# A variational finite volume scheme for Wasserstein gradient flows

## TL;DR

This paper introduces a novel variational finite volume scheme for approximating Wasserstein gradient flows, ensuring energy decay, non-negativity, and convergence, applicable to various energies including the linear Fokker-Planck equation.

## Contribution

It develops a new discretization method that preserves the variational structure of Wasserstein gradient flows at the discrete level, with proven convergence and robustness.

## Key findings

- Scheme guarantees non-negativity and energy decay.
- Proven convergence for linear Fokker-Planck equation.
- Numerical results show first-order accuracy and robustness.

## Abstract

We propose a variational finite volume scheme to approximate the solutions to Wasserstein gradient flows. The time discretization is based on an implicit linearization of the Wasserstein distance expressed thanks to Benamou-Brenier formula, whereas space discretization relies on upstream mobility two-point flux approximation finite volumes. Our scheme is based on a first discretize then optimize approach in order to preserve the variational structure of the continuous model at the discrete level. Our scheme can be applied to a wide range of energies, guarantees non-negativity of the discrete solutions as well as decay of the energy. We show that our scheme admits a unique solution whatever the convex energy involved in the continuous problem, and we prove its convergence in the case of the linear Fokker-Planck equation with positive initial density. Numerical illustrations show that it is first order accurate in both time and space, and robust with respect to both the energy and the initial profile.