The Convergence of an Exact Desingularization for Vortex Methods

Expanding upon an observation of Hou [Math. Comp., submitted], exact desingularizations are presented of the Euler equations in two and three dimensions for which the singularity within the Biot-Savart integrand is reduced by one order. The reformulated equations are then solved numerically using either the point vortex method or the vortex blob method. The increased smoothness of the Biot-Savart integrand allows us to prove convergence of these methods in the maximum norm. Our numerical experiments show that discretization of the reformulated equations display increased stability relative to discretizations of the original equations. The improvement in stability is manifested as a more slowly growing error in time. Key words, vortex method, exact desingularization AMS(MOS) subject classifications, primary 65M25; secondary 76C05