Vortex patch equilibria using a two-domain AAA-least squares algorithm

Over the past seven years, the adaptive Antoulas-Anderson (AAA) algorithm for complex, rational approximation and so-called 'lightning least-squares' methods for solving Laplace problems in simply connected domains have been developed by Trefethen, Costa and their colleagues. These methods were then combined into a single 'AAA-least squares algorithm', which computes the rational approximation of a harmonic function in seconds on a standard laptop. In this talk, a new extension to the algorithm developed for solving general, inhomogeneous two-domain interface problems is presented. Its efficacy is then demonstrated in a scenario in vortex dynamics, where a simply connected domain of fluid in rotational flow, known as a vortex patch, is surrounded by fluid in irrotational flow. Vortex patch equilibria are sought – solutions where the shape of the vortex patch is invariant over time in a rotating or translating frame. The two-domain AAA-least squares algorithm can then be used to reproduce known analytical and numerical vortex patch equilibria. It is speculated that this method could be used to find further equilibrium solutions, such as scenarios involving configurations of vortex patches, sheets and point vortices. A further application of the algorithm to a problem in sensory electrostatics is also briefly explored.