## Section: Scientific Foundations

### Absorbing Boundary Conditions and Perfectly Matched Layers

In most of the problems of wave propagation we have to deal with unbounded domains. It is then helpful to define (artificial) boundaries defining the numerical model under study in order to reduce the computational costs.

Since the innovative work of [56] , this question was addressed by using Absorbing Boundary Conditions (ABC) [52] or damping zones (sponge layers) [50] . In both cases, results are often not very satisfactory because spurious phases are reflected inside the computational domain, in particular at grazing incidence in the case of paraxial conditions, and at low frequency in the case of sponge layers. However these conditions are easy to include in numerical schemes and can be constructed such that they preserve the sparsity of the discrete matrix. This justifies Magique-3D  works on the improvement of ABCs.

More recently, Bérenger [48] introduced an innovative condition for Maxwell's equations, which has the property of being perfectly adapted to the model, in the sense that no spurious phase is produced in the domain before discretization and use of a numerical scheme. The resulting model is called a Perfectly Matched Layer (PML). Because of its efficiency, the PML method quickly became very popular in electromagnetics. Next, based on an analogy between Maxwell's equations and linear elasticity written as a first order system in velocity and stress, several authors adapted the PML approach to the propagation of elastic waves in infinite domains (see e.g. [51] , [53] ). However, for instance in the case of elastodynamics (and even more for porous media) and for shallow-water equations, the classical PMLs are known to be unstable. Moreover the layers do not properly handle grazing rays which gives rise to spurious reflections. This is why Magique-3D  develops new PML models both for elastodynamics, porous media and geophysical fluids.

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