This paper introduces discrete-holomorphic Perfectly Matched Layers (PMLs) specifically designed for high-order finite difference (FD) discretizations of the scalar wave equation. In contrast to standard PDE-based PMLs, the proposed method achieves the remarkable outcome of completely eliminating numerical reflections at the PML interface, in practice achieving errors at the level of machine precision. Our approach builds upon the ideas put forth in a recent publication [Journal of Computational Physics 381 (2019): 91-109] expanding the scope from the standard second-order FD method to arbitrary high-order schemes. This generalization uses additional localized PML variables to accommodate the larger stencils employed. We establish that the numerical solutions generated by our proposed schemes exhibit an exponential decay rate as they propagate within the PML domain. To showcase the effectiveness of our method, we present a variety of numerical examples, including waveguide problems. These examples highlight the importance of employing high-order schemes to effectively address and minimize undesired numerical dispersion errors, emphasizing the practical advantages and applicability of our approach.

翻译：本文提出了专为标量波动方程高阶有限差分离散格式设计的离散全纯完美匹配层。与基于偏微分方程的标准完美匹配层不同，所提方法实现了消除完美匹配层界面处数值反射的显著效果，实际误差可达到机器精度水平。我们的方法基于近期文献[Journal of Computational Physics 381 (2019): 91-109]提出的思想，将其应用范围从标准二阶有限差分法扩展到任意高阶格式。该推广通过引入额外的局部完美匹配层变量来适应更大的差分模板。我们证明了所提格式生成的数值解在完美匹配层区域内传播时呈现指数衰减率。为展示方法的有效性，我们给出了包括波导问题在内的多种数值算例。这些算例凸显了采用高阶格式以有效抑制并最小化非期望数值色散误差的重要性，同时强调了该方法在实际应用中的优势与可行性。