On the analysis of Bérenger's Perfectly Matched Layers for Maxwell's equations

Bécache, Éliane; Joly, Patrick

doi:10.1051/m2an:2002004

Cited by 101 publications

(100 citation statements)

References 12 publications

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“…Analogous exponential instabilities have been also observed in the simulation of non-destructive testing experiments [19]. This has motivated us to look at the question of the well-posedness and the stronger notion of stability of PMLs, introduced in [6], for anisotropic elastic waves. It is important here to make precise the distinction (see Section 3) between these two notions (see also [33] for similar considerations): by well-posedness, we mean that there exists a unique solution and that the L 2 norm of the solution can be bounded by some norm of the initial conditions multiplied by a constant CðtÞ.…”

Section: Introductionmentioning

confidence: 69%

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Stability of perfectly matched layers, group velocities and anisotropic waves

Bécache

Fauqueux

Joly

2003

Journal of Computational Physics

306

355

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Perfectly matched layers (PML) are a recent technique for simulating the absorption of waves in open domains. They have been introduced for electromagnetic waves and extended, since then, to other models of wave propagation, including waves in elastic anisotropic media. In this last case, some numerical experiments have shown that the PMLs are not always stable. In this paper, we investigate this question from a theoretical point of view. In the first part, we derive a necessary condition for the stability of the PML model for a general hyperbolic system. This condition can be interpreted in terms of geometrical properties of the slowness diagrams and used for explaining instabilities observed with elastic waves but also with other propagation models (anisotropic MaxwellÕs equations, linearized Euler equations). In the second part, we specialize our analysis to orthotropic elastic waves and obtain separately a necessary stability condition and a sufficient stability condition that can be expressed in terms of inequalities on the elasticity coefficients of the model.

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Section: Introductionmentioning

confidence: 69%

“…This type of mathematical questions has already been widely investigated by several authors [6,27,29,30,38] in the case of MaxwellÕs equations. For elastodynamics equations, it is easy to show that the PML model is well-posed (cf.…”

Section: Introductionmentioning

confidence: 99%

Stability of perfectly matched layers, group velocities and anisotropic waves

Bécache

Fauqueux

Joly

2003

Journal of Computational Physics

306

355

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“…Therefore the PMLs are high-frequency stable. A deeper analysis of PMLs in this context shows that they are actually stable (not only for high frequencies), see [4].…”

Section: Application To Some Non-dispersive Wave Modelsmentioning

confidence: 98%

“…[1][2][3][4], and [15] for a recap). For systems with constant coefficients, these two notions are both related to the solutions of the dispersion relation.…”

Section: A Necessary Criterion Of Stabilitymentioning

confidence: 99%

Perfectly matched layers in negative index metamaterials and plasmas

Bécache¹,

Joly²,

Kachanovska³

et al. 2015

ESAIM: Proc.

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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.

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“…This completes the proof. We remark that our stability analysis is inspired by the energy analysis in [3] in which the stability is proved separately for the PML system in the unbounded domain parallel to the axises and in the unbounded corner domain. Here we show the stability of the PML system in the whole truncated PML layer which will be useful for the convergence analysis of the time domain PML method in this paper.…”

mentioning

confidence: 99%

Long-Time Stability and Convergence of the Uniaxial Perfectly Matched Layer Method for Time-Domain Acoustic Scattering Problems

Chen¹,

Wu²

2012

SIAM J. Numer. Anal.

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Abstract. The uniaxial perfectly matched layer (PML) method uses rectangular domain to define the PML problem and thus provides greater flexibility and efficiency in dealing with problems involving anisotropic scatterers. In this paper we first derive the uniaxial PML method for solving the time-domain scattering problem based on the Laplace transform and the complex coordinate stretching in the frequency domain. We prove the long-time stability of the initial-boundary value problem of the uniaxial PML system for piecewise constant medium property and show the exponential convergence of the time-domain uniaxial PML method. Our analysis shows that for fixed PML absorbing medium property, any error of the time-domain PML method can be achieved by enlarging the thickness of the PML layer as ln T for large T > 0. Numerical experiments are included to illustrate the efficiency of the PML method.

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On the analysis of Bérenger's Perfectly Matched Layers for Maxwell's equations

Cited by 101 publications

References 12 publications

Stability of perfectly matched layers, group velocities and anisotropic waves

Stability of perfectly matched layers, group velocities and anisotropic waves

Perfectly matched layers in negative index metamaterials and plasmas

Long-Time Stability and Convergence of the Uniaxial Perfectly Matched Layer Method for Time-Domain Acoustic Scattering Problems

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