Perfectly Matched Absorbing Layers for the Paraxial Equations

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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“…Considering (15) as an equation in the frequency x for a given value of the wave vector k, we obtain four solutions …”

Section: Harmonic Plane Waves and Slowness Diagramsmentioning

confidence: 99%

“…[11,12,28,37,39]). The method has been extended to various propagation models (the paraxial wave equations [15], the linearized Euler equations [22,24,32], etc. ), including in particular elastic wave propagation in isotropic [21] and anisotropic media [17].…”

Section: Introductionmentioning

confidence: 99%

Stability of perfectly matched layers, group velocities and anisotropic waves

Bécache

Fauqueux

Joly

2003

Journal of Computational Physics

309

355

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“…To deal with the transparent boundary conditions on Γ ⊥ from a numerical point of view, the so called Perfectly Matched Layers (PML) approach leads to a very efficient method which has been proposed for Schrödinger equation in [17]; this method has also been implemented in the numerical code HERA (cf. [3]), with which the following numerical results are obtained.…”

Section: Numerical Schemementioning

confidence: 99%

Mathematical models for laser-plasma interaction

Sentis¹

2005

ESAIM: M2AN

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Abstract. We address here mathematical models related to the Laser-Plasma Interaction. After a simplified introduction to the physical background concerning the modelling of the laser propagation and its interaction with a plasma, we recall some classical results about the geometrical optics in plasmas. Then we deal with the well known paraxial approximation of the solution of the Maxwell equation; we state a coupling model between the plasma hydrodynamics and the laser propagation. Lastly, we consider the coupling with the ion acoustic waves which has to be taken into account to model the so called Brillouin instability. Here, besides the macroscopic density and the velocity of the plasma, one has to handle the space-time envelope of the main laser wave, the space-time envelope of the stimulated Brillouin backscattered laser wave and the space envelope of the Brillouin ion acoustic waves. Numerical methods are also described to deal with the paraxial model and the three-wave coupling system related to the Brillouin instability.Mathematics Subject Classification. 35Q55, 35Q60, 65M06, 34E20, 82D10.

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“…Collino [10] showed that the PML can be interpreted as a change of variable in the complex plane. The change of variable results in a simple modification of the differential operator a 'iW a %+ iu + a(z) z"…”

Section: Perfectly Matched Layer4mentioning

confidence: 99%

“…encoding parm Chirp encoding requires a stretching parameter which may vary between the approximate range [2][3][4][5][6][7][8][9][10]. Linear phase encoding requires the desired time shift between the shots, -to,in seconds.…”

Section: Salvo User's Guide -mentioning

confidence: 99%