I. Introduction
This paper deals with the numerical simulation of time-harmonic electromagnetic (EM) wave propagation. In such problems the time-harmonic Maxwell’s equations in the medium are complemented with suitable boundary conditions. In finite difference or finite element calculations of EM wave propagation, **Perfectly** **Matched** **Layers** (PMLs) aim at emulating radiation at infinity inside a bounded simulation domain. For some applications, the EM waves are fully absorbed at finite distance from the wave launchers. But this distance is still too large to include the damping region in the simulation domain with reasonable computing resources, or the damping mechanism cannot be simulated easily. In these cases PMLs also apply, but they can be introduced at unusual locations, e.g. the inner part of the simulation domain instead of its outer boundary. This unusual setting will be met in the paper, but the results obtained also apply to more standard PMLs after minor adaptation.

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Voluceau – BP 105- F78153 Le Chesnay (e-mail: patrick.joly@inria.fr)
Abstract: We show how Cagniard de Hoop method can be used, first to obtain error estimates
for the **Perfectly** **Matched** **Layers** in acoustics (PML), then to understand the instabilities of the PML when applied to aeroacousics. The principle of the methods consists in applying to the equations a Laplace transform in time and a Fourier transform in one space variable to obtain an ordinary differential equation which can be explicitely solved. This solution contains

Since introduced by Berenger [1], the Perfectly Matched Layers method (PML) has become a popular approach for non reflecting Absorbing Boundary Conditions (ABC) in the numerical solution[r]

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This work deals with **Perfectly** **Matched** **Layers** (PMLs) in the context of dispersive media, and in particular for Negative Index Metamaterials (NIMs). We first present some properties of dispersive isotropic Maxwell equations that include NIMs. We propose and analyse the stability of very general PMLs for a large class of dispersive systems using a new change of variable. We give necessary criteria for the stability of such models that show in particular that the classical PMLs applied to NIMs are unstable and we confirm this numerically. For dispersive isotropic Maxwell equations, this analysis is completed by giving necessary and sufficient conditions of stability. Finally, we propose new PMLs that satisfy these criteria and demonstrate numerically their efficiency.

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Abstract. The analysis of Cartesian **Perfectly** **Matched** **Layers** (PMLs) in the context of time-domain electro-
magnetic wave propagation in a 3D unbounded anisotropic homogeneous medium modelled by a diagonal dielectric tensor is presented. Contrary to the 3D scalar wave equation or 2D Maxwell’s equations some diag- onal anisotropies lead to the existence of backward waves giving rise to instabilities of the PMLs. Numerical experiments confirm the presented result.

1 Introduction
Nowadays, the numerical resolution of wave-like problems set on infinite or very large domains remains a challenging task. When using classical schemes based on finite diﬀerence, finite volume or finite element methods, a common strategy consists in computing the numerical solution only on a truncated domain, and using an adequate treatment at the artificial boundary to preserve the original solution. This treatment is supposed to simulate the outward propagation of signals and perturbations of all kinds generated inside the truncated domain, even if they are not a priori known. For this purpose, a lot of artificial boundary conditions, artificial **layers** and alternative techniques have been developed, studied and used for decades (see e.g. the review papers [ 5 , 25 , 34 , 36 , 38 , 39 , 75 ] and references therein). Among them, the high-order absorbing boundary conditions [ 4 , 35 , 39 , 40 , 67 ] and the **perfectly** **matched** **layers** (PMLs) [ 6 , 13 , 15 , 18 , 43 ,

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Abstract . In this work we consider a problem of modelling of 2D anisotropic dispersive wave propa- gation in unbounded domains with the help of **perfectly** **matched** **layers** (PML). We study the Maxwell equations in passive media with a frequency-dependent diagonal tensor of dielectric permittivity and magnetic permeability. An application of the traditional PMLs to this kind of problems often results in instabilities. We provide a recipe for the construction of new, stable PMLs. For a particular case of non-dissipative materials, we show that a known necessary stability condition of the **perfectly** **matched** **layers** is also sufficient. We illustrate our statements with theoretical and numerical arguments.

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In this work, we make use of absorbing **layers** in the manner of **perfectly** **matched** **layers** (PML), which were introduced by Bérenger [5] in the context of computational electromagnetics. A major difference between the PML and other buffer zones is that they are designed in such a way that, at any angle of incidence, waves are transmitted with (theoretically) no reflection. While PMLs have already been used in aeroacoustics (see, for instance, [19, 18, 24, 1, 20]), the applications were concerned with the linearized Euler equations solved in the time domain. In a previous paper [3], we dealt with the PML for the convected Helmholtz equation (that is, in the frequency domain) in a waveguide and proved the well-posedness and convergence of the method. However, that model being scalar, acoustic waves were the only ones taken into account, whereas the main difficulty when applying the PML in aeroacoustics lies in the appropriate treatment in the **layers** of the entropy and vorticity waves, which are both convected downstream of the mean flow.

