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

## TL;DR

This paper introduces a novel mesh-free particle method for advection-diffusion problems based on optimal transportation theory, achieving geometrically exact updates and demonstrating robustness through 3D examples.

## Contribution

The paper develops an innovative mesh-free scheme derived from optimal transportation principles, integrating entropy and Wasserstein distance for accurate advection-diffusion modeling.

## Key findings

- Geometrically exact advection and volume preservation.
- Robustness demonstrated through 3D numerical examples.
- Effective mesh-free max-ent interpolation for density updates.

## Abstract

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.