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

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 $Ω$ is decomposed into a bounded computational domain $Ω^{-}$ and an exterior unbounded domain $Ω^{+}$. 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 $Ω^{+}$. As a result, we obtain a new local absorbing boundary condition (ABC) for a bounded problem on $Ω^{-}$, 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 $Ω^{-}$. 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.