The design of modern aircrafts involves a balance competing cost and performance requirements, usually combining experimental and theoretical approaches. The propagation of waves around aircrafts is one of many components studied in this context, and this talk will focus on numerical simulations of both acoustic and electromagnetic waves. The presentation will start with modeling aspect for aeroacoustic problems, leading to partial differential equations (PDEs) with variable coefficients, before turning to numerical methods specifically tailored to handle such PDEs: the so-called quasi-Trefftz methods.

Trefftz methods rely, in broad terms, on the idea of approximating solutions to Partial Differential Equation (PDEs) via Galerkin methods using basis functions which are exact solutions of the PDE, making explicit use of information about the ambient medium. They are of particular interest for wave propagation problems. But in general, for problems modeled by PDEs with variable coefficients, no exact solutions are available. Hence quasi-Trefftz methods have been introduced to address this problem: they rely not on exact solutions to the PDE but instead of high order approximate solutions constructed locally. We will discuss some of the fundamental properties of these numerical methods for aeroacoustic problems. The goal here will be to emphasize why they represent promising alternative to more standard methods to tackle an important challenge related to the boundary layer of air flow around planes.
Finally we will highlight the main difficulty in developing quasi-Trefftz methods for electromagnetic wave propagation.