1.3 Lagrangian methods for diffusion

A number of numerical schemes model diffusion in vortex methods without using a mesh. Such methods are based on the Lagrangian approach and use vortices only. Often the vortices are unevenly and sparsely distributed; this makes it difficult to compute vorticity gradients and to represent diffusion in regions depleted of vortices. We will describe ways to handle such difficulties while discussing various methods. Some of the current methods model diffusion by changing the parameters of the vortices: their positions (Random Walk method); their sizes (Core Expansion method); or their circulations (Deterministic Particle method). Other methods model diffusion using smooth interpolants to approximate the actual vorticity distribution (Smoothed Particle Hydrodynamics method and Fishelov's method); or even by altering the character of the diffusion process (Diffusion Velocity method). We will briefly review the above methods in the following sections.