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

Thomas Gallouët (1) , Andrea Natale (2) , Gabriele Todeschi (3)

1 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
2 RAPSODI - Reliable numerical approximations of dissipative systems
3 ISTerre - Institut des Sciences de la Terre

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.

Keywords: Optimal transport; Wasserstein extrapolation; Wasserstein gradient flows; BDF2

Domains: Analysis of PDEs [math.AP]; Numerical Analysis [math.NA]

Thomas Gallouët, Andrea Natale, Gabriele Todeschi. From geodesic extrapolation to a variational BDF2 scheme for Wasserstein gradient flows. 2022. ⟨hal-03790981v1⟩