Vortex methods. I - Convergence in three dimensions. II - Higher order accuracy in two and three dimensions

Attention is given to recent approaches to simulating three-dimensional incompressible fluid flows using vortex methods, noting that some require only crude information, such as the vortex blobs of the two-dimensional case. The question whether such 'crude' algorithms can accurately account for vortex stretching and converge is considered. The question is answered affirmatively by constructing a new class of crude three-dimensional vortex methods and then proving that these methods are stable and convergent. It is also shown that they can even have arbitrarily high order accuracy without being more expensive than other crude versions of the vortex algorithm. The accuracy, however, requires that the consistency of a discrete approximation to the Biot-Savart law is assumed. This consistency statement is then proved, and substantially sharper results are derived for two-dimensional flows. Also given is complete, simplified proof of convergence in two dimensions.