6.2 Numerical implementation of diffusion

In this section we describe the numerical implementation of the new redistribution method formulated in section 4.1. Basically, the redistribution method is a matter of finding the fractions $f^n_{ij}$ from the system of linear equations (4.8) and following, and redistributing the circulation of the each vortex according to these fractions. Assuming that valid fractions $f^n_{ij}$ exist, they are found using linear programming techniques, as explained in subsections 6.2.1 and 6.2.2 below. However, it is possible that no valid fractions exist using the available vortices in a neighborhood. In that case, we create new vortices until there is a solution, see subsection 6.2.3. Further, due to convection effects, some vortices may move sufficiently close to another vortex that they are no longer useful for computational purposes. In subsection 6.2.4, we discuss how to remove those vortices. Also, near the edges of a diffusing region, the vorticity is exponentially small. To avoid excessive vortices, some cut-off strength is needed below which vortices are ignored. However, choosing this cut-off can be quite tricky, as we will explain in subsection 6.2.5. Finally, in subsection 6.2.6 we discuss evaluating the vorticity to compare with analytical solutions and other numerical computations in the literature.