边界节点法的计算稳定性研究  

Stability analysis of boundary knot method

作　　者：姜欣荣[1] 陈文[1] 林继[1] 王福章[1] 

机构地区：[1]河海大学工程力学系,工程与科学数值模拟软件中心,南京210098

出　　处：《计算力学学报》2013年第2期242-248,共7页Chinese Journal of Computational Mechanics

基　　金：国家杰出青年基金(1077403);水利部公益性行业专项经费(201101014);科技部国家重点基础研究发展计划(973)(2010CB832702)资助项目

摘　　要：边界节点法利用满足控制方程的非奇异通解作为基函数,半解析边界数值离散偏微分方程,具有精度高、收敛快、易编程等优点,是一种纯无网格配点方法。但是在求解具体问题时,随着节点数的增加,边界节点法经常得到严重病态的插值矩阵。本文利用有效条件数评价边界节点法求解Helmholtz问题线性方程组的计算稳定性;然后利用三种正则化方法处理其病态的线性方程组,并与高斯消元法比较计算精度和收敛性。通过数值实验,本文研究了有效条件数、误差和正则化方法之间的关系。The boundary knot method （BKM） uses the non-singular general solution,which satisfies the governing equation,as its basis functions for the semi-analytical numerical solution of partial differential equation on boundary collocation discretization and is a truly meshfree collocation method with merits of high accuracy, rapid convergence, easy-to-program. But with the increasing number of boundary knots, the method often results in an ill-conditioned interpolation matrix equation. In this study,we employ the effective condition number （ECN） approach to estimate the computational stability and accuracy of the BKM solution of Helmholtz problems. And then this paper employs the three regularization techniques to damp the ill-posedness of the BKM interpolation matrix in conjunction with the Gaussian elimination method for the ~olution of discretization algebraic equations. Numerical experiments illustrate the rela- tionship among the ECN, accuracy and regularization techniques with respect to the BKM solution of Helmholtz problems.

关 键 词：边界节点法 病态矩阵 有效条件数 正则化方法 HELMHOLTZ方程 

分 类 号：O175.13[理学—数学] O317[理学—基础数学]