The perfectly matched layers (PML) and exterior complex scaling (ECS) methods for absorbing boundary conditions are analyzed using spectral decomposition. Both methods are derived through analytical continuations from unitary to contractive transformations. We find that the methods are mathematically and numerically distinct: ECS is complex stretching that rotates the operator's spectrum into the complex plane, whereas PML is a complex gauge transform which shifts the spectrum. Consequently, the schemes differ in their time-stability. Numerical examples are given.
On perfectly matched layers for discontinuous Petrov-Galerkin methods
The Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin（德克萨斯大学奥斯汀分校计算工程与科学研究所）；MSC Software Corporation（MSC软件公司）
This note is intended as a brief introduction to the theory and practice of perfectly matched layer (PML) absorbing boundaries for wave equations, originally developed for MIT courses 18.369 and 18.336. It focuses on the complex stretched-coordinate viewpoint, and also discusses the limitations of PML.
Wrapping a computation domain with a perfectly matched layer (PML) is one of the most effective methods of imitating/approximating the radiation boundary condition in Maxwell and wave equation solvers. Many PML implementations often use a smoothly-increasing attenuation coefficient to increase the absorption for a given layer thickness, and, at the same time, to reduce the numerical reflection from the interface between the computation domain and the PML. In discontinuous Galerkin time-domain (DGTD) methods, using a PML coefficient that varies within a mesh element requires a different mass ma…
Perfectly-Matched Layers (PML) are widely used in Particle-In-Cell simulations, in order to absorb electromagnetic waves that propagate out of the simulation domain. However, when charged particles cross the interface between the simulation domain and the PMLs, a number of numerical artifacts can arise. In order to mitigate these artifacts, we introduce a new PML algorithm whereby the current deposited by the macroparticles in the PML is damped by an analytically-derived, optimal coefficient. The benefits of this new algorithm is illustrated in practical simulations.
Dispersive Perfectly Matched Layer and high order Absorbing Boundary Conditions for the computation of Quasinormal modes of open electromagnetic structures
For numerical simulations of highly relativistic and transversely accelerated charged particles including radiation fast algorithms are needed. While the radiation in particle accelerators has wavelengths in the order of 100 um the computational domain has dimensions roughly 5 orders of magnitude larger resulting in very large mesh sizes. The particles are confined to a small area of this domain only. To resolve the smallest scales close to the particles subgrids are envisioned. For reasons of stability the alternating direction implicit (ADI) scheme by D. N. Smithe et al. (J. Comput. Phys. 22…
The conventional Perfectly Matched Layer (PML) is unstable for certain kinds of anisotropic media. This instability is intrinsic and independent of PML formulation or implementation. The Multi-axial PML (MPML) removes such instability using a nonzero damping coefficient in the direction parallel with the interface between a PML and the investigated domain. The damping ratio of MPML is the ratio between the damping coefficients along the directions parallel with and perpendicular to the interface between a PML and the investigated domain. No quantitative approach is available for obtaining thes…
A Finite-Difference Time-Domain (FDTD) scheme with Perfectly Matched Layers (PMLs) is considered for solving the time-dependent Schrödinger equation, and simulate the ionization of an electron initially bound to a one-dimensional $δ$-potential, when applying a strong time-oscillating electric field. The performance of PMLs based on different absorption functions are compared, where we find slowly growing functions to be preferable. PMLs are shown to be able to reduce the computational domain, and thus the required numerical resources, by several orders of magnitude. This is demonstrated by tes…
We present a stable discontinuous Galerkin (DG) method with a perfectly matched layer (PML) for three and two space dimensional linear elastodynamics, in velocity-stress formulation, subject to well-posed linear boundary conditions. First, we consider the elastodynamics equation, in a cuboidal domain, and derive an unsplit PML truncating the domain using complex coordinate stretching. Leveraging the hyperbolic structure of the underlying system, we construct continuous energy estimates, in the time domain for the elastic wave equation, and in the Laplace space for a sequence of PML model probl…
Implementing computational boundary conditions, such as perfectly matched layers PML does have advantages for forwarding modeling of the earth's crust. The mathematical modeling of many physical problems encountered in industrial applications often leads to a system of linear partial differential equations-PDEs. It considerably improves the visualization of seismic events relevant to oil and gas exploration. In this work, we present an efficient numerical scheme for hyperbolic partial differential equations where a computational technique takes care of reflections at the borders domain using a…
acousticHelmholtzFoam: an MPI-parallel unstructured finite-volume solver for frequency-domain acoustics in heterogeneous two-phase media with perfectly matched layers
We study and implement a simple method, based on the Perfectly Matched Layer approach, to treat non reflecting boundary conditions with the Smoothed Particles Hydrodynamics numerical algorithm. The method is based on the concept of physical damping. We illustrate how it works in the case of 1D and 2D time dependent waves propagating in a finite domain.
