The acoustic scattering problem is modeled by the exterior Helmholtz equation, which is challenging to solve due to both the unboundedness of the domain and the high dispersion error, known as the pollution effect. We develop high-order compact finite difference methods (FDMs) in polar coordinates to numerically solve the problem with multiple arbitrarily shaped scatterers. The unbounded domain is effectively truncated and compressed via perfectly matched layers (PMLs), while the pollution effect is handled by the high order of our method and a novel pollution minimization technique. This technique is easy to implement, rigorously proven to be effective and shows superior performance in our numerous numerical results. The FDMs we propose in regular polar coordinates achieve fourth consistency order. Yet, combined with exponential stretching and mesh refinement, we can reach sixth consistency order by slightly enlarging the stencil at certain locations. Our numerical examples demonstrate that the proposed FDMs are effective and robust under various wavenumbers, PML layer thickness and shapes of scatterers.

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