Convergence of Vortex Methods for Euler’s Equations. II

Abstract

The vortex method is a numerical technique for approximating the flow of a two-dimensional incompressible, inviscid fluid. The method amounts to approximating the vorticity of the fluid by a sum of delta functions (point vortices) and to follow the movement of the point vortices. It is shown that the velocity field computed by the vortex method converges toward the velocity of the fluid in the least squares sense. The result is established for flows without boundaries and with vorticity which have compact support and is valid for arbitrary long time intervals. The basic idea in the proof is to partition the support of vorticity into blobs at time $t = 0$ and to show that the path of a point vortex approximates the path of the center of gravity of the blob with which it agree’s initially. The error in the vortex method is proportional to the square of the initial diameter of the blobs.