## Section: New Results

### Absorbing boundary conditions and absorbing layers

#### High order Absorbing Boundary Conditions for Anisotropic Models

Participant : Éliane Bécache.

Our collaboration with Dan Givoli (Technion, Israel), Tom Hagstrom (Southern Methodist University) on high order ABC goes on.

The first topic of the collaboration concerns the extension of high-order ABC (the so-called Hagstrom-Warburton or H-W ABC) that were introduced by Givoli et al, and Hagstrom et al for isotropic models, to a very general anisotropic scalar model, which includes in particular the anisotropic scalar wave equation and the convective (dispersive and non-dispersive) wave equation. We have proposed the construction of high order ABC for outgoing waves (in the sense of the sign of the group velocity). We have worked on the analysis of these ABC (well-posedness showed with the Kreiss theory and in some cases with energy estimates, reflection coefficients....). The efficiency of these conditions have been shown with numerical results obtained for the anisotropic wave equation. A paper containing this work has been submitted and accepted for publication.

The second topic has been done in collaboration with Daniel Rabinovich (post-doc at Technion, Israel). It is a numerical work which compares the accuracy of high-order ABC and PMLs for the Helmholtz equation. The comparison between the two methods is done for the same cost of computation, which is determined either by the order of the ABC or by the number of elements in the absorbing layer. A paper on this topic has been submitted for publication.

We are now working (still in collaboration with D. Rabinovich) on the extension of the H-W high-order ABC to the isotropic elastic waves.

#### Leaky modes and PML techniques for non-uniform waveguides

Participants : Anne-Sophie Bonnet-Ben Dhia, Benjamin Goursaud, Christophe Hazard.

This topic was initiated in the framework on the ANR SimNanoPHot (with the
Institut d'Electronique Fondamentale, Orsay), about the simulation of tapers
in integrated optics, or more generally varying cross section open
waveguides. Our motivation was to study the possible use of the so-called
*leaky modes* in the numerical simulation of such devices. The work
presented in the previous report concerned the case of two-dimensional
waveguides. The generalization to the three-dimensional case was recently
studied. Using an infinite PML surrounding the core of the waveguide, which
amounts to a complex stretching of spatial coordinates, the leaky modes
appear as the eigenvalues of the transverse component of the stretched
Helmholtz operator defined in a two-dimensional section. We have achieved the
spectral analysis of this operator in the case of radial PMLs, but not for
the more familiar cartesian PMLs for which open questions remain. Using a PML
of finite width yields a numerical approximation of the leaky
modes. Numerical results were obtained for both radial and cartesian PMLs. As
for 2D waveguides, an instability related to the exponentially increasing
spatial behaviour of leaky modes was observed. Again, the remedy consists in
reducing the intermediate zone between the core and the PML. Two kinds of 3D
numerical computations have been made. On one hand, the knowledge of the
leaky modes furnishes a very efficient way of computing the Green's function
of a 3D uniform waveguide. On the other hand, these leaky modes can be used
to express a diagonal form of the Dirichlet-to-Neumann (DtN) operator of a
semi-infinite uniform waveguide. We have dealt with the case of the junction
of two such waveguides: the problem can be reduced to a bounded region
containing the junction using PMLs in the transverse direction and the DtN
operators at both ends of the junction.

#### Perfectly matched layers for one-way wave equations

Participants : Anne-Sophie Bonnet-Ben Dhia, Christophe Hazard, Patrick Joly, Jérôme Le Rousseau.

Perfectly matched layers are used as absorbing boundary layers to simulate the wave equation as they yield no reflection. When one is interested in the part of the wavefield that propagates in a preferred direction, one-way wave equations can be derived. The wave equation is diagonalized according to this direction. The one-way operators are of pseudo-differential type away from transverse propagation. The issue of designing proper absorbing layers is also important for these one-way wave equations. We investigate how the PMLs introduced by Berenger can be transposed to the one-way setting. Even in constant media, the PML-one-way operator is intricate and its numerical evaluation is rather involved. We are studying ways to obtain efficient numerical methods.

This is a joint project in collaboration with Alison Malcolm of the Massachusetts Institute of Technology (Cambridge, MA, USA), with application to seismic imaging as performed in exploration geophysics.