Numerical analysis
On a surprising instability result of Perfectly Matched Layers for Maxwell’s equations in 3D media with diagonal anisotropy
Comptes Rendus. Mathématique, Volume 359 (2021) no. 3, pp. 249-256.

The analysis of Cartesian Perfectly Matched Layers (PMLs) in the context of time-domain electromagnetic 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 diagonal anisotropies lead to the existence of backward waves giving rise to instabilities of the PMLs. Numerical experiments confirm the presented result.

Dans cette note nous nous intéressons à l’analyse de stabilité de la méthode de couches absorbantes parfaitement adaptées (PMLs) pour la propagation d’ondes électromagnétiques en régime transitoire dans un milieu anisotrope décrit par un tenseur diélectrique diagonal. Contrairement aux cas de l’équation d’ondes scalaire 3D et des équations de Maxwell 2D, certaines anisotropies diagonales mènent à l’existence d’ondes inverses qui provoquent des instabilités de la méthode PML. Ce résultat est illustré par des simulations numériques.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.165
Bécache, Éliane 1; Fliss, Sonia 1; Kachanovska, Maryna 1; Kazakova, Maria 2

1 Labratoire POEMS, CNRS, Inria, ENSTA Paris Institut Polytechnique de Paris, 828 boulevard des Maréchaux, 91762 Palaiseau, France
2 Normandie Univ, INSA de Rouen Normandie, LMI (EA 3226 - FR CNRS 3335), 76000 Rouen, France, 685 Avenue de l’Université, 76801 St Etienne du Rouvray cedex, France
@article{CRMATH_2021__359_3_249_0,
     author = {B\'ecache, \'Eliane and Fliss, Sonia and Kachanovska, Maryna and Kazakova, Maria},
     title = {On a surprising instability result of {Perfectly} {Matched} {Layers} for {Maxwell{\textquoteright}s} equations in {3D} media with diagonal anisotropy},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {249--256},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {3},
     year = {2021},
     doi = {10.5802/crmath.165},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.165/}
}
TY  - JOUR
AU  - Bécache, Éliane
AU  - Fliss, Sonia
AU  - Kachanovska, Maryna
AU  - Kazakova, Maria
TI  - On a surprising instability result of Perfectly Matched Layers for Maxwell’s equations in 3D media with diagonal anisotropy
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 249
EP  - 256
VL  - 359
IS  - 3
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.165/
DO  - 10.5802/crmath.165
LA  - en
ID  - CRMATH_2021__359_3_249_0
ER  - 
%0 Journal Article
%A Bécache, Éliane
%A Fliss, Sonia
%A Kachanovska, Maryna
%A Kazakova, Maria
%T On a surprising instability result of Perfectly Matched Layers for Maxwell’s equations in 3D media with diagonal anisotropy
%J Comptes Rendus. Mathématique
%D 2021
%P 249-256
%V 359
%N 3
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.165/
%R 10.5802/crmath.165
%G en
%F CRMATH_2021__359_3_249_0
Bécache, Éliane; Fliss, Sonia; Kachanovska, Maryna; Kazakova, Maria. On a surprising instability result of Perfectly Matched Layers for Maxwell’s equations in 3D media with diagonal anisotropy. Comptes Rendus. Mathématique, Volume 359 (2021) no. 3, pp. 249-256. doi : 10.5802/crmath.165. http://www.numdam.org/articles/10.5802/crmath.165/

[1] Appelö, Daniel; Hagstrom, Thomas; Kreiss, Gunilla Perfectly matched layers for hyperbolic systems: general formulation, well-posedness, and stability, SIAM J. Appl. Math., Volume 67 (2006) no. 1, pp. 1-23 | DOI | MR | Zbl

[2] Bécache, Éliane; Fauqueux, Sandrine; Joly, Patrick Stability of perfectly matched layers, group velocities and anisotropic waves, J. Comput. Phys., Volume 188 (2003) no. 2, pp. 399-433 | DOI | MR | Zbl

[3] Bécache, Éliane; Joly, Patrick; Kachanovska, Maryna Stable perfectly matched layers for a cold plasma in a strong background magnetic field, J. Comput. Phys., Volume 341 (2017), pp. 76-101 | DOI | MR | Zbl

[4] Bécache, Éliane; Joly, Patrick; Vinoles, Valentin On the analysis of perfectly matched layers for a class of dispersive media and application to negative index metamaterials, Math. Comp., Volume 87 (2018) no. 314, pp. 2775-2810 | DOI | MR | Zbl

[5] Bécache, Éliane; Kachanovska, Maryna Stability and Convergence Analysis of Time-domain Perfectly Matched Layers for The Wave Equation in Waveguides (2020) (submitted, https://hal.archives-ouvertes.fr/hal-02536375)

[6] Bérenger, Jean-Pierre A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., Volume 114 (1994) no. 2, pp. 185-200 | DOI | MR | Zbl

[7] Berenger, Jean-Pierre Three-dimensional perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., Volume 127 (1996) no. 2, pp. 363-379 | DOI | MR | Zbl

[8] Born, Max; Wolf, Emil Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Cambridge University Press, 1999 | DOI

[9] Demaldent, Edouard; Imperiale, Sébastien Perfectly matched transmission problem with absorbing layers : application to anisotropic acoustics in convex polygonal domains, Int. J. Numer. Meth. Engng., Volume 96 (2013) no. 11, pp. 689-711 | DOI | MR | Zbl

[10] Diaz, Julien; Joly, Patrick A time domain analysis of PML models in acoustics, Comput. Methods Appl. Mech. Eng., Volume 195 (2006) no. 29-32, pp. 3820-3853 | DOI | MR | Zbl

[11] Hu, Fang Q. A stable, perfectly matched layer for linearized Euler equations in unsplit physical variables, J. Comput. Phys., Volume 173 (2001) no. 2, pp. 455-480 | DOI | MR | Zbl

[12] Kreiss, Heinz-Otto; Lorenz, Jens Initial-boundary value problems and the Navier–Stokes equations, 47, Society for Industrial and Applied Mathematics, 1989 | DOI

Cited by Sources: