Fast Triangulated Vortex Methods for the 2-d Euler Equations

Vortex methods for inviscid incompressible two-dimensional uid ow are usually based on blob approximations. This paper presents a vortex method in which the vorticity is approximated by a piecewise polynomial interpolant on a Delaunay triangulation of the vortices. An e cient reconstruction of the Delaunay triangulation at each step makes the method accurate for long times. The vertices of the triangulation move with the uid velocity, which is reconstructed from the vorticity via a simpli ed fast multipole method for the Biot-Savart law with a continuous source distribution. The initial distribution of vortices is constructed from the initial vorticity eld by an adaptive approximation method which produces good accuracy even for discontinuous initial data. Numerical results show that the method is highly accurate over long time intervals. Experiments with single and multiple circular and elliptical rotating patches of both piecewise constant and smooth vorticity indicate that the method produces much smaller errors than blob methods with the same number of degrees of freedom, at little additional cost. Generalizations to domains with boundaries, viscous ow and three space dimensions are discussed.