On cubic spline approximations for the vortex patch problem

Based on the recent theoretical advances on the CDE (Bertozzi, 1991; Bertoz zi and Constantin, 1993) we introduce a numerical method for solving the CD E by means of a global cubic spline interpolation between nodes. This metho d is shown to be convergent for all time and numerically tested against exa ct solutions for the CDE, the well-known flows of the Kirchoff ellipses (La mb, 1932). We compare this method with the one obtained using the building blocks of the method designed by Dritschel in 1989. Without the use of any node redistribution technique, we find a better performance of our method i n several error estimates such as node position, tangent and curvature. Thi s performance improves as the curvature of the contour increases. (C) 2001 IMACS. Published by Elsevier Science B.V. All rights reserved.