  We develop an Optimal Transportation Meshfree (OTM) particle method for
advection-diffusion in which the concentration or density of the diffusive
species is approximated by Dirac measures. We resort to an incremental
variational principle for purposes of time discretization of the diffusive
step. This principle characterizes the evolution of the density as a
competition between the Wasserstein distance between two consecutive densities
and entropy. Exploiting the structure of the Euler-Lagrange equations, we
approximate the density as a collection of Diracs. The interpolation of the
incremental transport map is effected through mesh-free max-ent interpolation.
Remarkably, the resulting update is geometrically exact with respect to
advection and volume. We present three-dimensional examples of application that
illustrate the scope and robustness of the method.
