无网格RBF插值方法在偏微分方程计算中的应用

发布时间：2018-05-08 17:14

本文选题：无网格方法 + 径向基函数　；参考：《浙江理工大学》2017年硕士论文

【摘要】：无网格方法是继有限差分法与有限元法等传统的数值方法之后兴起的一种很有前景的数值方法。相比传统的数值方法,无网格方法对网格没有较强的依赖性,自适应性较强等优点。随着近年来国内外学者的研究,无网格方法也日渐成熟、多样。基于径向基函数()无网格法是一种真正的无网格法,具有形式简单、数值精度高等优点,逐渐成为近年研究的热门课题之一。本文将基于插值应用于解决偏微分方程数值解问题。正文首先介绍了插值,分别模拟高阶导数插值与多重积分插值的数值实验,并给出了形状参数(8的取值范围,数值实验表明:多重积分插值相比高阶导数插值更加稳定、精度高,对于形状参数(8的选择更加灵活多变。文中给出了常微分方程导数插值与积分插值方法的数值格式。在全域导数插值方法中,针对系数矩阵具有较强的奇异性,学者们提出了局部插值方法,并给出了迎风格式以提高数值解的稳定性,我们分析并比较全域与局部两种导数插值方法对数值实验的影响。另外,本文结合Tikhonov正则化法给出径向基插值有限积分法的误差估计,该估计表明:函数的光滑性越高,数值精度越高。数值实验表明:结合插值的有限积分方法具有数值精度高、结构简单、形状参数(8选取灵活等特点。
[Abstract]:Meshless method is a promising numerical method after the traditional numerical methods such as finite difference method and finite element method. Compared with the traditional numerical method, the meshless method has no strong dependence on the mesh, and is more adaptive. With the research of scholars at home and abroad in recent years, meshless methods are becoming more and more mature and diverse. Meshless method based on radial basis function (RBF) is a real meshless method with the advantages of simple form and high numerical accuracy. It has gradually become one of the hot topics in recent years. In this paper, we apply interpolation to solve numerical solutions of partial differential equations. First of all, the interpolation is introduced, the numerical experiments of high-order derivative interpolation and multi-integral interpolation are simulated, and the range of shape parameter is given. The numerical experiments show that the multi-multiple integral interpolation is more stable than the high-order derivative interpolation. High precision, for shape parameters of the selection of more flexible and changeable. In this paper, the numerical schemes of derivative interpolation and integral interpolation for ordinary differential equations are given. In the global derivative interpolation method, the local interpolation method is proposed to improve the stability of the numerical solution in view of the strong singularity of the coefficient matrix, and the upwind scheme is given to improve the stability of the numerical solution. We analyze and compare the effects of global and local derivative interpolation methods on numerical experiments. In addition, this paper gives the error estimate of the radial basis interpolation finite integration method combined with Tikhonov regularization method. This estimate shows that the higher the smoothness of the function, the higher the numerical accuracy. Numerical experiments show that the finite integral method combined with interpolation has the advantages of high numerical accuracy, simple structure and flexible selection of shape parameters.
【学位授予单位】：浙江理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

【相似文献】