Algorithms for numerical solution of integral equations of three-dimensional scalar diffraction problem

Abstract

The three-dimensional diffraction problem of stationary acoustic waves on a homogeneous inclusion is considered. It is reduced to the weakly singular boundary Fredholm integral equations of the first kind with one unknown function, each of which is conditionally equivalent to the original problem. By using the original method of averaging the integral operators kernels, these equations are approximated by systems of linear algebraic equations. The resulting systems are solved numerically by the generalized minimal residual method (GMRES). Then the solution of the initial problem is calculated. To find the solution on the spectrum of integral operators, where the condition of equivalence of integral equations to the original problem is violated, the interpolation solution method is proposed. It does not require knowledge of the spectrum and allows us to find the approximated solutions with high accuracy. The proposed algorithms have been implemented in the computing cluster of Computing Center FEB RAS. The results of the calculations that allow us to assess the possibilities of this approach are presented.