Absorbing Layers for Wave-Like Time-Dependent Problems - Design, Discretization and Optimization

The numerical simulation of wave-like phenomena occurring in large or infinite domains is a great challenge for a wide range of technological and scientifical problems. A classical way consists in considering only a limited computational domain with an artificial boundary that requires a specific treatment. In this thesis, \textit{absorbing layers} are developed and studied for time-dependent problems in order to deal with such artificial boundary.

A large part of this thesis is dedicated to the \textit{perfectly matched layers} (PMLs), which exhibit appealing properties. They are first studied in a fundamental case with non-dispersive linear scalar waves. A procedure for building PMLs is proposed for convex domains with regular boundary. It permits a great flexibility when choosing the shape of the computational domain. After, the issue of choosing PML parameters is addressed with the aim of optimizing the PML effectiveness in discrete contexts. The role of each parameter, including the so-called \textit{absorption function}, is highlighted by means of analytical and numerical results. A systematic comparison of different kinds of absorption functions is performed for several classical numerical schemes (based on finite differences, finite volumes or finite elements). Then, while the PMLs do not a priori account for incoming signals generated outside the computational domain, different problem formulations that account for such forcing are detailed and compared. The interest of the whole approach is finally illustrated with two- and three-dimensional numerical examples in electromagnetism and acoustics, using a discontinuous finite element scheme.

In regional oceanic models, modeling open-sea boundaries brings new difficulties. Indeed, additional linear/nonlinear dynamics are involved and the external forcing is generally poorly known. In this context, different absorbing layers and the widely used Flather boundary condition are compared by means of classical benchmarks. The choice of the absorption function and the way of prescribing the external forcing are discussed in specific marine cases.