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.

翻译：我们提出了一种基于经典二阶向后差分公式的Wasserstein梯度流时间离散化方法。该格式的主要构建模块是Wasserstein空间中的测地线外推概念，这一概念通常不具有唯一性。针对该操作，我们提出了若干可能的定义，并在福克-普朗克方程情形下证明了相应格式收敛于极限偏微分方程。当采用特定的外推方式时，我们还证明了一个更一般的结果，即收敛于EVI流。最后，我们提出了该格式的变分有限体积离散化方法，在数值上实现了空间和时间上的二阶精度。