By Shankar Subramaniam

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3) fijn Γni φδ (x − xj ) . 4) into ω n+1 = i j at the next time level n + 1. We would like this change to approximate the true diffusion over the time-step in some way. Our approach will be to demand that all finite wave numbers of the Fourier transform are correctly damped. This is similar to a weak formulation in which Fourier modes are used as weighting functions. 37 The Fourier transform of the new vorticity distribution is ω n+1 = φ(kδ) Γni e−ik·xi fijn e−ik·(xj −xi ) . 5) j This is to be compared with the Fourier transform of the exactly diffused vorticity: ωen+1 = φ(kδ) Γni e−ik·xi e−k 2 ν∆t .

3 Deterministic particle method A deterministic method to simulate diffusion has been developed by Raviart [180], Choquin & Huberson [45], and Cottet & Mas-Gallic [64]. They use viscous/inviscid splitting of the vorticity equation and then solve the diffusion equation exactly using the fundamental solution of the heat equation. Recently the ‘Deterministic Particle (or Vortex) Method’ has been developed along different lines by Degond & MasGallic [72], and Mas-Gallic & Raviart [147]. 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.

We assume that any body forces on the fluid are derived as a gradient of a scalar function. The governing equations for the motion of the fluid are the conservation of mass and linear momentum [14]. 1) where u is the velocity and ∇ is the gradient operator. We also denote x = (x, y) to be any point in the plane and xˆ and yˆ to be the unit vectors along the axes. 2) where t is time; p is mechanical pressure; F is body force per unit mass of the fluid; ν is kinematic viscosity, defined as the ratio of the dynamic viscosity and the density of the fluid and ∇2 is the Laplacian operator.