High order local absorbing boundary conditions for acoustic waves in terms of farfield expansions

TL;DR

This paper introduces a high order local absorbing boundary condition for acoustic wave problems that uses farfield expansions, providing improved accuracy and flexibility over existing methods without enlarging the computational domain.

Contribution

The paper presents a novel high order local ABC based on truncated farfield expansions, allowing arbitrary accuracy matching the numerical method's order without increasing boundary size.

Findings

Numerical results show improved accuracy in 2D and 3D cases.

The new ABC simplifies implementation compared to existing conditions.

Error order can be increased arbitrarily with more expansion terms.

Abstract

We devise a new high order local absorbing boundary condition (ABC) for radiating problems and scattering of time-harmonic acoustic waves from obstacles of arbitrary shape. By introducing an artificial boundary $S$ enclosing the scatterer, the original unbounded domain $\Omega$ is decomposed into a bounded computational domain $\Omega^{-}$ and an exterior unbounded domain $\Omega^{+}$. Then, we define interface conditions at the artificial boundary $S$, from truncated versions of the well-known Wilcox and Karp farfield expansion representations of the exact solution in the exterior region $\Omega^{+}$. As a result, we obtain a new local absorbing boundary condition (ABC) for a bounded problem on $\Omega^{-}$, which effectively accounts for the outgoing behavior of the scattered field. Contrary to the low order absorbing conditions previously defined, the order of the error induced by this ABC can easily match the order of the numerical method in $\Omega^{-}$. We accomplish this by simply adding as many terms as needed to the truncated farfield expansions of Wilcox or Karp. The convergence of these expansions guarantees that the order of approximation of the new ABC can be increased arbitrarily without having to enlarge the radius of the artificial boundary. We include numerical results in two and three dimensions which demonstrate the improved accuracy and simplicity of this new formulation when compared to other absorbing boundary conditions.