regridding techniques [17]. Even before the flow becomes disorganized, however, obtaining high-order accuracy with
We present a new approach to vortex methods for the 2D Euler equations. We obtain long-time high-order accuracy at almost opti-a single quadrature rule requires smoothing of the singular mal cost by using three tools: fast adaptive quadrature rules, a free-Biot-Savart kernel. Smoothing gives high-order accuracy Lagrangian formulation, and a useful new analysis of the consisfor short times but slows down fast velocity evaluation tency error. Our error analysis halves the order of differentiability techniques and halves the order of accuracy of the method required of the flow and suggests an efficient new balance of relative to the order of differentiability of the flow. smoothing parameters which works well with fast summation In Section 3, we contrast two free-Lagrangian vortex schemes. Numerical experiments with our methods confirm our theoretical predictions and display excellent long-time accuracy. methods, the triangulated vortex method of [20], and the แฎ 1997 Academic Press quadrature-based method of [23]. The triangulated vortex method is robust, practical, and efficient but limited to second-order accuracy. The quadrature-based method This section gives an overview of 2D vortex methods Mathematical Sciences Subprogram of the Office of Energy Research, U.S. Department of Energy under Contract DE-AC03-76SF00098.
for incompressible inviscid flow. First, we describe how the 108