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We develop an Optimal Transportation Meshfree (OTM) particle method foradvection-diffusion in which the concentration or density of the diffusivespecies is approximated by Dirac measures. We resort to an incrementalvariational principle for purposes of time discretization of the diffusivestep. This principle characterizes the evolution of the density as acompetition between the Wasserstein distance between two consecutive densitiesand entropy. Exploiting the structure of the Euler-Lagrange equations, weapproximate the density as a collection of Diracs. The interpolation of theincremental transport map is effected through mesh-free max-ent interpolation.Remarkably, the resulting update is geometrically exact with respect toadvection and volume. We present three-dimensional examples of application thatillustrate the scope and robustness of the method.

Geometrically-exact time-integration mesh-free schemes for  advection-diffusion problems derived from optimal transportation theory and  their connection with particle methods