Mathematical Modelling and Numerical Analysis on the Analysis of B ´ Erenger's Perfectly Matched Layers for Maxwell's Equations

In this work, we investigate the Perfectly Matched Layers (PML) introduced by B erenger (3) for designing ecient numerical absorbing layers in electromagnetism. We make a mathematical analy- sis of this model, rst via a modal analysis with standard Fourier techniques, then via energy techniques. We obtain uniform in time stability results (that make precise some results known in the literature) and state some energy decay results that illustrate the absorbing properties of the model. This last technique allows us to prove the stability of the Yee's scheme for discretizing PML's. R esum e. Dans ce travail, nous consid erons le mod ele de couches parfaitement adapt ees, dit PML (Perfectly Matched Layers), introduit par B erenger (3) pour la mod elisation de fronti eres absorbantes en electromagn etisme. Nous menons une analyse math ematique de ce mod ele, d'une part par une analyse modale par transformation de Fourier, d'autre part par des techniques energ etiques. Nous obtenons ainsi des r esultats de stabilit e uniforme en temps (qui pr ecisent des r esultats d ej ac onnus dans la litt erature) et etablissons des r esultats de d ecroissance d' energie qui illustrent les propri et es d'absorption du mod ele. Cette derni ere technique permet aussi de d emontrer la stabilit ed u sch ema de Yee pour discr etiser les couches absorbantes.

[1]  David Gottlieb,et al.  A Mathematical Analysis of the PML Method , 1997 .

[2]  F. Hu.  On Absorbing Boundary Conditions for Linearized Euler Equations by a Perfectly Matched Layer , 1995 .

[3]  Thomas Hagstrom,et al.  A COMPARISON OF TWO ACCURATE BOUNDARY TREATMENTS FOR, COMPUTATIONAL AEROACOUSTICS , 1997 .

[4]  Allen Taflove,et al.  Computational Electrodynamics the Finite-Difference Time-Domain Method , 1995 .

[5]  Caractère bien posé du problème de Cauchy pour le système de Bérenger , 1999 .

[6]  E. Turkel,et al.  Absorbing PML boundary layers for wave-like equations , 1998 .

[7]  Andreas C. Cangellaris,et al.  A Reflectionless Sponge Layer Absorbing Boundary Condition for the Solution of Maxwell's Equations with High-Order Staggered Finite Difference Schemes , 1998 .

[8]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[9]  Andreas C. Cangellaris,et al.  A general approach for the development of unsplit-field time-domain implementations of perfectly matched layers for FDTD grid truncation , 1996 .

[10]  Jean-Pierre Berenger,et al.  A perfectly matched layer for the absorption of electromagnetic waves , 1994 .

[11]  H. Kreiss,et al.  Initial-Boundary Value Problems and the Navier-Stokes Equations , 2004 .

[12]  J. Hesthaven.  On the Analysis and Construction of Perfectly Matched Layers for the Linearized Euler Equations , 1998 .

[13]  David Gottlieb,et al.  On the construction and analysis of absorbing layers in CEM , 1998 .

[14]  Richard W. Ziolkowski,et al.  Time-derivative Lorentz material model-based absorbing boundary condition , 1997 .