Unfitted Lattice Green's Function Method for Exterior Scattering in Complex Geometry

TL;DR

This paper introduces an unfitted lattice Green's function method that combines boundary algebraic equations with finite differences to efficiently solve exterior scattering problems in complex geometries without singularities.

Contribution

The paper develops a novel finite-difference approach using the lattice Green's function for exterior scattering, avoiding singularities and simplifying implementation for arbitrary shapes.

Findings

Accurate results for various scatterers including circles and triangles.

Robustness demonstrated for multiple-body configurations.

Method retains advantages of boundary integral methods without singularities.

Abstract

This paper develops a finite-difference analogue of the boundary integral/element method for the numerical solution of two-dimensional exterior scattering from scatterers of arbitrary shapes. The discrete fundamental solution, known as the lattice Green's function (LGF), for the Helmholtz equation on an infinite lattice is derived and employed to construct boundary algebraic equations through the discrete potentials framework. Unlike the continuous fundamental solution used in boundary integral methods, the LGF introduces no singularity, which simplifies numerical implementation. Boundary conditions are incorporated through local Lagrange interpolation on unfitted cut cells. The resulting method retains key advantages of boundary integral approaches-including dimension reduction and the absence of artificial boundary conditions--while enabling finite differences for complex geometries. Numerical results demonstrate the accuracy and robustness of the method for various scatterers, including circular, triangular, and multiple-body configurations.