ON THE ROKHLIN-GREENGARD METHOD WITH VORTEX BLOBS FOR PROBLEMS POSED IN ALL SPACE OR PERIODIC IN ONE DIRECTION

In this paper we consider the Rokhlin-Greengard (R-G) fast multipole algorithm when used to evaluate vortex blob interactions in a two-dimensional fluid. We use exact solutions of the incompressible Euler equations to demonstrate that the R-G algorithm can compute vortex blob interactions accurately. However, we also show that the structure of vortex blobs forces a practical limitation on the highest (finest) bisection level one can use in the R-G algorithm, a restriction which does not apply when point vortices are used. If this maximum bisection level is exceeded, then the accuracy of the R-G algorithm with blobs may be significantly reduced. A similar constraint should hold in three dimensions. We also extend the R-G algorithm with blobs to problems which are periodic in one spatial dimension and unbounded in the other, and we document the performance of the resulting algorithm using some exact periodic solutions of the incompressible Euler equations. (C) 1995 Academic Press, Inc.