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Applied Mathematics and Mechanics (English Edition) ›› 2011, Vol. 32 ›› Issue (12): 1505-1514.doi: https://doi.org/10.1007/s10483-011-1519-7

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Mechanical quadrature methods and extrapolation for solving nonlinear boundary Helmholtz integral equations

程攀1 黄晋2 王柱2,3   

  1. 1. School of Science, Chongqing Jiaotong University, Chongqing 400074, P. R. China;
    2. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, P. R. China;
    3. Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA
  • 收稿日期:2010-11-16 修回日期:2011-10-11 出版日期:2011-12-09 发布日期:2011-12-01

Mechanical quadrature methods and extrapolation for solving nonlinear boundary Helmholtz integral equations

CHENG Pan1, HUANG Jin2, WANG Zhu2,3   

  1. 1. School of Science, Chongqing Jiaotong University, Chongqing 400074, P. R. China;
    2. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, P. R. China;
    3. Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA
  • Received:2010-11-16 Revised:2011-10-11 Online:2011-12-09 Published:2011-12-01

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摘要： This paper presents mechanical quadrature methods (MQMs) for solving nonlinear boundary Helmholtz integral equations. The methods have high accuracy of order O(h3) and low computation complexity. Moreover, the mechanical quadrature methods are simple without computing any singular integration. A nonlinear system is constructed by discretizing the nonlinear boundary integral equations. The stability and convergence of the system are proved based on an asymptotical compact theory and the Stepleman theorem. Using the h3-Richardson extrapolation algorithms (EAs), the accuracy to the order of O(h5) is improved. To slove the nonlinear system, the Newton iteration is discussed extensively by using the Ostrowski fixed point theorem. The efficiency of the algorithms is illustrated by numerical examples.

Abstract: This paper presents mechanical quadrature methods (MQMs) for solving nonlinear boundary Helmholtz integral equations. The methods have high accuracy of order O(h3) and low computation complexity. Moreover, the mechanical quadrature methods are simple without computing any singular integration. A nonlinear system is constructed by discretizing the nonlinear boundary integral equations. The stability and convergence of the system are proved based on an asymptotical compact theory and the Stepleman theorem. Using the h3-Richardson extrapolation algorithms (EAs), the accuracy to the order of O(h5) is improved. To slove the nonlinear system, the Newton iteration is discussed extensively by using the Ostrowski fixed point theorem. The efficiency of the algorithms is illustrated by numerical examples.

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