From geodesic extrapolation to a variational BDF2 scheme for Wasserstein gradient flows

TL;DR

This paper develops a second-order time discretization scheme for Wasserstein gradient flows using geodesic extrapolation, proving convergence to PDE solutions and introducing a variational finite volume method for numerical implementation.

Contribution

It introduces a novel BDF2-based scheme for Wasserstein gradient flows utilizing geodesic extrapolation, with proven convergence and a new finite volume discretization.

Findings

Proved convergence of the scheme to the Fokker-Planck PDE.

Established convergence towards EVI flows for specific extrapolation.

Achieved second order accuracy in space and time numerically.

Abstract

We introduce a time discretization for Wasserstein gradient flows based on the classical Backward Differentiation Formula of order two. The main building block of the scheme is the notion of geodesic extrapolation in the Wasserstein space, which in general is not uniquely defined. We propose several possible definitions for such an operation, and we prove convergence of the resulting scheme to the limit PDE, in the case of the Fokker-Planck equation. For a specific choice of extrapolation we also prove a more general result, that is convergence towards EVI flows. Finally, we propose a variational finite volume discretization of the scheme which numerically achieves second order accuracy in both space and time.