Abstract: A journey in applied mathematics: from two physical problems to numerical methods
The two physical problems are related to wave propagation around planes for one, and to fusion reactors for the other.
Nuclear fusion, the energy powering the stars, requires extreme conditions of temperature and pressure to produce a plasma. Confining a plasma under such conditions in an experiment is very challenging. Magnetic confinement fusion (MCF) is concerned with the use of strong magnetic fields to confine plasmas. Tokamaks and stellarators are two types of toroidal MCF reactors. The former has a toroidal symmetry that considerably simplifies the geometry, but leads to plasma instabilities. The latter, in order to limit instabilities, does not have a toroidal symmetry and offers much more flexibility in choosing its shape. The corresponding part of this presentation will touch on the basics of mathematical modeling for stellarator design, leading to recent challenges in optimization problems. The importance of setting a common language in interdisciplinary collaborations will also be discussed.
The design of modern aircrafts involves a balance of 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. The corresponding part of this presentation will start with modeling aspects 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.