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1991 Mathematics Subject Classification. 65M12, 35Q60.
.
Introduction
The method of the **perfectly** **matched** **layers** (PML), introduced by B´ erenger [1, 2], is used ubiquitously in engineering and physics communities to compute a solution to a problem posed in an unbounded domain. However, it is well-known that for some classes of problems (e.g. the wave propagation in anisotropic and/or dispersive media) the classical PML method may result in instabilities [3,4]. For a class of anisotropic dispersive models PMLs were stabilized [5].

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To deal with the unboundedness of the computational domain we wish to employ the **perfectly** **matched** layer (PML) technique suggested by B´ erenger in his seminal articles [5, 6], which consists of surrounding the domain with an artificial layer. Inside this layer the original equations are modified so that the solution decays rapidly, and the truncation of the layer with zero bound- ary conditions would result in a negligible reflection. Crucially, the **perfectly** **matched** property ensures that there is no reflection on the interface between the physical domain and the **perfectly** **matched** layer. However, this method is known [7, 8, 9] to suffer of instabilities, connected to the presence of so-called backward propagating modes [10], which are the consequence of the anisotropic [10] or dispersive [9, 11] nature of the system. Although much effort had been dedicated to the construction and analysis of stable **perfectly** **matched** **layers** for general hyperbolic systems, see e.g. [12, 13], so far, to our knowledge, there exists no general recipe to deal with such instabilities, and each of the problems requires a separate treatment, see e.g. [14, 15, 16, 17, 11].

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loc if the negative permittivity does not lie in some critical interval whose definition
depends on the shape of the device. When the latter has corners, for values inside the critical interval, unusual strong singularities for the electromagnetic field can appear. In that case, well-posedness is obtained by imposing a radiation condition at the corners to select the outgoing black-hole plasmonic wave, that is the one which carries energy towards the corners. A simple and systematic criterion is given to define what is the outgoing solution. Finally, we propose an original numerical method based on the use of **Perfectly** **Matched** **Layers** at the corners. We emphasize that it is necessary to design an

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In IUTAM Symposium on Computational Methods for Unbounded Domains (Boulder, CO, 1997), volume 49 of Fluid Mech. Appl., pages 245–254. Kluwer Acad. Publ., Dordrecht, 1998.
[MPV98] R. Mittra, ¨ U. Pekel, and J. Veihl. A theoretical and numerical study of Berenger’s **perfectly** **matched** layer (PML) concept for mesh truncation in time and frequency domains. In Approxima- tions and numerical methods for the solution of Maxwell’s equa- tions (Oxford, 1995), volume 65 of Inst. Math. Appl. Conf. Ser. New Ser., pages 1–19. Oxford Univ. Press, New York, 1998. [Nat06] Fr´ed´eric Nataf. A new approach to **perfectly** **matched** **layers** for

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R´ esum´ e. Ce travail porte sur la stabilit´ e des Couches Absorbantes Parfaitement Adapt´ ees (**Perfectly** **Matched** **Layers**, PMLs). La premi` ere partie est un r´ ecapitulatif de r´ esultats ant´ erieurs pour les milieux non dispersifs (construction et condition n´ ecessaire de stabilit´ e). La seconde partie concerne quelques extensions de ces r´ esultats. Nous donnons un nouveau crit` ere n´ ecessaire de stabilit´ e valable pour une grande classe de mod` eles dispersifs et pour des PMLs plus g´ en´ erales que celles classiques. Ce crit` ere est appliqu´ e ` a deux mod` eles dispersifs : les m´ etamat´ eriaux ` a indice n´ egatif et les plasmas anistropiques uniaxiaux. Dans les deux cas, les PMLs classiques sont instables mais le crit` ere nous permet de concevoir de nouvelles PMLs stables. Des simulations num´ eriques illustrent notre propos.

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In this paper, we are more specificially interested in the interaction of relativistic interac- tion of atoms, molecules or wavepackets with intense and short laser pulses. The key point is that the laser field actually delocalizes the wavefunction, and the latter can actually interact with the domain boundary. In order to avoid artificial reflection it is necessary to impose absorbing boundary conditions [5], absorbing complex potentials or **perfectly** **matched** lay- ers [43]. Theoretically, this approach allows to benefit from the spectral convergence and simplicity of Fourier-based methods, more specifically pseudospectral, on bounded domains reducing the periodic boundary condition effect thanks to artificial wave absorption at the domain boundary. **Perfectly** **Matched** **Layers** are now widely used in many engineering and physics simulations codes [1, 12, 13, 15, 16, 17, 19, 34, 35, 42, 48, 49, 51, 52] to model ex- terior domains and to avoid any unphysical reflection at the boundary. PML for the Dirac equation were developed in [43] and approximated with a real space method. The derivation of high-order ABCs for the Dirac equation were proposed in [5]. We also refer to [6] for an overview of PMLs and ABCs for quantum wave equations including the Dirac equation and to [4, 7, 8, 9, 11, 14, 22, 26, 28, 32, 33, 38, 39, 41] for different approaches for solving the Dirac equation in real or Fourier space.

