## Abstract

As soon as people have been interested in the numerical simulation of wave propagation phenomena, particularly in the time domain, they had to face the delicate question of realizing “transparent” artificial boundaries. Indeed, many problems of this type are posed in a unbounded domain (or in a domain which is very large with respect to a zone of interest in which one wishes to observe the solution): one thus has to bound artificially the computational domain in such a way that the non physical boundaries are “transparent” for the solution. Due to the crucial importance of this subject, but also because of its difficulty, this question has aroused the curiosity from many researchers, first from engineers, next from applied mathematicians. It would be much too ambitious to give a complete panorama of a research that began almost forty years ago (see however the article by T. Hagstrm [20] which constitutes a quite complete review of the domain) but one can now measure how much the many ingenious ideas proposed by the one or the others have permitted to accomplish impressive progress in this matter. Nowadays, if one restricts oneself to linear wave propagation problems, one can consider that, even though some questions remain open, satisfactory solutions are available for a large number of applications, in particular in the “canonical case” I will have in mind throughout this paper, namely the case where one wishes to reduce numerical computations to a convex bounded domain, assuming that the exterior domain is homogeneous and source free.

Recently, in particular for the applications in acoustics and electromagnetism, important progress has been achieved in the use of “exact” boundary conditions associated to homogeneous exterior domains. One usually introduces a boundary operator called “Dirichlet to Neumann” (DtN) operator that can be represented by using separation of variables in the exterior domain (in 3D, one will use typically spherical harmonics [19], which requires a spherical artificial boundary) or through the coupling with an integral representation formula of the exterior solution (which relies on the explicit knowledge of the Green’s function of the problem) [4].

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Joly, P. An elementary introduction to the construction and the analysis of perfectly matched layers for time domain wave propagation.
*SeMA* **57, **5–48 (2012). https://doi.org/10.1007/BF03322599

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### Keywords

- Wave Equation
- Cauchy Problem
- Hyperbolic System
- Perfectly Match Layer
- Unperturbed System