偏微分方程 拟尺度函数 基函数 最小二乘法 

分 类 号：O241.6[理学—计算数学] O241.82[理学—数学]