On cubic spline approximations for the vortex patch problem

Abstract

Based on the recent theoretical advances on the CDE we introduce a numerical method for solving the CDE by means of a global cubic spline interpolation between nodes. This method is shown to be convergent for all time and numerically tested against exact solutions for the CDE, the well-known flows of the Kirchoff ellipses. 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 in several error estimates such as node position, tangent and curvature. This performance improves as the curvature of the contour increases.