A New Theoretical Approach to Absorbing Layers

We consider the construction of absorbing boundary layers using asymp-totic expansions, the small parameter being the width of the layer. This allows us to deene the order of the layer. We compute layers of various orders and analyse their stability properties. Finally we perform some numerical tests to evaluate the practical utility of our layers. 1. Introduction Classically when one wants to simulate numerically the propagation of waves in an unbounded domain in space, there are (at least) two methods for limiting the computational domain. One is to use absorbing boundary conditions which consist in prescribing boundary conditions speciically derived in order to minimize the artiicial reeections inside the computational domain. Such conditions are now very well known and have been studied intensively from the physical, mathematical and numerical point of view in the last 15 years; see for instance 1], 3], 4], 7], 15] and references therein. A technically simpler and older approach is to use absorbing layers. This is equivalent to surrounding the domain of interest by some geometrical layer where one adds physical attenuation terms to the propagation equations in order to absorb the waves penetrating this region. Of course to get good absorption the thickness of the absorbing layer must be suuciently large and to avoid increasing too much the size of the computational domain and therefore the computational cost it must be suuciently small. Moreover, the transition between the propagation medium and the absorbing region should not be too abrupt in order to allow the waves to penetrate the absorbing layer. The design of such absorbing layers being essentially based on physical or engineering considerations, it is not surprising that most of the articles concerning this subject can be found in the physical literature, see for instance 2], 8], 9], 10] and 13]. On the other hand, this domain has not retained too much the attention of the applied mathematicians (perhaps because of the development of the more attractive