Boundary integral equation solving in vortex method using the Barnes-Hut/multipole algorithm

The problem of efficient solution of the boundary integral equation arising at each time step in problems of 2D outer flow simulation by using the Vortex Particle Method is considered. It is necessary to develop an efficient approach to the corresponding linear system solving with the dense and non-symmetric matrix. For this purpose, an adaptation of the Barnes-Hut-based fast method, which has been initially developed for vortex particles velocities calculation, is suggested. The proposed modification is applied to the second-order numerical scheme with piecewise-linear solution representation since it allows considering vorticity distribution over far-placed panels, and performing integration of the local expansion coefficients over control panels that provides the satisfaction of the boundary condition in Galerkin sense. Besides the speedup effect, it does not require full matrix computation and storage, thus only about 1 % of matrix coefficients should be stored, which are calculated directly.