As discussed in section 3.2,
the convection is modeled by moving the vortices
according to the equation
To integrate (6.1), we used the fourth order Runge-Kutta time stepping scheme proposed by Blum [27].
Unfortunately, the cost of computing the velocity of all the vortices using the summation (6.1) is proportional to , where is the number of vortices. Such computational effort would be unrealistic for flows where significant small-scale motion requires a fine vortex spacing, in other words, large .
To solve this dilemma, `fast' algorithms were developed by, among others, Greengard and Rokhlin [97] and Carrier et al. [36], and independently by Van Dommelen and Rundensteiner [233,240]. For the computations in this work the latter of these three schemes was used; it seems to have been the first available scheme that was solution adaptive [240], but it can be noticeably slower than the first two schemes. For all these schemes, the required amount of work is roughly proportional to . More recent variations have been proposed by a number of authors, such as [2,5,10,74,75].
As explained in section 3.1,
the velocity field
of a vortex is obtained by convolving the velocity field
of a point vortex
with a smoothing function
which
is usually taken to be of the form