  The dynamical formulation of optimal transport, also known as Benamou-Brenier
formulation or Computational Fluid Dynamics formulation, amounts to write the
optimal transport problem as the optimization of a convex functional under a
PDE constraint, and can handle \emph{a priori} a vast class of cost functions
and geometries. Several discretizations of this problem have been proposed,
leading to computations on flat spaces as well as Riemannian manifolds, with
extensions to mean field games and gradient flows in the Wasserstein space.
  In this article, we provide a framework which guarantees convergence under
mesh refinement of the solutions of the space-time discretized problems to the
one of the infinite-dimensional one for quadratic optimal transport. The
convergence holds without condition on the ratio between spatial and temporal
step sizes, and can handle arbitrary positive measures as input, while the
underlying space can be a Riemannian manifold. Both the finite volume
discretization proposed by Gladbach, Kopfer and Maas, as well as the
discretization over triangulations of surfaces studied by the present author in
collaboration with Claici, Chien and Solomon fit in this framework.
