  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.
