4.3 General justification

We would certainly not suggest that our redistribution method, and its detailed implementation, is the only possible approach to diffusion in vortex computations. We merely want to explain the reasoning that led us to formulate this particular procedure. Hopefully, this will explain why our method does have a number of advantages that may be of importance.

We wanted a scheme to replace the random walk method in our computations. Like this method, it should not require ordered vortices: the method should not be based on associations between individual vortices such as a uniform or regular distribution of the vortices, a numerical quadrature rule using the vortices, or any partitioning of the domain. Our motivation for this demand was that in a Lagrangian computation convection effects eventually decouple vortices initially associated with each other. Any order introduces complications: it needs to be decided how long the computation can proceed without restoring a new order, and how to restore it. It causes uncertainty about the possible errors introduced by each of those decisions. A scheme that merely identifies neighboring vortices avoids these difficulties. It also simplifies the computation of flows about complex geometries.

We did not want a partitioning of the domain as in `unstructured' computations. Such a partitioning is still a form of structure that must be regenerated. It brings in complicating aspects such as the geometry of triangles that are not found in the physical flow. Our scheme uses all available vortices within some reasonable distance, rather than a selected subset, to find a suitable discretization for the diffusion of each vortex.

Yet, variations on our scheme remain possible. For example, instead of attempting to describe the diffusion of each individual vortex separately as we do, it would be possible to divide the domain into small square or hexagonal regions and demand only that the net diffusion of all vortices within each region is correctly represented to some order.

However, the work involved in diffusing the individual vortices does not seem to be prohibitive. This is certainly true theoretically, since the work for the redistribution process is asymptotically negligible compared to the work needed to find the velocity field. (We do not consider the machine precision finite as other authors, since this does not allow convergence to occur). In our experimental results for limited number of vortices, the actual work is acceptable but still significant. As explained in subsection 6.2.1, we believe that this is due to our brute force approach to finding the redistribution fractions.

Our requirement that the redistribution weights are positive was motivated in part by the standard five-point explicit finite difference scheme for the diffusion process. For that finite difference scheme, the transition from a stable to an unstable scheme occurs when one of the fractions becomes negative. Therefore, at least when the vortices are located on a uniform mesh, and the redistribution radius includes five vortices at a time, the positivity constraint needs to be satisfied. Furthermore, the positivity condition is sufficient: in chapter 5 we will prove for the linear Stokes equation that it ensures convergence of the method for any arbitrary point distributions.

On the other hand, there are certainly stable finite difference schemes with negative fractions that will be excluded by the positivity constraint. Yet this does not appear to be an unacceptable loss; suitable positive solutions can always be found. As discussed in subsection 6.2.3, for any order of approximation our redistribution equations can be solved using only a finite number of points within a finite scaled radius $R$.