Dispersive Waves and PML Performance for Large-Stencil Differencing in the Parabolic Equation

The discrete reflection/absorption performance of perfectly matched layer (PML) boundary conditions is a function of the field discretization scheme throughout the computational domain and the form of the complex coordinate extension that characterizes the PML region. Most previous work analyzing PML reflection performance has assumed use of low-order (second- or fourth-), small-stencil (three- or five-point) spatial discretizations whose dispersion is poor outside a narrow low-wavenumber band. Large-stencil differencing schemes, regardless of order, are easily created with physically realistic, wide-band dispersive characteristics, but the effect their adoption has on PML performance has not been analyzed previously. In the context of a semi-discrete approximation to the parabolic/paraxial equation, this work derives the analytic form of the discrete PML reflection coefficient for an arbitrary-order/arbitrary-stencil interior differencing scheme coupled to an arbitrary-order/arbitrary-stencil PML region. This result is then used to show that the absorption performance of the PML regions is largely insensitive to changes in interior differencing. As a consequence, the benefits of superior dispersion characteristics provided by use of large-stencil differencing in the interior physical domain can be had without incurring discrete reflection penalties at the PML.