机械求积法及其外推算法求解Helmholtz非线性边界积分方程

程攀; 王柱; 黄大荣

doi:10.3879/j.issn.1000-0887.2011.12.002

机械求积法及其外推算法求解Helmholtz非线性边界积分方程

doi: 10.3879/j.issn.1000-0887.2011.12.002

程攀¹,
王柱^2,3,
黄大荣⁴

重庆交通大学理学院, 重庆 400074;
电子科技大学数学科学院, 成都 611731;
维吉尼亚综合技术大学数学系, 布莱克斯堡, VA 24061, 美国;

基金项目: 国家自然科学基金资助项目(10871034);重庆市自然科学基金资助项目(CSTC20-10BB8270);AFOSR(FA9550-08-1-0136);NSF(OCE-0620464)的部分资助项目

详细信息

作者简介:
程攀(1976- ),男,重庆人,讲师,博士(联系人.E-mail:cheng_pass@sina.com).

中图分类号: O24;O39
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- 被引次数: 0
出版历程
- 收稿日期: 2010-11-18
- 修回日期: 2011-11-02
- 刊出日期: 2011-12-15

Mechanical Quadrature Methods and Extrapolation for Solving Nonlinear Boundary Integral Equations of Helmholtz Equation

School of Science, Chongqing Jiaotong University, Chongqing 400074, P. R. China;
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, P. R. China;
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U. S. A

摘要

摘要: 当Helmholtz微分方程转化为非线性边界积分方程后,可以利用机械求积法求得近似解,此方法具有较高的收敛精度阶O(h3)和较低的计算复杂度．构造机械求积法时,一个非线性方程系统通过离散非线性积分方程得到．此外,每个矩阵元素的值都不需要计算任何奇异积分．根据渐近紧理论和Stepleman定理,整个系统的稳定性和收敛性得到了证明．利用h3-Richardson外推算法,收敛精度阶可以提高到O(h5)．为了求解非线性方程组,利用Ostrowski不动点定理研究了Newton的解的收敛性．几个算例从数值上说明了本算法的有效性．
- Helmholtz方程 /
- 机械求积法 /
- Newton迭代法 /
- 非线性边界条件
Abstract: Mechanical quadrature methods(MQMs) for solving nonlinear boundary integral equations of Helmholtz equation, which possessed high accuracy order O (h³) and low computing complexities, were presented. Moreover, the mechanical quadrature methods were simple without computing any singular integration. A nonlinear system was constructed by discretizing the nonlinear boundary integral equations. The stability and convergence of the system were proved based on asymptotical compact theory and Stepleman theorem. Using the h³-Richardson extrapolation algorithms (EAs), the accuracy order to O (h⁵) was improved. For solving the nonlinear system, Newton iteration was discussed extensively by Ostrowski fixed point theorem. The efficiency of the algorithms was illustrated by numerical examples.
- Helmholtz equation /
- mechanical quadrature method /
- Newton iteration /
- nonlinear boundary condition

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参考文献(22)

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