# Homogenisation of one-dimensional discrete optimal transport

## TL;DR

This paper investigates how one-dimensional discrete optimal transport metrics, derived from spatial discretisations of the Benamou--Benamou formula, converge to a limiting metric influenced by mesh geometry and non-local mobility, revealing the impact of microstructure.

## Contribution

It proves convergence of discrete transport metrics to a limiting metric with effective mobility in a 1D periodic setting, highlighting the role of mesh geometry and non-local effects.

## Key findings

- Discrete transport metrics converge to a non-trivial limiting metric.
- Mesh geometry and non-local mobility influence transport cost.
- Microstructure can reduce transport costs in discretised models.

## Abstract

This paper deals with dynamical optimal transport metrics defined by spatial discretisation of the Benamou--Benamou formula for the Kantorovich metric $W_2$. Such metrics appear naturally in discretisations of $W_2$-gradient flow formulations for dissipative PDE. However, it has recently been shown that these metrics do not in general converge to $W_2$, unless strong geometric constraints are imposed on the discrete mesh. In this paper we prove that, in a $1$-dimensional periodic setting, discrete transport metrics converge to a limiting transport metric with a non-trivial effective mobility. This mobility depends sensitively on the geometry of the mesh and on the non-local mobility at the discrete level. Our result quantifies to what extent discrete transport can make use of microstructure in the mesh to reduce the cost of transport.