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.

翻译：声学散射问题由外部亥姆霍兹方程建模，该方程由于计算域的无界性和高色散误差（称为污染效应）而难以求解。我们开发了极坐标系下的高阶紧致有限差分法，用于数值求解具有多个任意形状散射体的问题。无界域通过完美匹配层被有效截断和压缩，而污染效应则通过方法的高阶性及一种新颖的污染最小化技术来处理。该技术易于实现，经严格证明有效，并在大量数值结果中展现出优越性能。我们在规则极坐标下提出的有限差分法达到了四阶一致性精度。然而，结合指数拉伸和网格细化，通过在某些位置略微扩大模板，我们能够实现六阶一致性精度。数值算例表明，所提出的有限差分法在不同波数、完美匹配层厚度和散射体形状下均表现出有效性和鲁棒性。