Development of the discrete vortex method in combination with the fast multipole method in hydrodynamic problems

In this paper, the flow of a non-viscous incompressible fluid is discussed in terms of vorticity. In the framework of the discrete vortex method, each material particle of the fluid is considered in Lagrange variables; in this case, the velocities are determined by the Biot-Savard law. Thus, the influence of vortices on each other is taken into account. The aim of the work is to construct a numerical method of different orders of accuracy in the problems of vortex dynamics. The fast multipole method used in combination with the standard midpoint and fourth order Runge-Kutta methods significantly reduces the algorithmic complexity. In the fast multipole method, any vortex system is represented by discrete vortices. The fluid domain, determined by the motion of vortices, is divided into several ring-type subdomains, in each of which the velocities are calculated sequentially. To verify the combinability of the numerical methods, three test cases are considered: the dynamics of the symmetric and asymmetric Lamb-Chaplygin dipoles, as well as the rotation of the fluid occupying a cylindrical region of finite radius. It is known that the latter example is rather complex for direct numerical calculations in contrast to the elementary representation of its analytical solution. In fact, the performed calculations confirm that, without the Fast Multipole Method, the numerical treatment for this test case is hardly possible at a sufficiently large number of discrete vortices within a reasonable amount of time. The results of the test calculations are presented in the form of graphs and tables. The application of the standard discrete vortex methods combined with the fast multipole method shows that, due to the optimal number of subdomains and discrete vortices, the time of calculations can be significantly reduced.