mathematics of computation volume 58, number 197 january 1992, pages 103-117 A NEW DESINGULARIZATION FOR VORTEX METHODS THOMAS Y. HOU Abstract. A new desingularization is introduced for the vortex method. The idea is to subtract off the most singular part in the discrete approximation to the velocity integral and replace it by the velocity of a vortex patch of constant vorticity, which can be evaluated explicitly. Stability and convergence of the method are obtained in the maximum norm. Preliminary numerical results are presented. 1. Introduction In this paper, we introduce a new desingularization for the vortex method. The idea is to subtract off the most singular part in the discrete approximation to the velocity integral and replace this singular summation by the velocity of a vortex patch of constant vorticity, which can be evaluated explicitly (see the Ap- pendix). As a consequence, the desingularized vortex method is asymptotically one order less singular than the original vortex method if the vorticity field is Lipschitz continuous. This desingularization technique applies to both the point vortex method and the vortex blob method. Preliminary numerical experiments seem to indicate that the desingularized vortex method provides more accurate approximations for large-time calculations than the original vortex method. Because of our desingularization, we can prove stability of the method in the maximum norm. This allows us to analyze convergence of the method in the case when a local regridding procedure is introduced for the method [11]. Moreover, our analysis only uses the fact that the Biot-Savart kernel is LXoc. This implies that the method is also convergent for the analogue of the incompressible Euler equations (l)-(2) (see below) with a kernel K which is more singular than the Biot-Savart kernel. Consider the incompressible 2-D Euler equations in the vorticity-stream func- tion formulation: (1) Co, + (u- V)w = 0, <y(x, 0) = Wo(x). Here the velocity u is related to the vorticity a> by the Biot-Savart law (2) u(x,t)= [ K(x-y)ca(y,t)dy, J& Received December 18, 1989; revised June 21, 1990. 1991MathematicsSubject Classification. Primary 65M25; Secondary76C05. Key words and phrases. Vortex method, desingularization, large time accuracy. Research supported in part by the Air Force Office of Scientific Research under URI grant AFOSR 86-0352. ©1992 American Mathematical Society 0025-5718/92 $1.00+ $.25 per page 103 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use