A Singularity-free Boundary Equation Method for Wave Scattering

TL;DR

This paper introduces a novel boundary difference equation method for wave scattering that avoids singularities by discretizing differential equations first and then converting to boundary equations, enhancing accuracy and applicability.

Contribution

The paper presents a singularity-free boundary difference equation approach for wave scattering, differing from traditional integral methods by discretizing differential equations prior to boundary conversion.

Findings

Method demonstrates high accuracy and convergence in numerical tests

Effective for 2D scattering problems and generalizable to 3D and other linear problems

Avoids singular Green's functions, simplifying computations

Abstract

Traditional boundary integral methods suffer from the singularity of Green's kernels. The paper develops, for a model problem of 2D scattering as an illustrative example, singularity-free boundary difference equations. Instead of converting Maxwell's system into an integral boundary form first and discretizing second, here the differential equations are first discretized on a regular grid and then converted to boundary difference equations. The procedure involves nonsingular Green's functions on a lattice rather than their singular continuous counterparts. Numerical examples demonstrate the effectiveness, accuracy and convergence of the method. It can be generalized to 3D problems and to other classes of linear problems, including acoustics and elasticity.