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

Abstract

This paper presents mechanical quadrature methods (MQMs) for solving nonlinear boundary Helmholtz integral equations. The methods have high accuracy of order O(h 3 ) 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 h 3 -Richardson extrapolation algorithms (EAs), the accuracy to the order of O(h 5 ) 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.

Sixth-order finite difference scheme for the Helmholtz equation with inhomogeneous Robin boundary condition

“…When the wave number k is large, the solution of the problem becomes highly oscillating and efficient numerical methods are required in order to get high performance simulation results. In this topic, various numerical methods were developed in the past decades, such as the finite difference method (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]), the finite element method (see, e.g., [17][18][19][20][21][22][23][24][25]), the boundary element method (see, e.g., [26][27][28][29][30]), and other techniques (see, e.g., [17,[31][32][33]). For the finite difference method, two common methods are considered in the literature, namely the parameter method and the high-order method.…”