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Finally, let us note that these boundary conditions cannot be a priori implemented into a Fourier pseudospectral approximation scheme [2, 3, 7, 14, 33, 40] based on FFTs since the enforced boundary conditions are not periodic.
An alternative method to avoid spurious unphysical reflection at the domain boundary consists in using the method of **Perfectly** **Matched** **Layers** (PMLs) which has been extensively studied for many integer order PDE models (see e.g. [8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 25, 26, 43, 44, 46]). PMLs usually enjoy some good mathematical and numerical properties: easiness of implementation, flexibility, accuracy, stability,... which made them extremely popular since their introduction in the seminal paper by B´ erenger [9] in 1994 for simulating electromagnetic waves. Even if PMLs are very attractive, to the best of our knowledge, the extension to the space FPDEs has never been studied. The aim of this paper is to contribute to deriving PMLs for some classes of FPDEs, resulting in Fractional PMLs (FPMLs). As for the integer case, these FPMLs are easy to integrate into the FPDE mathematical formula- tion. Of course, they also need to be fixed for each FPDE, as for the standard case, and in particular concerning the absorption profiles and their tuning parameters. Therefore, in the present paper, instead of focusing on one specific FPDE, we explain the general idea and illustrate the accuracy of FPMLs through explicit examples. For a better understanding of the full approach, more focused future investigations are required to develop optimal FPMLs for a given FPDE. In addition, we also introduce a specific Fourier based pseudospectral dis- cretization scheme for approximating the FPMLs model which appears to be very flexible and accurate for the resulting modified problem. However, it is also clear that many other discretization approaches could be investigated, which should be again further studied. In particular, one can freely choose the boundary condition imposed at the outer domain bound- ary. Therefore, this allows for adapting many already existing discretization schemes. Even if we start with Riemann-Liouville operators [24, 35, 36], the extension to other definitions of fractional derivative operators should be possible, as we notice for Caputo-type deriva- tive operators. Moreover, we also prospect the extension to higher-dimensional problems and FPDEs involving fractional laplacians [31, 34]. The method can be adapted but some nontrivial questions remain open for further improvements. Finally, well-posedness of the truncated problem is partially addressed here as well as a stability analysis of the scheme. Finally, extending the method to time-fractional PDE seems possible.

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6 Conclusion
We have designed absorbing boundary conditions and **perfectly** **matched** **layers** for some scalar second order partial differential operators: the Helmholtz equation and the wave equation. The construction of the ABC is based on Fourier symbol compu- tations of an exact ABC which is then approximated by a rational fraction. By intro- ducing auxiliary unknowns, this approximation yields, in the physical space, a system of partial differential equations. This technique is quite general and can be applied to systems of partial differential equations, like the isotropic or anisotropic elasticity sys- tem with or without anisotropy, Oseen equations (linearized Navier-Stokes equations), Maxwell system, . . .. As for PML, their designs rely on a complex coordinate coor- dinate change that will be stable in the sense that propagative modes are turned into evanescent ones and that evanescent modes remain evanescent. For complex systems of PDEs with anisotropy, it is not always easy to find such a change of coordinates. Thus, although PMLs are usually better fitted to numerical computations than ABCs, their designs are not always feasible and are still an active area of research.

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b Universit´ e de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France c Universit´ e de Li` ege, Institut Montefiore B28, 4000 Li` ege, Belgium
Abstract
**Perfectly** **Matched** **Layers** (PMLs) appear as a popular alternative to non-reflecting boundary conditions for wave-type problems. The core idea is to extend the computational domain by a fictitious layer with specific absorption properties such that the wave amplitude decays significantly and does not produce back reflections. In the context of convected acoustics, it is well-known that PMLs are exposed to stability issues in the frequency and time domain. It is caused by a mismatch between the phase velocity on which the PML acts, and the group velocity which carries the energy of the wave. The objective of this study is to take advantage of the Lorentz transformation in order to design stable **perfectly** **matched** **layers** for generally shaped convex domains in a uniform mean flow of arbitrary orientation. We aim at presenting a pedagogical approach to tackle the stability issue. The robustness of the approach is also demonstrated through several two-dimensional high-order finite element simulations of increasing complexity.

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Unité de recherche INRIA Rocquencourt Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex France Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois -[r]

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We develop new numerical anisotropic **perfectly** **matched** layer (PML) boundaries for elastic waves in Cartesian, cylindrical and spherical coordinate systems. The elasticity tensor of this absorbing boundary is chosen to be anisotropic and complex so that waves from the computational domain are attenuated in the boundary layer without reflection. The new PMLs are easy to formulate for both isotropic and anisotropic solid media. They utilize fewer unknowns in a general three-dimensional problem than the existing elastic wave PMLs using the field splitting scheme. Moreover, it can be implemented directly to the finite element method (FEM), as well as the finite difference time domain (FDTD) method. The high efficiency of these PMLs is illustrated by some numerical samples in FEM.

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