Generalized Hamiltonian point vortex dynamics on arbitrary domains using the method of fundamental solutions

A new algorithm (VOR-MFS) is presented for the solution of a generalized Hamiltonian model of point vortex dynamics in an arbitrary two-dimensional computational domain. The VOR-MFS algorithm utilizes the method of fundamental solutions (MFS) to obtain an approximation to the model Hamiltonian by solution of an appropriate boundary value problem. Unlike standard point vortex methods, VOR-MFS requires knowledge only of the free-space (R2R2) Green’s function for the problem as opposed to the domain-adapted Green’s function, permitting solution of a much wider range of problems. VOR-MFS is first validated against a vortex image model for the case of (2D Euler) multiple vortex motion in both circular and ‘Neumann-oval’ shaped domains. It is then demonstrated that VOR-MFS can solve for quasi-geostrophic shallow water point vortex motion in the same domains. The exponential convergence of the MFS method is shown to lead to good conservation properties for each of the solutions presented.