By Shankar Subramaniam
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Additional resources for A NEW MESH-FREE VORTEX METHOD
In computations, Leonard  for example, φ is taken to be an axisymmetric smoothing function for simplicity in evaluating the velocity field. 13); they also show how to construct such functions, for both two and three-dimensions, to approximate the vorticity to high orders of accuracy. Winckelmans and Leonard  also give a list of smoothing functions in both two and three-dimensions. Beale & Majda , Perlman , and Daleh  have studied the choice of smoothing function and core size based on the errors in the computed velocity and vorticity fields.
Recently the ‘Deterministic Particle (or Vortex) Method’ has been developed along different lines by Degond & MasGallic , and Mas-Gallic & Raviart . The basic ingredients in this approach are: (a) to consider the strength (circulation) of each particle (vortex) as an unknown coefficient that changes with time due to diffusion effects, (b) to approximate the diffusion operator by an integral operator, and (c) to discretize the integral using the particle positions as quadrature points. 13 In practice, such methods start out with a fixed number of particles, distributed uniformly over the domain, each with a prescribed initial strength [119, 170].
This allows us to choose this function after the actual computation has already been completed. 8) and following will lead to a convergent approximation. For example, a consistent but unstable explicit finite difference scheme would satisfy the equations. Some form of stability condition needs to be imposed; following Van 39 Dommelen , we will demand that all fractions are positive: fijn ≥ 0 . 13) |Γni | of the circulation cannot grow. In the next two sections we will further justify the above conditions using physical and mathematical arguments.