Perfectly matched layers in negative index metamaterials and plasmas

Abstract

Abstract. This work deals with the stability of Perfectly Matched Layers (PMLs). The first part is a survey of previous results about the classical PMLs in non-dispersive media (construction and necessary condition of stability). The second part concerns some extensions of these results. We give a new necessary criterion of stability valid for a large class of dispersive models and for more general PMLs than the classical ones. This criterion is applied to two dispersive models: negative index metamaterials and uniaxial anisotropic plasmas. In both cases, classical PMLs are unstable but the criterion allows us to design new stable PMLs. Numerical simulations illustrate our purpose.Résumé. Ce travail porte sur la stabilité des Couches Absorbantes Parfaitement Adaptées (Perfectly Matched Layers, PMLs). La première partie est un récapitulatif de résultats antérieurs pour les milieux non dispersifs (construction et condition nécessaire de stabilité). La seconde partie concerne quelques extensions de ces résultats. Nous donnons un nouveau critère nécessaire de stabilité valable pour une grande classe de modèles dispersifs et pour des PMLs plus générales que celles classiques. Ce critère est appliquéà deux modèles dispersifs : les métamatériauxà indice négatif et les plasmas anistropiques uniaxiaux. Dans les deux cas, les PMLs classiques sont instables mais le critère nous permet de concevoir de nouvelles PMLs stables. Des simulations numériques illustrent notre propos.

Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: necessary and sufficient conditions of stability

“…Consider the plasma model ( 8), with the PML in y-direction chosen as ψ(s) = ε 1 (s) −1 . The stability of this PML was confirmed in [6,7]. In this case in Corollary 3.7 m = µ = 0.…”

“…Example 3.2 (Instability of classical PMLs for anisotropic Drude material). We model an anisotropic Drude material, which extends the isotropic model (6), where…”

“…One of the ways to bound the computational domain is offered by the use of the perfectly matched layers (PMLs), a method introduced by Bérenger in [1,2]. For an application of this technique to some dispersive materials, for example, plasmas, see [3][4][5][6][7], and for the use of the method in isotropic metamaterials see the papers by Cummer [8] and Bécache et al [9]. However, it is well known that the PMLs often exhibits instabilities in the presence of dispersion and/or anisotropy [9,10].…”

Stable Perfectly Matched Layers with Lorentz transformation for the convected Helmholtz equation

“…and similarly with e y for a PML in the y-direction. We refer to [22,38,39] for additional details. In other words, the instability occurs when the phase and group velocities of the wave colliding the PML interface have opposite signs.…”

Energy Decay and Stability of a Perfectly Matched Layer For the Wave Equation

“…Appelö, Hagstrom and Kreiss [3] also derived necessary and sufficient stability conditions for first order constant coefficient Cauchy problems; they require verifying a number of algebraic inequalities in Fourier-Laplace domain, but also yield an energy in physical space that involves combinations of higher order derivatives of the unknowns and decays with time -see also [27]. In recent years, stable PML formulations were proposed for linearized Euler equations [29,34], anisotropic acoustics [20], aeroacoustics [21], short water waves [5] and electromagnetic dispersive media [9,10,11,8].…”