p-Wasserstein distances on networks and 3D to 1D convergence

TL;DR

This paper investigates transport distances on gas network graphs, extending Wasserstein distances to networks and analyzing their convergence from 3D domains to 1D graphs, supported by theoretical proofs and numerical examples.

Contribution

It introduces a framework for Wasserstein distances on networks, including mass storage, and proves convergence from 3D domains to metric graphs.

Findings

Wasserstein distances can be extended to networks with mass storage.

The static Wasserstein distance on 3D networks converges to that on metric graphs.

Numerical examples illustrate the convergence results.

Abstract

We study transport distances on metric graphs representing gas networks. Starting from the dynamic formulation of the Wasserstein distance, we review extensions to networks, with and without the possibility of storing mass on the vertices. Next, we examine the asymptotic behavior of the static Wasserstein distance on a three-dimensional network domain that converges to a metric graph. We show convergence of the distance with a proof that is based on the characterization of optimal transport plans as $c$-cyclically monotone sets. We conclude by illustrating our finding with several numerical examples.