机械求积法及其外推算法求解Helmholtz非线性边界积分方程

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

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.

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.