Computation of the Schwarz function and its application to vortex dynamics

The efficient and accurate rational approximation of the Schwarz function in the complex plane by means of the recently developed AAA (adaptive Antoulas-Anderson) algorithm is applied here to the dynamics of uniform vorticity patches. The key observation is realising that the AAA algorithm readily computes the poles of the rational approximation, and that this enables a natural partitioning of the Schwarz function in such a way that the velocity field exterior to a patch is given in terms of the poles interior to the patch. In particular, knowing the velocity on the patch boundary and exterior to the patch allows both vortex patch equilibria, including configurations with and without satellite point vortices surrounding the vortex patch, and time-dependent motions to be computed. Examples are presented and the performance of the new numerical approach compared to previously established numerical methods, including the method of contour dynamics and the recently developed AAA-least squares algorithm.