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：本文介绍了专门为标量波动方程的高阶有限差分离散化设计的离散全纯完美匹配层。与标准的基于偏微分方程的完美匹配层相比，所提方法实现了在完美匹配层界面处完全消除数值反射的显著效果，在实践中达到了机器精度级别的误差。我们的方法基于近期一篇出版物[Journal of Computational Physics 381 (2019): 91-109]中提出的思想，将适用范围从标准的二阶有限差分方法扩展到任意高阶格式。该推广通过引入额外的局部化完美匹配层变量来适应所采用的大模板。我们证明了所提方案生成的数值解在完美匹配层域内传播时呈现指数衰减率。为展示方法的有效性，我们提供了包括波导问题在内的多种数值算例。这些算例突显了采用高阶格式以有效处理和最小化不期望的数值色散误差的重要性，从而强调了所提方法的实用优势与适用性。