High order methods for acoustic scattering: Coupling Farfield Expansions ABC with Deferred-Correction methods

TL;DR

This paper develops high order numerical methods for time-harmonic acoustic scattering on unbounded domains by coupling advanced boundary conditions with finite difference schemes using deferred correction techniques, achieving high convergence rates.

Contribution

It introduces a novel coupling of farfield expansion-based ABCs with deferred-correction finite difference methods for high order accuracy in acoustic scattering problems.

Findings

High order convergence demonstrated through numerical experiments.

Rigorous proof of scheme consistency with Helmholtz equation and ABCs.

Construction of DC finite difference schemes detailed and validated.

Abstract

Arbitrary high order numerical methods for time-harmonic acoustic scattering problems originally defined on unbounded domains are constructed. This is done by coupling recently developed high order local absorbing boundary conditions (ABCs) with finite difference methods for the Helmholtz equation. These ABCs are based on exact representations of the outgoing waves by means of farfield expansions. The finite difference methods, which are constructed from a deferred-correction (DC) technique, approximate the Helmholtz equation and the ABCs, with the appropriate number of terms, to any desired order. As a result, high order numerical methods with an overall order of convergence equal to the order of the DC schemes are obtained. A detailed construction of these DC finite difference schemes is presented. Additionally, a rigorous proof of the consistency of the DC schemes with the Helmholtz equation and the ABCs in polar coordinates is also given. The results of several numerical experiments corroborate the high order convergence of the novel method.