Applied Math Seminar: Stretching, damping and perfectly matched layers
Event Description:
Abstract: This talk will give an overview of different techniques to truncate unbounded domains by layers. Truncation of unbounded domains is an alternative to the use of so called non-reflecting or artificial boundary conditions (discussed by Stephen in one of the seminars last semester).
In particular we will talk about the perfectly matched layer (PML) introduced in 1994 by Berenger in the paper A perfectly matched layer for the absorption of electromagnetic waves Journal of computational physics 114 (2), 185-200 1994. This is one of the most cited (7718 times on Google Scholar!!!) papers in computational electro-magnetics (CEM), and although the technique is not the most optimal it is by now the standard technique in CEM.
The invention of PML for CEM initiated a very active period of research on how to adopt the PML to other problems. Some of the fundamental questions were:
- Is PML well-posed in general?
- For what problems is PML stable?
- What is the most general model for a PML?
In the talk I will show how a basic PML can be constructed and when it can be expected to be stable.
Unfortunately there are many important problems, some linear some non-linear, for which a stable PML does not yet exist (and may never be found.) For such problems the only alternative to truncate an unbounded domain may be to add layer designed to ensure stability rather than ensuring perfect matching. The two basic ingredients for such layers are stretching through a change of coordinates (or equivalently “slowing down”) and damping. For some time (see e.g. Grosch and Orszag 77, Israeli and Orzag 81, Kosloff and Kosslof 86, Karni 96) stretching and sponge layers were designed without taking the numerical approximation into account. This reduced the efficiency and thus the popularity of this simpler but more robust approach. If however the design process does take the discretization in to consideration it turns out that the performance of the simple absorbing layers can be quite good, often on par with the performance of the numerical discretization method itself.
